The material is a little on the technical side for a blog. In future I hope to do some short and simple posts on the topic, but it seemed helpful to establish a reference point first. For convenience, a list of notation is provided at the end. The images can be opened in a different tab or window by right-clicking on them. If text size is an issue, holding down the control button while pressing plus or minus can adjust the size.

**1. Income Determination with a Job Guarantee**

It will be convenient to think of the economy as comprising just two sectors: the ‘job guarantee sector’ or sector j and the ‘broader economy’ or sector b. The broader economy is taken to include households, firms, the rest of the public sector and non-residents.

Suppose that most planned expenditure is exogenous and can be lumped together as autonomous demand A. The exceptions are induced consumption net of endogenous imports, which rises with income, and job guarantee spending, which varies countercyclically. Steady state income in the absence of a job guarantee would be A_{ }/_{ }α, where α is the marginal propensity to leak to taxes, saving and imports. With a job guarantee, the system becomes:

Equation (1) is the steady state condition. It says that income Y must equal demand Y_{d} for the level of activity to be stable. Equation (2) shows demand as the sum of induced net consumption (1 – α)Y, autonomous demand and job guarantee spending G_{j}.

Income for the economy as a whole is composed of the sectoral incomes. The income of sector j can be regarded as the wages paid to job guarantee workers. This is in keeping with the national accounting treatment of public sector activity in general. If a fraction ϕ of job guarantee spending goes to wages, sector j income will be ϕG_{j}. It is assumed that the remaining fraction 1 – ϕ of job guarantee spending goes on materials supplied by the broader economy and so contributes to sector b income. These considerations are summarized as:

A preliminary expression for the steady state level of total income can be obtained by solving the system of equations (1) for Y. The result can be plugged in to the third equation in (2) to give a corresponding expression for sector b income:

To progress any further, we need an expression for job guarantee spending.

Let w_{j} be the exogenously set job guarantee wage. With ϕ defined as the fraction of job guarantee spending going to wages, w_{j}_{ }/_{ } ϕ must be the amount of job guarantee spending per unit of sector j employment L_{j}. This implies:

Sector j employment accounts for whatever employment is not located in the broader economy. This means we can write the above as

In the present discussion, total employment L is simply taken as given. This simplifying assumption means that variations in one sector’s employment translate one-for-one into inverse variations in the other sector: ΔL_{j} = -ΔL_{b}. A more sophisticated approach would factor in procyclical variation in labor force participation.

Unlike other entries on the right-hand side of (4), L_{b} is endogenous. It depends on the sector’s income and productivity ρ_{b}. Specifically, L_{b} = Y_{b}_{ }/_{ }ρ_{b}. Although it is possible to endogenize ρ_{b} to allow for procyclical variation in sector b’s productivity, here it will be considered exogenous. Additional complications can always be introduced at a later time.

Since L_{b} = Y_{b}_{ }/_{ }ρ_{b}, we can take the expression for Y_{b} in (3) and divide by ρ_{b}. The result can then be substituted for L_{b} in (4) to obtain an equation solvable for G_{j}. The solution is the steady state level of job guarantee spending:

Our expression for G_{j} can now be used in (3) to find the steady state levels of Y and Y_{b}:

Corresponding expressions for the other endogenous variables are easily obtained. The steady state level of Y_{j} is simply the expression for G_{j} multiplied by ϕ. The sectoral employment levels in a steady state will be Y_{b}_{ }/_{ }ρ_{b} and Y_{j}_{ }/_{ }w_{j}.

An exogenous change in autonomous demand will cause multiplied changes in the steady state levels of the endogenous variables. These multipliers can be obtained by differentiating or taking the first difference of the relevant expression with respect to A:

Exogenous changes in total employment also have multiplier effects, which can be found by differentiating the various steady state expressions with respect to L. An individual decision to enter the labor force and take a job guarantee position will activate government expenditure on the program, with a multiplier effect on the economy. This is a notable aspect of the job guarantee in that an increase in supply-side potential automatically translates into higher demand. An earlier post considers a couple of effects along these lines. The present focus, however, will be on the expenditure multipliers.

The expressions in (5) and (6) can be arranged in various ways. The above arrangement is chosen for its succinctness and to provide a common denominator. As displayed, the expressions tell us the steady state levels of the endogenous variables given the values of the exogenous variables and parameters. But the form of presentation is not conducive to understanding the likely dynamics involved in moving from one steady state to another. This is a different question and the focus of subsequent sections of the post.

Before turning to that question, though, the meaning of q requires some explanation. A cursory glance at (5) and (6) reveals that the term appears in both the numerator and denominator of all the main expressions. It is a positive constant defined as

The margin ρ_{b} – w_{j} is the additional income that is generated by having a unit of employment located in sector b rather than sector j. This margin is created because productivity in sector b is assumed to be higher than in sector j, where productivity is taken to equal the job guarantee wage on the basis that ρ_{j} = Y_{j}_{ }/_{ }L_{j} = w_{j}. The margin, then, is just the productivity differential.

Another way to think of the productivity differential is as the extra gap that is opened up between maximum possible income Y_{max} and income Y for every unit of employment that is located in sector j. In this interpretation, Y_{max} is taken to be the level of income that notionally could be generated if all employment were located in sector b. Looked at this way, the first term in q, which is the reciprocal of ρ_{b} – w_{j}, can be understood as the marginal response of sector j employment to a unit widening of the income gap.

The second term in q, w_{j}_{ }/_{ }ϕ, has already been encountered. As observed earlier, it represents job guarantee spending per unit of sector j employment.

Combining these observations, q can be expressed in terms of changes:

So q measures the endogenous response of job guarantee spending to a widening of the income gap. Under present assumptions, it also expresses job guarantee spending as a fraction of the income gap:

**2. A Representation of Dynamic Adjustment**

Here is a way to think about the dynamic adjustment process between two steady states that is analogous to the way the standard model’s dynamics are usually interpreted.

Beginning from a steady state, imagine a one-off exogenous change in demand that causes the economy to leave the steady state. Conceptually, we can suppose that this has an immediate impact on total income and sector b income but that the effects on job guarantee spending and consumption are delayed.

Specifically, let ΔA denote the change in autonomous demand. At time t = 0, it is assumed that total income and sector b income change by the amount ΔA. At time t = 1, job guarantee spending and consumption respond to the events of time t = 0. This affects total income and sector b income in the same step. These changes in the level of income and its composition at time t = 1 then impact on job guarantee spending and consumption at time t = 2, with immediate impacts once again on income and its composition, and so on. The objective is to arrive at expressions describing the behavior of total income, sector b income, job guarantee spending and consumption (and, by implication, other endogenous variables) over time.

In general terms, the following process is envisaged.

The abbreviations are to save space and make the algebra easier, with c = 1 – α being the net marginal propensity to consume. It is also convenient to let m = 1 – ϕ represent the fraction of job guarantee spending going to materials.

The negative fraction k is constant, given the values of the parameters. It measures the inverse response of job guarantee spending to variations in sector b income. A unit increase in sector b income results in 1/ρ_{b} units of employment switching from sector j to sector b and so causes a reduction in job guarantee employment of 1/ρ_{b}. Since government spends w_{j}_{ }/_{ }ϕ per unit of sector j employment, the unit increase in sector b income causes job guarantee spending to change (negatively) by -(1/ρ_{b})(w_{j}_{ }/_{ }ϕ), or k. In terms of changes:

The immediate aim is to express changes in Y, Y_{b}, G_{j} and C in terms of exogenous variables and parameters. This can be done by writing down appropriate equations for the first step, t = 0, and then using these initial expressions to write down appropriate equations for the second step, t = 1, and so on, until a clear pattern emerges. Excel or a similar spreadsheet package is extremely useful for checking that the equations actually work. Setting up a spreadsheet is much easier than deriving the equations. The simple recursive rules outlined in (8) are all that are required for that purpose.

Applying the first of the rules in (8) for time t = 0 gives:

The endogenous expenditures will begin to respond in the next step, with further ramifications for income and its composition.

At time t = 1:

In the last line, use is made of the definition m = 1 – ϕ to express k – ϕk as mk.

Before continuing, it is helpful to note that

An effective strategy is to group c + mk terms together and let M = c + mk. When a c + k term appears, it can be split into an M term and a ϕk term in accordance with (9). The ϕk terms will get multiplied by c to generate ϕkc = v terms. So for convenience:

In economic meaning, M is the spending induced by an increment in sector b income, taking into account both induced consumption and countercyclical variation in job guarantee spending on materials. v is the (negative) change in sector j workers’ consumption caused by a one unit increase in Y_{b} due to the latter’s effect on sector j employment and wages. This makes M + v the net marginal propensity to spend out of sector b income, taking into account both induced consumption and the inverse effects on consumption and materials expenditure of endogenous variation in job guarantee spending. So, in generating the equations, the net marginal propensity to spend out of sector b income will get decomposed into separate M and v terms. For given parameter values, M and v are constants.

Following this strategy, the expressions for ΔY_{1} and ΔY_{b1} can be restated as:

At time t = 2:

The overall pattern is not clear at this stage but emerges if we continue with the algebra for some more steps. Here, we will jump ahead a little and look at some of the expressions that emerge for early steps in the process. Energy can be conserved by focusing on the equations for Y_{b} and G_{j} since these are pivotal and can be used to obtain corresponding expressions for the other variables.

To save a bit more space, changes in Y_{b} and G_{j} will be divided by ΔA with the following notation adopted:

Continuing to work through the algebra gives the following changes in Y_{b} per unit change in A for steps 0 to 9. The purpose here is to notice a pattern.

The pattern followed by the coefficients in (12) relates to Pascal’s triangle. The integers involved are figurate numbers that can be read off the diagonals of the triangle. The first terms in each row are preceded by a 1. The second terms, from t = 2 onward, are preceded by ascending natural numbers (1, 2, 3, …). The third terms, from t = 4 onward, are preceded by ascending triangular numbers (1, 3, 6, …). The fourth terms, from t = 6 onward, are preceded by ascending tetrahedral numbers (1, 4, 10, …). The fifth terms, from t = 8 onward, are preceded by ascending pentatope numbers (1, 5, 15, …). The pattern continues through the simplex numbers.

With the pattern now evident, it is possible to form an equation that describes the behavior of Y_{b}. Focusing on the right-hand side of (12), the rows (or equations) can be numbered from t = 0 and the columns (or terms) from n = 0. The term in row t and column n will be

The first term in brackets is the binomial coefficient C(t – n, n). The change in Y_{b} with respect to A at time t will equal the sum of the terms in row t:

Summing across all time steps gives the total multiplier impact on Y_{b} of an exogenous change in demand:

Together, these last two expressions give a description of Y_{b}‘s behavior over time. Expression (13) enables us to calculate the change in Y_{b} at a given time step, while (14) can be used to find the cumulative change over all or a subset of time steps.

The behavior of the other variables can be considered in the same way. Fortunately, it is only really necessary to retread the above steps for job guarantee spending. Once the behavior of G_{j} becomes clear, it will be easy to infer the behavior of the other variables from the combined behavior of Y_{b} and G_{j}.

Working through time steps 1 to 9 for the changes in job guarantee spending gives:

The pattern is very similar to the behavior displayed by Y_{b} except that it is lagged a step, with each term multiplied by k. In view of the lag, we can number the rows from t = 1. To avert possible confusion, the columns can also be numbered from n = 1.

The value of the term in row t and column n will be

The change in G_{j} with respect to A at time t is given by the sum of the terms in row t:

Summing across all time steps gives the full multiplier impact:

The corresponding expressions for Y can be obtained using the following relationships:

The expressions for C are most easily obtained from the resulting expressions for Y:

**3. Conformity of the Dynamics to Steady State Requirements**

We now have expressions for the steady state levels of the endogenous variables. These were obtained in section 1. We also have expressions that describe the behavior of the endogenous variables at each step in time when they follow a simple set of recursive rules. These were obtained in section 2. But for the assumed dynamic behavior to be consistent with the underlying assumptions of the model, we need it to generate results that are the same as those implied by the steady state relationships. Specifically, the full multiplier impacts generated by the dynamic behavior should match the multipliers that are implicit in the steady state expressions and listed in (7). We need a way to compare the two sets of multipliers and verify that they are actually the same. For this purpose, it will be helpful to have formulas for the series sums developed in section 2 that are easy to compare with the multipliers in (7). Obtaining such formulas and showing that they are equivalent to the multipliers in (7) will verify that the assumed dynamics conform to the steady state requirements of the model.

A key to the exercise is to keep in mind that the sum of a power series with common coefficient *a* and factor r is

So long as the absolute value of r is less than one, the terms in the series approach zero as t becomes large, and the sum of the series converges on a finite point.

It will be possible to obtain summation formulas for the endogenous variables in the form of (20). As in section 2, most effort can go toward obtaining formulas for Y_{b} and G_{j} because of the way they are connected with the other variables.

The full impact on Y_{b} of an exogenous change in demand has been found to be:

Recall that this sum relates to a system of equations for ΔY_{bt}_{ }/_{ }ΔA. The first equations in this system are shown in (12). In (14), terms in a given row of (12) are summed horizontally with the result added to the total of all previously summed rows. Each row pertains to a particular time step. This choice was useful for describing the behavior of Y_{b} over time, which was our aim in section 2, but it is not so helpful for obtaining a formula for the sum of the power series. For that purpose, it is more convenient to sum vertically down columns and add the column sums together.

Some of the equations for y_{bt} = ΔY_{bt}_{ }/_{ }ΔA are reproduced below for easy reference:

To sum vertically, we fix the value of n (which denotes a column) and sum over all values of t. The same can be done for all other values of n to obtain an expression for the total change in Y_{b}.

In summing the columns, care is needed in setting up the summation. In (14), the summation only applies to values of n that are less than or equal to t/2. This implies that we need to take the column sums for values of t greater than or equal to 2n:

The sum of column n will be

The limits of this summation can be shifted so as to start each column sum from t = 0:

As was noted earlier, the coefficients in (12) are figurate numbers. These numbers can be represented in terms of rising and falling factorials. To match our notation, let n represent the “type” of number in Pascal’s triangle, with n = 0 denoting ones, n = 1 denoting natural numbers, n = 2 denoting triangular numbers and so on. The *t _{ }*th number (t = 0, 1, 2, …) of type n will be given by the formula

Here, n bar is the *n* th rising factorial of t + 1 and n! is the factorial of n. By definition, zero factorial equals 1.

To give a couple of examples, the fourth triangular and seventh tetrahedral numbers are

This way of representing the numbers in Pascal’s triangle can be applied to the equations in (12). Column n in that system of equations contains numbers of type n. The *t _{ }*th nonzero entry in the column will be the

Using this notation, (21′) can be written as

As has been noted, this is the sum of column n. We want to find a simple formula for this sum. A way to find the formula is to substitute successive values for n into (22) and look for a pattern. The strategy is as follows. Each time a value for n is substituted into (22), treat the resulting expression as a function of M. By integrating the function n times with respect to M, it will be possible to isolate a ΣM^{t} term. The formula for the sum of a power series given in (20) can then be applied, with *a* = 1 and r = M, to replace ΣM^{t} with 1/(1 – M). With this replacement made, the function can be differentiated n times with respect to M to eliminate any constants of integration and solve for *f*_{ }(M) = ΔY_{bn}_{ }/_{ }ΔA.

When n = 0, (22) becomes:

When n = 1,

Since n =1, we now integrate once with respect to M. The t + 1 term will cancel:

K_{1} is a constant of integration. Differentiating with respect to M and solving:

A pattern is emerging. We can go over some of the procedure one more time to check.

Letting n = 2,

Without actually performing the integrations it is clear that the first integration will eliminate the t + 1 term and the second integration will eliminate the t + 2 term. M will be raised to the power of t + 2. M^{2} can be taken outside the summation, leaving only ΣM^{t} on the inside, which can be replaced with 1/(1 – M). This will give:

where K_{1} and K_{2} are constants of integration. Differentiating twice with respect to M gives:

The procedure gets more laborious as n is increased because of the need to integrate and differentiate n times, but the pattern is clear. All terms in the numerator other than v^{n} cancel out after the *n*_{ }th differentiation and we are left with an expression in the form:

This expression gives the sum of column n in the set of equations abbreviated in (12).

The total multiplier impact on Y_{b} is obtained by summing the totals of all the columns:

Conveniently, (24) is in the form of a power series, as shown in (20), with *a* = 1/(1 – M) and r = v/(1 – M). Plugging into the formula in (20) gives:

So long as the system converges to a steady state, this formula gives the full multiplier impact on Y_{b} of a change in autonomous demand when behavior follows the simple recursive rules outlined in (8).

For Y_{b} to converge, it is necessary that:

Keeping in mind that M + v is the net marginal propensity to spend out of sector b income, convergence will occur for appropriate values of the parameters.

Mathematically, divergence would occur if either M + v > 1 or M + v < -1. But neither of these cases is plausible. The first case is impossible so long as there is a positive marginal propensity to leak (α > 0) because M always takes a value less than one and v is defined to be negative. The second case would make no sense from a policy perspective. For M + v to be negative at all, let alone less than -1, a unit increase in sector b income would need to be accompanied by a withdrawal of job guarantee spending that more than offset the extra induced consumption. The main way it could happen mathematically would be for the fraction of job guarantee spending on wages (ϕ) to be unrealistically small. For example, for α = 1/2, ρ_{b} = 2 and w_{j} = 1/2, divergence would occur for a value of ϕ less than 2/15. For ϕ of exactly 2/15, the system would oscillate from one time step to the next with sector b income showing no tendency either to rise or fall. Or with α = 1/5, ρ_{b} = 3 and w_{j} = 1/2, the system would diverge for values of ϕ less than 5/63. These choices for ϕ would be inapplicable from the policy perspective. On economic considerations, the system will always converge.

We are now in a position to compare the multiplier impact generated by our assumed dynamic behavior, given in (25), with the multiplier for Y_{b} that was derived from the steady state relationships and listed in (7). If the two multipliers happen to be the same, the assumed dynamics will not violate steady state requirements. Our conceptualization of the dynamics will be consistent with the underlying assumptions of the model.

The steady state relationships imply a multiplier for Y_{b} that is shown in (7):

We need to compare our formula in (25) with (26). To do so, it is necessary to substitute in the actual expressions for M and v, since these terms are abbreviations chosen to conserve space. It is also necessary to recall the definition of q. For convenience, these are reproduced below:

It needs to be shown that:

This verifies that the multiplier impact on Y_{b} generated by the assumed dynamic process matches the multiplier impact implied by the corresponding steady state expression. In other words, the dynamics of sector b income conform to steady state requirements. For given values of the exogenous variables and parameters, the assumed system behavior will cause Y_{b} to converge on its steady state level, as depicted in (6).

**Convergence of Other Variables**

The approach just taken for sector b income can also be applied in the case of job guarantee spending. By summing vertically down each column of the system of equations for g_{jt} = ΔG_{jt}_{ }/_{ }ΔA it is possible to arrive at a simple formula for the sum of a power series. This formula will be equivalent to the multiplier derived for G_{j} on the basis of steady state relationships. Once equivalence is established for the multipliers of both Y_{b} and G_{j}, it is straightforward to handle Y, C and, by implication, Y_{j} and other endogenous variables because their expressions can all be obtained from those for Y_{b} and G_{j}. Specifically,

It is unnecessary to go into the same detail for G_{j} as we did for Y_{b} as the steps and algebra are almost identical. But there are a few points that might be unclear without some elaboration.

The g_{jt} for t = 1 to t = 9 are reproduced below:

Before summing vertically, it is once again important to consider the limits of summation in the expression for the total change in G_{j}:

The summation only applies to values of n that are less than or equal to (t + 1)/2. So we need to take the column sums for values of t greater than or equal to 2n – 1:

The sum of column n will be

Left as it is, this would be inconvenient for integrating and differentiating n times. It helps to shift the limits of the summation so as to start each column sum from t = 0:

Replacing the binomial coefficients with equivalent expressions involving rising and falling factorials and treating ΔG_{jn}_{ }/_{ }ΔA as a function of M gives us a convenient starting point:

From here there are no surprises. For each n, we integrate n times to get a ΣM^{t} term by itself that can be replaced with 1/(1 – M). We then differentiate n times to eliminate the constants of integration and solve for *f*_{ }(M).

As might be guessed, the end result is a formula that is almost identical to the one for the total change in Y_{b} except that the numerator is multiplied by k.

Similar expressions for the other variables can be obtained from those for Y_{b} and G_{j}. The resulting formulas are all equivalent to the corresponding multipliers implied by the steady state expressions. The various multipliers resulting from the assumed dynamics are shown below.

**4. Concluding Remark**

The focus has been on a model economy with a job guarantee. Adopting the income-expenditure model as a base, we identified characteristics of a steady state and described a dynamic process – analogous to the usual dynamic interpretation of the base model – that would tend to bring the system, when outside a steady state, back toward it.

**Appendix**

A fair bit of notation has been used. For easy reference, it is summarized below.

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The standard model can be summarized in the following set of equations:

For the system to be in a steady state, output and income (both denoted Y) must equal demand Y_{d}. Demand is defined as the sum of planned expenditures. These expenditures include household consumption C, private investment I, government expenditure G and net exports (exports X minus imports M). The other expressions are behavioral equations. In the simplest versions of the model, private investment, government spending and exports are all considered exogenous (denoted with a 0 subscript). Private consumption, taxes T and imports are endogenous. They are largely induced by income, though with autonomous components (C_{0}, T_{0} and M_{0}). The marginal propensities to consume, tax and import (c, t and m, respectively) are all assumed to be constants that take values between zero and one.

Through a series of substitutions the system can be solved for Y to obtain the steady state level of income:

The numerator of this expression is net planned autonomous expenditure, or ‘autonomous demand’. The denominator is the ‘marginal propensity to leak’. It is the fraction of an increment in income that drains to taxes, saving and imports. The model therefore depicts steady state income as a multiple of autonomous demand, where the ‘expenditure multiplier’ is the reciprocal of the marginal propensity to leak.

To condense the model, let A represent autonomous demand and α the marginal propensity to leak. The model can then be depicted more succinctly as

Demand, in (2), is the sum of induced consumption net of endogenous imports, represented by (1 – α)Y, and autonomous demand.

Substituting the expression for demand into steady state condition (1) and solving for Y gives:

As before, steady state income depends on the level of autonomous demand (now denoted A) and the marginal propensity to leak α. The multiplier, found by differentiating (3) with respect to A, is 1/α.

The model can be used to consider how a change in either the exogenous variable A or the parameter α will affect the steady state level of Y. If autonomous demand changes by the amount ΔA, the steady state level of income will change by ΔY = (1/α)ΔA.

As it stands, the model says nothing about how the system might get from one steady state to another. There are many conceivable ways this could occur. The one that is usually taken to make most sense goes along the following lines. An exogenous change in demand of ΔA will be the result of either an ‘injection’ (from one or more of government spending, private investment and/or exports) or autonomous household consumption, or some combination of the two. The extra exogenous spending will be received as income. This will induce additional household consumption on domestic output, as well as result in leakage or ‘withdrawal’ from the circular flow of income to taxes, saving and imports. In this way, a given increase in exogenous spending will initiate a multiplier process in which the newly created income induces consumption which, in turn, creates still more income, again inducing consumption, and so on. But the multiplier process associated with a particular act of autonomous spending eventually runs out of steam because withdrawals to taxes, saving and imports occur on each round of the process. Given the new value of autonomous demand and the marginal propensity to leak, there will be a stable (steady state) level of activity to which income converges.

The process just described suggests a power series. The idea is implicit in the expression for the multiplier. Since α is always between zero and one, 1/α can be interpreted as the sum of a convergent series. The sum of the series will be

The change in income caused by a change in autonomous demand can now be found by multiplying through by ΔA:

So, in the economic process envisaged, a change in autonomous demand of ΔA occurring at time t = 0 will be received as income. At time t = 1, a fraction 1 – α of this new income will go to consumption and be received as new income of (1 – α)ΔA. At time t = 2, a fraction 1 – α of this new income will go to consumption and create still further new income of (1 – α)^{2}ΔA. And so on.

The basic logic of the model will remain fundamentally the same when a job guarantee is included. But interaction between the broader economy and job guarantee program – which entails endogenous, countercyclical spending – modifies the steady state relationships and can especially complicate plausible representations of the system’s behavior when outside of a steady state.

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At the present time, of course, there is no job guarantee in the comprehensive form articulated by its proponents. The job guarantee is a proposal rather than an existing policy. In today’s economies, some people are relegated to unemployment for the sake of containing inflation. So long as there is excess capacity and unemployment, it is possible to expand production through an increase in employment and the rate of capacity utilization. When appropriate, expansionary demand-side policies can be implemented to this end. At a certain point, however, bottlenecks are likely to emerge due to shortages of particular kinds of workers and raw materials. Although these bottlenecks are often possible to address through additional investment in capacity, this takes time. Meanwhile, the presence of bottlenecks can cause a bidding up of some wages and prices. Potentially, this could also intensify distributional conflict between workers and capitalists as both attempt to protect real income shares. In any case, as the economy gets closer to full employment, the proportion of sectors affected by bottlenecks is likely to rise, adding to the risk of inflation.

In the absence of an incomes policy involving direct wage and price controls, or a job guarantee, the likely government response to an inflation threat will be to adopt contractionary macroeconomic policies aimed at weakening demand in general, and with it, demand for labor-power and raw materials. Given time, this policy approach can succeed in eliminating inflationary pressure. But it comes at the considerable economic and social cost of unemployment. To the extent the policy approach works through a generalized slowdown in demand rather than attempting to address specific areas of the economy directly, it is likely to require more pronounced macroeconomic contraction and job loss than might be necessary under better designed policy.

The job guarantee is proposed as a way of making full employment compatible with price stability. Rather than sacrifice some employment as a means of moderating inflation, it is contended that price stability can be achieved without inflicting the costs of involuntary unemployment. Under a job guarantee, there would still be voluntary unemployment in cases where laid off skilled workers preferred to wait for a similar position to become available rather than accept a job guarantee position. But anybody who was willing and able to take a job at minimum wage (including defined benefits) would be employed. There would be zero involuntary unemployment: a situation described as ‘loose full employment’.

According to the logic of the proposal, the pay and conditions of job guarantee employment would serve as a floor under the economy’s relative wage structure and set the value of the currency in terms of simple labor. A minimum wage (with benefits factored in) of $w per hour would mean that one unit of the currency commanded 1/w hours of simple labor power. Other employers would need to offer pay and conditions that were competitive with the job guarantee as well as commensurate with the complexity of labor involved in the roles required to be performed. Given the economy’s average level of productivity and on an assumption that firms largely set prices as a mark up over wage and materials costs, the wage floor set by the job guarantee would act as a nominal price anchor for the economy’s wage and price structure.

It is argued that the anchoring effect will be strongest when job guarantee employment as a proportion of total employment (referred to as the ‘buffer employment ratio’ or BER) is at its highest. During booms, the BER will tend to decline as some job guarantee workers are attracted into better paying positions in the broader economy. As the job guarantee sector contracts, the nominal price anchor will lose some of its influence on the broader economy, and other wages and prices might rise relative to the job guarantee wage as employers compete for workers. Because government would not compete on wages in an attempt to retain job guarantee workers, these wage payments would not contribute to the inflationary pressures developing elsewhere in the economy. Even so, a boom that continued long enough might result in demand-side inflationary pressure.

Much like under the current policy approach, this situation would be addressed by contractionary policies to moderate activity. But importantly, these policies would not cause involuntary unemployment. The negative impact on employment in the broader economy would instead cause a net migration of workers into job guarantee employment. The consequent increase in the BER would reinforce the nominal price anchor and help to keep inflation in check.

In terms of the discretionary policy response to inflation, clearly there is a resemblance between the current policy approach and the way policy would be conducted in the presence of a job guarantee. Under the current regime, policymakers think in terms of a rate of unemployment (the ‘non-accelerating inflation rate of unemployment’ or NAIRU) at which inflationary pressures first become problematic. With a job guarantee in place, the focus would shift to the proportion of job guarantee employment in total employment (the BER) instead of the unemployment rate. In principle, reference could be made to a ‘non-accelerating inflation buffer employment ratio’ (NAIBER). The NAIBER would be the lowest level of the BER still consistent with stable inflation.

Although the notion of the NAIBER is conceptually clear, Modern Monetary Theorists do not propose attempting to pinpoint its actual value. In reality, and as is true of the NAIRU, its actual value would be difficult to determine. Structural changes, by affecting the likelihood and distribution of bottlenecks, would affect the value of the NAIBER. Institutional changes, by affecting whether bottlenecks were likely to initiate wage-price spirals, could also alter the NAIBER. For instance, combining a job guarantee with an incomes policy might lower the NAIBER by enabling overall demand to be kept stronger without causing inflation. Pronounced distributional changes, by altering the pattern of demand relative to the economy’s present structure, could also alter the likelihood of bottlenecks appearing. Rather than trying to identify the precise value of the NAIBER, the point of relevance to policy is simply that demand-side inflation would be taken as evidence that the BER was too low and that contractionary policies were required. Of course, in the reverse scenario of a generalized slump or deflation, discretionary expansionary policies would be appropriate.

Modern Monetary Theorists maintain that the introduction of a job guarantee would give the economy a more effective nominal price anchor than currently exists. In comparison to the current policy approach, they point out that:

**Job guarantee spending would be highly targeted.**Anyone wanting a job but unable to find one in the broader economy would be able to register for a job guarantee position. This individual act would automatically trigger the minimum level of government spending necessary to generate employment for the worker, given the fixed job guarantee wage. It would be the minimum necessary public expenditure to get the worker back in employment because other forms of spending would only generate the same employment, if at all, indirectly through a multiplier process in which, at each step of the process, some of the newly created income drained to taxes, saving and imports. Since the spending would be kept to a minimum, the inflation risk would likewise be minimized relative to less direct methods of generating the extra job. This argument is not about financial considerations, but demand pressure and potential for inflation.**Much job guarantee spending would be at a fixed price.**In contrast to other spending, which is typically conducted at market prices, the hiring of job guarantee workers would occur at a fixed wage. This spending, unlike most other public and private spending, would not directly participate in any bidding up (or bidding down) of wages and prices that might occur elsewhere in the economy. This spending rule would provide a stable floor under other wages and, given the economy’s level of productivity and a prevalence of mark up pricing, influence prices in general. During booms, other wages might spread higher above the floor, but they would do so to a lesser degree than if government competed on wages to retain workers in the job guarantee program. In a slump, other wages might decline relative to the wage floor, but the floor would limit their fall. In this way, the fixed job guarantee wage would be a source of stability in the economy’s nominal wage and price structure. There would be some job guarantee spending undertaken at market prices. Most notably, materials used in the production processes of the job guarantee sector might mostly or entirely be purchased at market prices from the broader economy. But with wages being a substantial fraction of total job guarantee spending, the stability provided by the wage floor would still be significant.**A more job-ready labor force.**Unemployment, especially when long term, can affect the employability of individuals due to skill atrophy and the loss of work habits, self-esteem and social networks. Employers, for their part, display a marked preference for hiring employed rather than unemployed individuals. It is likely that the job guarantee, by enabling people to maintain continuous employment, would make it easier for laid off workers to transition back into the broader economy as well as make the process more cost-efficient for firms. The relevance of this argument from the perspective of price stability is that job-guarantee workers would be more competitive in job applications than unemployed workers, and so exert more competitive pressure on wages in the broader economy. In other words, the nominal price anchor built in to the job guarantee would exert more influence on other wages and prices than if employers instead were hiring from a pool of unemployed workers.

**A Simple Depiction of the Theory**

The theory just described emphasizes two main sources of inflation. First, there are structural, institutional and supply-side factors (such as the skill profile of the workforce) that affect the likelihood and consequences of bottlenecks. These factors determine the NAIBER. Second, inflationary pressure is conceived as inversely related to the buffer employment ratio rather than the unemployment rate. When the BER gets too low, bottlenecks can arise, resulting in inflation.

A third potentially relevant factor, not touched on above, is a role for inflation expectations. These can conceivably influence the pricing behavior of firms and wage demands of workers. It is not clear, though, that expectations really play an independent causal role in the inflation story that has been outlined. Workers or firms might well expect inflation of five percent, but if they lack the capacity to raise their own wages and prices by five percent, prices in the end will not mirror their expectations. Conversesly, if workers or firms find that they can raise their wages or prices by ten percent, they will hardly be constrained in their actions by an expectation that inflation will only be five percent. In either scenario, the critical factor will not actually be expectations. The critical factor will be whether workers and firms actually have the capacity to raise their wages or prices, and this capacity – in the theory briefly outlined – will depend on demand conditions (summarized as the value of the BER relative to the NAIBER) along with the structural, institutional and distributional aspects of the economy that influence the likelihood and consequences of bottlenecks.

The job guarantee would put an end to the inverse Phillips Curve relationship between inflation and unemployment for the simple reason that there would be no involuntary unemployment. Rather than accepting some unemployment as the price of containing inflation, loose full employment would be maintained at all times.

Although there would be no Phillips Curve, we could, if we wished, draw a similar curve to depict the situation with a job guarantee. I’m not sure if such an exercise would be exactly MMT kosher but it seems that we could define a function with inflation inversely related to the buffer employment ratio (BER) rather than the unemployment rate. A simple representation would be:

with π_{t} the inflation rate of the current period, π_{0} exogenous inflation and *f* a function describing the sensitivity of inflation to excess or deficient demand.

With BER < NAIBER, there would be excess demand and inflationary pressure. Demand deficiency and decelerating inflation would occur with BER > NAIBER.

Graphically it might look something like this:

The curve gets flatter as the BER increases to reflect the view that the influence of the job guarantee’s nominal price anchor will be at its weakest when the BER is near zero and stronger when the BER is high.

During a boom, the economy would move along the curve, up and to the left, as the job guarantee program was deprived of workers and the BER decreased. If policymakers did nothing to constrain demand, the theory suggests inflation would rise. If, instead, policymakers implemented contractionary policy, as appropriate, this would help to bring the economy back down along the curve toward the NAIBER.

An increase in exogenous inflation emanating from the supply side of the economy (for instance, an oil shock) would cause an upward shift of the entire curve. For any given BER, there would then be a higher rate of inflation. Here, too, it might be necessary for policymakers to implement contractionary policy to compel a less inflationary resolution to conflict heightened by real income losses associated with the supply shock.

A significant change in the economy’s structure or institutions could cause the NAIBER to shift either left or right, depending on whether the structural change made the economy less or more prone to inflation.

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The answers to these questions depend, above all, on whether the government in question is a currency issuer or a mere currency user. The simple tale traced below illustrates some basic aspects of a society in which government issues its own currency. It is important to understand that the illustration does not apply to governments that use a currency issued by some other entity. For example, it does not apply to a state or local government that is required to transact in a currency issued by its national government. Such governments are mere currency users, much like households and private businesses. They do not issue their own currencies. Equally, the illustration does not apply to a national government that has given up its prerogative to issue a currency and has instead agreed to use what is essentially a foreign currency. This is currently the predicament of those European governments who have committed to using the euro. At least for now, such governments have reduced themselves to the status of mere currency users. If, at some point in the future, these European governments decide to go off the euro and reintroduce their own currencies, the following illustration will then become applicable to them, but not until then.

Most national governments, however, do issue their own currencies. This is true of the national governments of countries such as China, the US, Russia, the UK, Japan, Brazil, India and too many others to mention.

To illustrate in very simple terms the position of such governments and the nature of their currencies, consider Buckwell Island. It is an imaginary island. Its people have just voted to form a nation. Elected representatives from each region of the island have gathered to discuss some basic needs of the new nation.

After much discussion and consultation it is agreed that there need to be some common rules governing access to natural resources, ownership or stewardship of property, and the conduct of business activities. It is decided that the newly formed government will need to see to the effective defense of the island and develop key public services in education, child and aged care, health care and civil administration as well as build or oversee the building of roads, a public transport system and a modern communications network. All this will take time and much effort.

The island government realizes that it will need to hire some people to perform roles in the public sector. It also needs a way for private sector activity to be integrated into the economy. To this end, it decides to introduce a national currency called the ‘buck’.

The currency is established in two basic steps:

First, the government officials return to their local electorates and announce that all citizens of at least working age will be required to pay taxes. A tax authority will be established to monitor and enforce payment. Significantly, it is specified that all taxes and other government charges will have to be paid in bucks. No other currency or item will be accepted. In addition, all income and wealth generated on the island will be evaluated in bucks, with taxes imposed accordingly. Court settlements will also be specified in bucks.

Since people require an income, and since income and wealth will be evaluated in bucks and subject to tax, this first step in establishing a national currency creates a need within the community for its members to earn or otherwise obtain bucks.

The effect of this first step is to ensure at least some demand for the currency. Since people need bucks to pay taxes and other government charges, they will accept the currency in payment for goods and services that they are able and willing to supply either to government or to fellow Buckwellians.

With a basic demand for the currency now created, the second step in establishing the currency is for the government to open up various channels through which members of the community can actually get hold of the currency. In partial fulfillment of this objective, nominated government representatives release a list of public sector job positions. It is announced that there will be jobs for teachers, health practitioners, administrators, police officers, defense and peacekeeping forces and other important roles. Suitable applicants are free to apply for these jobs. If successful, they will be paid in bucks.

The officials also announce a list of goods and services that the government would like to purchase from private businesses. This opens up opportunities for prospective businesses to sell output to a large customer (the government). To obtain finance, competing businesses will be able to approach a newly established public bank for loans. These loans will be issued in bucks. In time, private banks may also be permitted to operate. For now it is left as a future question for the voters to decide.

The effect of these measures is to ensure that citizens can get hold of the currency. Some people will accept a job offer from the government to work in the public sector. Business operators, financed at least initially by public loans, will be able to start production and sell their products or services to customers, generating revenues in bucks. The rest of the working-age population will be able to seek employment in the private sector in exchange for bucks.

The government also announces a Job Guarantee program. Anyone who cannot find a job elsewhere can accept a job-guarantee position financed by government and administered by local government or perhaps community organizations. Those who are unable to work due to age or sickness will receive a pension paid by government. For accounting purposes, the pension payments are treated as negative taxes rather than spending, because they do not constitute direct spending on goods and services.

With these policies in place, everyone can get hold of bucks. They will be obtained either as newly issued currency when the government spends, pays pensions or lends, or as circulating currency when households or businesses make payments with bucks yet to be extinguished by taxation.

To facilitate economic activity as well as saving, citizens are granted the right to hold accounts at the public bank. Payments for goods and services can be made through the direct debiting and crediting of bank accounts or the exchange of hard currency.

To summarize, there is a basic two-step logic involved in Buckwell Island’s introduction of a currency. First, by requiring taxes and similar charges to be paid in bucks, the government ensures that people are willing to accept the currency. This step ensures at least some demand for bucks since people need them, at minimum, to pay taxes. Second, the currency is issued. The government issues bucks by hiring workers, providing pensions, extending loans to businesses and purchasing some of their output.

It may be noticed that it would be impossible for anyone to pay taxes before the currency had actually been issued. Until government spent on goods and services, or paid pensions, or the public bank lent, there would be no way for anyone to obtain bucks. An implication is that government spending and public lending are logically prior to the payment of taxes. The government’s spending and lending are not constrained by the amount of taxes paid in the past. To the contrary, the capacity of citizens to pay taxes originates from government spending and lending. A currency-issuing government, such as Buckwell Island’s government, can always afford to purchase whatever is available for sale in its own currency. In other words, if something has a sale price in bucks, the island government can always afford to purchase it, since it is the sole issuer of bucks.

A system of accounting and basic bookkeeping is put in place with the aid of spreadsheets maintained by the public bank. Whenever the government decides to spend or lend bucks, the public bank types in the appropriate new numbers in its spreadsheets. If the government pays 100 bucks for office stationary, the public bank marks up the account of the stationary supplier by that amount. The new deposit will be an asset of the stationary supplier and a liability of government.

The stationary supplier is one entity in what can be called the non-government sector. This sector includes all households and private businesses on the island as well as any foreigners making transactions in bucks.

When the government spends 100 bucks on office supplies, the financial assets of the non-government sector, taken as a whole, increase by the same amount. This increase in non-government financial assets is offset exactly by an increase in government liabilities of the same amount. In this way, bucks (which are government liabilities and non-government assets) are created out of nothing.

Conversely, when the stationary supplier pays perhaps 30 bucks in tax, this amount is subtracted from its bank account. This action destroys 30 bucks. They no longer exist. The stationary supplier’s assets go down by 30 bucks as do government liabilities. As a whole, the financial assets of the non-government sector decrease by this amount, as do outstanding government liabilities.

In short, government spending and lending create bucks, which are a financial asset of non-government and financial liability of government. Taxation eliminates bucks. Loan repayments to the public bank also function as tax payments, and so eliminate bucks. The discharge of any other government charges, such as through the payment of licensing fees or fines, also eliminates bucks.

As a matter of logic, something cannot be destroyed before it exists. Bucks must be created through government spending or lending before they can be eliminated through taxes, the repayment of public loans or the discharge of other financial obligations to government. Or, to put it another way, the government must issue its financial liabilities before they can be extinguished.

Clearly, there is no limit to how many financial liabilities the government can issue in its own currency. For Buckwell Island’s government, there is never a question of financial affordability when it comes to anything that is available for sale in bucks. To issue its liabilities, the government simply makes a decision to spend, pay pensions or lend. With the decision made, the bucks are keystroked into existence when the public bank types them into the accounts of spending recipients, pensioners or borrowers.

However, the absence of a financial limit does not mean that there are no limits to what can be achieved in real terms. It would be pointless for Buckwell Island’s government to keep increasing its spending if there were no actual goods and services that could be produced to meet the extra demand. Acting in this way would merely bid up the prices of various goods and services. And since the prices of these goods and services might also enter into the costs of producing other goods and services, there could potentially be a bout of excessive inflation in which prices, on average, rose rapidly and continued to rise for some time. Inflation causes the currency to lose some of its purchasing power. A unit of the currency – one buck for the islanders – would purchase less than before.

Fortunately, Buckwell Island’s government and citizens understand that the level of total spending – which includes the spending of government, the island’s households and businesses, and foreigners – needs to be kept in sensible proportion to the capacity of the society as a whole to produce real goods and services. This does not necessarily mean that taxes must be as high as government spending. To the extent that households, businesses and foreigners wish to spend less than their incomes, this subtracts from the overall level of demand and makes it possible for government to spend somewhat more than it taxes without causing excess demand.

Although our tale is very simple compared with the complexities of modern economies, it does highlight some basic points that will help address the questions posed at the outset. Introducing more real-world factors, such as a private banking sector, would greatly complicate the details of the story but not change the basic issues presently under consideration. Returning to the questions:

**Question 1.** Where does a national currency come from?

**Answer.** In a society with a currency-issuing government, the currency originates from governmental decisions to spend or lend. Whenever Buckwell Island’s government follows through with its decision to spend or lend, bucks are created. As has been discussed, the bucks so created are a financial liability of government – a form of liability that the government can issue without limit – and a financial asset of non-government.

**Question 2.** How does a currency system basically work?

**Answer.** The currency originally comes from government. When the Buckwell Islanders sell goods or services (including labor services) to government, they receive bucks in exchange as income. When businesses borrow from the public bank to invest in production or hire workers, these expenditures likewise go to somebody as income. In this way, bucks enter the economy and can be used for various purposes within the economy. Some of the bucks received as income will be used by households and businesses to pay taxes. Some will be saved in accounts at the public bank or in hard currency. Some will be used to buy goods and services from foreigners. And some will be used to buy goods and services from others in the island community, creating additional income in the process. In this way, bucks circulate from one household or business to another until at some point they are eliminated through taxation.

**Question 3.** Why might people agree to accept a national currency in the first place?

**Answer.** An important reason is that people must obtain the currency in order to pay taxes or meet other financial obligations imposed by government. This ensures that the Buckwell Islanders will accept bucks in particular, in preference to some other currency. The imposition of taxes and other government charges is not necessarily the only reason people might accept the currency. But this factor is sufficient to ensure at least a base level demand for the currency.

**Question 4.** How can we be confident that a national currency won’t collapse and that people will continue to accept it in economic transactions?

**Answer.** A complete collapse of the currency would mean that it was no longer possible to buy goods and services with the currency. It would mean that nobody was willing any longer to accept the currency in payment for anything. A complete breakdown of the currency will not happen so long as the government effectively enforces taxes and other obligations denominated in its currency. It is the successful enforcement of these obligations that guarantees the currency remains acceptable in exchange at least to some extent.

Even so, we have also recognized that the currency can lose some of its purchasing power when the tax burden becomes too light relative to the level of government spending. If taxes were kept very low but the government chose to spend a great deal in comparison, this could cause (in combination with other spending) more demand than the economy could cope with in a timely fashion and result in a bidding up of prices for many goods and services. A unit of the currency – one buck for the islanders – would then buy less than before. The purchasing power of the currency would decline.

What matters, in this respect, is the strength of spending in general, whether by government or non-government. All spending carries some risk of inflation if it causes demand to outstrip the capacity of the economy to supply additional output. The more households, businesses and foreigners wish to save bucks (rather than spend them), the more it is possible for government spending to exceed tax payments without causing undue inflation.

The solution to an outbreak of severe inflation would be for government either to cut its spending, raise taxes or do a combination of both. These actions would constrain the level of public and/or private spending and help to keep inflationary pressures in check. In the absence of a Job Guarantee, the policies would almost certainly create some unemployment. Fortunately, Buckwell Island has a Job Guarantee. This program ensures that full employment can be maintained despite any reduction in total spending on goods and services that might occasionally be necessary to curb inflation.

**Question 5.** Can a government ever go broke and leave citizens footing the bill?

**Answer.** A currency-issuing government can never go broke so long as it sticks to operating in its own currency and refrains from borrowing in foreign currencies. (To the extent that a government becomes indebted in a foreign currency, it reduces itself to the status of a currency user and can run into financial difficulties.) A government that sticks to its own currency faces no financial constraint. It can always create more currency if and when this is deemed appropriate.

**Question 6.** Can financial affordability even be an issue for government?

**Answer.** A currency-issuing government can always purchase whatever is available for sale in its own currency. By the same token – and at the risk of stating the obvious – the government cannot purchase what is unavailable for sale. Put simply, there can be a shortage of natural resources. There can be a shortage of workers. There can be a shortage of knowledge and technical know-how that prevents certain goods or services from being supplied within a particular time frame. These constraints place limits on what can be done in real terms. But, importantly, anything that can be done within these real limits is affordable for a currency-issuing government.

**Related Posts**

Consider a currency-issuing government that requires itself either to deduct taxes from non-government accounts or to issue debt to non-government before it spends. This is quite typical of governments today. At first glance, these requirements may seem problematic. From inception, it would clearly be impossible for a currency issuer to receive tax payments in its own currency or to auction off public debt in exchange for its own currency before the currency itself had actually been issued. The resolution to this apparent conundrum is that government can always (and currently does) advance to non-government the currency it requires to purchase newly issued public debt.

More specifically, the central bank, as the monetary arm of government, issues currency in the form of reserves (by crediting reserve accounts) while requiring collateral in the form of previously issued government bonds. Non-government is then in a position to purchase newly issued government bonds, with reserve accounts debited in settlement.

Typically, there will be a bidding process at a Treasury auction. The bonds will go to the participants with successful bids – all those bids that offer an interest rate on government debt that is at or below the cutoff. If the central bank is prohibited from purchasing bonds directly from the Treasury (which may or may not be the case, depending on the country in question), it will be necessary to auction off all newly-issued debt to non-government. In that case, the cutoff between successful and unsuccessful bids will be determined in the auction itself.

Considering that the sale of public debt is typically subjected to a bidding process, it may be wondered how government can nonetheless maintain control over the terms on which it spends. In particular, it may be unclear what prevents the rate of interest on public debt from rising above the rate that government actually wants to pay.

The answer is simple. The central bank can always purchase public debt in the secondary market (the market for previously issued bonds) and signal an intention to do so at a particular price. This means that official bond dealers participating in the Treasury auction know that bonds purchased in the primary issue on slightly more attractive terms can be sold to the central bank in the secondary market at a profit. It also means that the central bank can control interest rates on all government debt (of varying duration) by purchasing as much of each type as necessary in the secondary market. This pushes up bond prices and lowers yields, making newly issued bonds more attractive than otherwise. There is no limit to the central bank’s capacity to create reserves. For this reason, it can always drive interest rates on public debt lower through secondary-market transactions.

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**Behavioral Assumption**

As was emphasized in the earlier post, a key behavioral assumption of the income-expenditure model is that, in setting production levels, firms attempt to eliminate unplanned investment, defined as the unanticipated change in inventories. Equivalently, unplanned investment can be defined as excess supply. In response to excess demand or supply, firms are assumed to vary the level of production in an effort to cater to the level of demand. The model presupposes that there is sufficient slack for output to be adjusted to demand in the assumed way.

In the context of a continually growing economy, the elimination of excess demand or supply requires firms not only to react to what has happened in the immediate past (as reflected in any unexpected change in inventories) but also to anticipate how fast demand is likely to grow in the immediate future. This suggests the following decision rule:

This says that firms will adjust actual output (Y) by an amount (ΔY) that is intended to cover the expected change in demand (ΔY_{d}^{e}) as well as eliminate part or all (a proportion λ) of unplanned investment (I_{u}).

If growth is taken to be continuous, with time divisible into tiny increments, the process can be represented in terms of elementary calculus. The above decision rule becomes:

The dots indicate time derivatives. Y dot (= dY/dt) is the instantaneous change in output. Y_{d}^{e} dot is the expected change in demand. It equals the expected growth rate of demand (g_{d}^{e}) multiplied by the level of total demand: Y_{d}^{e} dot = g_{d}^{e }Y_{d}. The level of unplanned investment (I_{u}) – or excess supply – is the difference between actual output and total demand: I_{u} = Y – Y_{d}. Substituting these alternative expressions for Y_{d}^{e} dot and I_{u} into the decision rule and rearranging gives:

**Expectations**

At times it is necessary to specify how expectations are formed. For present purposes, the simplest form of adaptive expectations will do. According to this assumption, firms expect demand in the present to grow at the same rate as it was growing in the recent past:

Here, g_{d}^{e} is the expected growth rate of demand at time t and g_{dt–θ} is the actual growth rate of demand at an earlier time t – θ, where θ is a phase of time.

Adaptive expectations result in systematic errors. When the actual growth of demand is accelerating, so that demand is growing faster now than in the recent past (g_{d} > g_{dt–θ}), firms that form expectations in the assumed way will underestimate the growth rate of demand. Conversely, when the growth of demand is decelerating, firms will overestimate the growth rate of demand. Expectations will only turn out to be correct when the growth rate of demand stabilizes for long enough that g_{d} = g_{dt–θ} = g_{d}^{e}.

This simple form of adaptive expectations implies that, whenever there are errors, expectations are adapted according to the rule:

The difference between g_{d}^{e} and g_{d} is the error in expectations. It is assumed that firms respond by adjusting expectations in the opposite direction to the error. For example, if firms overestimate the growth rate of demand prevailing at time t, the error will be positive. Expectations will be adjusted downwards. This is a special case of a more general rule of adjusting expectations by the amount -β(g_{d}^{e} – g_{d}), where the positive reaction parameter β is set to one.

A different assumption could be made, such as basing expectations on a weighted average of past outcomes, allowing for slower adjustment of expectations in the event of errors, or opting for rational expectations, which would allow for quicker correction of random, unbiased errors, but none of these choices would alter the basic results of the model. One way or another, expectations have to be modeled to give determinate results, but the precise choice does not matter much in the present context as long as the assumed rule is at least somewhat sensible. Accordingly, a simple option is preferred.

**Growth of Actual Output**

Expression (2), which describes the way firms respond to excess demand or supply, implies a growth equation for actual output. This is obtained by dividing both sides of the expression by actual output (Y) and noting that Y dot / Y is the instantaneous growth rate of actual output (g).

If there is excess demand, the fraction Y_{d }/ Y exceeds one and actual output grows faster than expected demand (g > g_{d}^{e}). Under excess supply, the reverse is true. Equilibrium requires that demand equal supply (Y_{d }/ Y = 1). In that case, the growth rate of actual output will coincide with the growth rate of expected demand (g = g_{d}^{e}).

As has already been noted, expectations can only become correct if the growth of total demand (g_{d}) first stabilizes. Otherwise, if g_{d} keeps changing, expectations will also keep changing (with a time delay of θ) and result in variations in the growth rate of actual output.

For equilibrium to persist, firms’ demand expectations would need to become – and then remain – correct (g_{d}^{e} = g_{d}). If this occurred, actual output would grow at the same rate as total demand (g = g_{d}) until firms made an error in predicting the trajectory of demand. Starting, hypothetically, from a point of equilibrium, any error would cause actual output to deviate from total demand (Y_{d }/ Y would differ from one) and result in diverging growth rates for actual output and total demand.

**Other Key Assumptions**

As has been discussed, the growth equation for actual output given by (5) is predicated on the behavioral assumption that firms, through their production decisions, attempt to eliminate unplanned investment and keep pace with demand. Two other key assumptions of the income-expenditure model are that: (i) there is a positive level of autonomous demand (A > 0); and (ii) there is a positive, and stable, marginal propensity to leak (α > 0) to taxes, saving and imports.

On the basis of these three assumptions, there is a tendency, within the model, for both actual output and demand to converge, at any point in time, toward a particular (but, in a growth context, evolving) level of output (Y*) at which supply would equal demand.

The characterization of this equilibrium level of output will depend on what is assumed about demand. As always, total demand can be considered as the sum of two parts: induced demand and autonomous demand. Here, all demand other than net induced consumption is assumed to be autonomous. This makes induced demand (1 – α)Y. Autonomous demand is A. Therefore:

**Equilibrium Output**

With the determinants of actual output and total demand specified, the level of output (Y*) to which supply and demand converge can be established. With actual output equal to total demand, Y and Y_{d} can be replaced with Y* in (6):

Solving for Y* gives:

Output level Y* is a particular multiple (equal to 1_{ }/_{ }α) of autonomous demand. It is a notional level of output (changing over time) that the economy does not necessarily reach. Its relevance hinges on it being an ‘attractor’ for both actual output and total demand. In the income-expenditure model, Y* does in fact play the role of an attractor under most circumstances. A way to verify this point is presented later.

**Growth of Equilibrium Output**

The growth rate of Y* can be obtained from (6′). The first step is to differentiate with respect to time:

All terms involving Y* dot can be collected to the left-hand side:

Dividing through by αY* will convert the left-hand side to a growth rate (Y* dot / Y*). Multiplying A dot by A / A will also enable this derivative to be converted to a growth rate:

Rewriting the expression in terms of growth rates and noting that A / α = Y* gives:

Under present assumptions, the growth rate of Y* simply follows the behavior of autonomous demand (g* = g_{A}), which is taken as exogenously given.

**Growth of Total Demand**

The growth behavior of total demand is implicit in (6). Following the same procedure as in the previous section, the expression can be differentiated with respect to time:

Dividing through by Y_{d} and multiplying Y dot and A dot by Y / Y and A / A, respectively, gives:

Expressed in terms of growth rates:

Here, g_{d}, g and g_{A} are the growth rates of total demand, actual output and autonomous demand, respectively. The growth rate of total demand is basically a weighted average of the growth rates of actual output (g) and autonomous demand (g_{A}). The first term on the right-hand side describes endogenous demand growth. The second term describes autonomous demand growth.

Substituting for g, using (5), and rearranging yields the following expression for the growth rate of total demand:

Terms 1 and 2 can be considered briefly in turn. If there is excess demand, Y / Y_{d} is less than one and term 1 will be greater than (1 – α) g_{d}^{e}. As Y / Y_{d} increases toward one, the term approaches (1 – α) g_{d}^{e}. If Y / Y_{d} actually equals one, the term will equal (1 – α) g_{d}^{e}. Now, it has already been mentioned, in relation to (5), that when Y / Y_{d} equals one, the growth rate of actual output equals both the expected growth rate of demand (g = g_{d}^{e}) and, by (8), the growth rate of autonomous demand (g = g_{A} = g*). So the tendency of term 1 is to approach (1 – α) g*.

Turning to term 2, in equilibrium A_{ }/_{ }Y_{d} equals α. This is clear because, from (7), the equilibrium level of income is Y = Y_{d} = Y* = A_{ }/_{ }α, implying A_{ }/_{ }Y = A_{ }/_{ }Y_{d} = α. When there is excess demand, both Y and Y_{d} are less than Y*, implying Y_{d} < A_{ }/_{ }α or, upon rearrangement, A_{ }/_{ }Y_{d} > α. So term 2 exceeds αg_{A} when there is excess demand but tends toward αg_{A} as A_{ }/_{ }Y_{d} shrinks toward α. In equilibrium, term 2 equals αg_{A}. And since g_{A} is the equilibrium growth rate, term 2 tends toward αg*.

Considering the right-hand side of (10) as a whole, the growth rate of demand (g_{d}) tends to approach (1 – α)g* + αg* = g*.

**Dynamic Stability**

The discussion so far has highlighted certain characteristics of ‘equilibrium’ – or what is often referred to as ‘steady state’ – growth. In a nutshell, a steady state would require actual output continually to equal total demand (Y = Y_{d} = Y* = A / α). Under these circumstances, autonomous demand would be a stable fraction (α) of actual output and total demand (A / Y = A / Y_{d} = α). In terms of growth rates, all key variables would be growing at the same rate (g = g_{d}^{e} = g_{d} = g_{A} = g*).

Of course, it is one thing to identify features of steady-state growth. It is another to establish whether the system has a tendency to move toward such a state. Basically, the question is this: if the exogenously given growth rate of autonomous demand is held constant for long enough, will the assumed behavior of firms and households tend to push the economy toward a situation in which actual output and total demand converge on output level Y* with all key variables growing at the same rate as autonomous demand?

It should be stressed that this is an exercise more in logic than empirical reality. There is no suggestion that autonomous demand would in fact grow at a constant rate for any significant length of time or that a real-world economy could ever attain a steady state. The purpose of the exercise is to establish the results and tendencies that follow, logically, from the stated assumptions of the model.

To consider the issue of dynamic stability, let:

Variables y and z, both ratios, are measures of excess demand. Variable y is greater than, equal to, or less than one depending on whether demand is greater than, equal to, or less than output. When Y = Y_{d} = Y*, y = 1.

Variable z, the share of autonomous demand in income, is an alternative indicator of excess demand. It has already been observed that when demand and supply are both at the level Y*, z = α. If, starting from a steady state, autonomous demand subsequently grows at a different rate than output, there will be excess demand or supply, with z diverging from α. Under excess demand, z > α; under excess supply, z < α.

It follows from the definitions of y and z that the share of autonomous demand in total demand (A / Y_{d}) is z / y. As with z, this share is greater than, equal to, or less than α depending on whether there is excess demand, equilibrium, or excess supply.

Most of our focus can be on variable y (= Y_{d }/ Y). Dynamic stability requires y to tend toward one and, if it gets there, to remain at one unless there is an exogenous change in either the growth rate of autonomous demand (g_{A}) or the marginal propensity to leak (α). Clearly, whether y will behave in the required way will depend on the growth behavior of total demand (Y_{d}) and actual output (Y). This can be made explicit by differentiating y with respect to time (applying the quotient rule):

The fractions in brackets on the far right are growth rates and Y_{d }/ Y is variable y. Modifying the expression accordingly gives:

This is a differential equation. It relates the instantaneous change in y (y dot = dy/dt) to the growth rates of demand and output (g_{d} and g) as well as to y itself.

For a steady state to persist, it is not enough that y = 1 at a given moment. It is also necessary for y to remain unchanged (y dot = 0). As (12) spells out, this can only occur if demand and output happen to grow at the same rate (g_{d} = g = g*).

Equation (12) is missing details that are needed to assess stability. One way to fill in these details is to express the equations for g_{d} and g in terms of y and substitute the resulting expressions into (12). An alternative way, which turns out to be a bit simpler, is to substitute an equation for g into an equivalent expression for y dot, making use of the definition of z. The first step is to differentiate z ( = A / Y) with respect to time to obtain:

It can then be observed that

Since both 1 and α are constants, it follows that y and z both change at the same rate:

This means that the expression for z dot in (13) can be substituted into the right-hand side of (12) to obtain:

To eliminate z, it can be noted from rearrangement of (14) that z = y + α – 1. Upon substituting for z, (12) becomes:

Since (12) and (12′) are equivalent expressions, either can be used to check stability.

It is still necessary to substitute an expression for g in (12′). Writing (5) in terms of y gives the required expression:

Substituting this into (12′) gives:

Formulated in terms of (12”), the question of dynamic stability comes down to how y dot responds to a small change in y. Starting from a situation in which y = 1 and y dot = 0, it is necessary that changes in y away from 1 set in motion forces that tend to cause y to move back toward 1. In considering this question, expectations will be taken as given. At time t they are already formed, based on what happened at time t – θ. And in a steady state, the expected growth rate of demand will equal the other growth rates under consideration. The other elements in (12”) are either parameters (α, λ) or an exogenous variable (g_{A}) and so can also be taken as given. Accordingly, we take the derivative of y dot with respect to y. Dynamic stability requires that the derivative be negative when evaluated at the critical point y = 1.

In undertaking this exercise, it is important to mention that equation (12”) is nonlinear in y because of a y-squared term (if the expression is expanded out). This means that taking the derivative with respect to y serves only as a test of ‘local stability’ as opposed to ‘global stability’. Basically, the derivative represents the best linear approximation to the behavior of y near the point y = 1.

Taking the relevant derivative (applying the product rule) gives:

Evaluating the derivative where y = 1 and g_{d}^{e} = g_{A} = g* reduces the expression to:

The derivative is negative provided:

Therefore, the modeled behavior is locally stable provided the growth rate of autonomous demand (g_{A} = g*) exceeds -λ, where λ is the positive reaction parameter that indicates the intensity of firms’ response to unanticipated changes in inventories. For positive growth regimes, the model always exhibits local stability.

To get an idea of the global characteristics of the model’s behavior, the relationship between y dot and y can be represented graphically, once again taking expectations and the growth rate of autonomous demand as given and equal to each other. In the diagram below, g_{d}^{e} = g_{A} = 0.02, λ = 1 and α = 0.5.

In reference to the diagram, we can take an arbitrary value for y and consider whether it is a stable point, and then ask the same question for other arbitrary values of y.

Not all values of y are permissible because of the assumption that the level of autonomous demand must be positive. This implies, in turn, that z (= A_{ }/_{ }Y) must also be positive. Since, by (14), z = y + α – 1, it follows that y must exceed 1 – α. In the diagram, y must exceed 0.5, reflecting the value chosen for α.

If there is excess supply, y is less than one. In the scenario considered, if y is between 0.5 and 1, y dot is positive, meaning that the assumed behavior of households and firms will push y higher, toward 1. This is indicated by the arrows pointing to the right. Conversely, when there is excess demand, y is greater than 1 and y dot is negative. The arrows situated to the right of y = 1 and pointing left indicate that y, when above one, is pushed lower, toward one. The only stable point is y = 1. This is indicated by the black dot.

There is another point at which y dot is zero, indicated by the white dot. But in most scenarios, this point is unstable. On and to the left of this dot, y is below the minimum permissible value. To the right of the white dot, y is pushed toward y = 1.

Assigning different values to the parameters (α and λ) affects the diagram in certain ways. As has been discussed, the value of α determines the minimum permissible value of y. Choosing a smaller value for λ flattens the curve, reflecting slower adjustment, without altering the positions of the two critical points. However, for some extreme cases, the positions of the white and black dots can switch.

The result that local stability requires g* > -λ comes into play in these extreme cases. If the growth rate is very negative and λ is small, it is possible for the point y = 1 to be unstable. For example, if the values of g_{d}^{e} and g_{A} are both set to -0.2 and λ is reduced to 0.1, with α kept at 0.5, the critical point y = 1 is unstable and repels the system, which is pushed either toward infinity (if y > 1) or toward y = 1 – α (if y < 1).

In extreme cases such as this, the model suggests that stability can be restored through an exogenous change in policy that lifts the growth rate of autonomous demand.

**Description of the Model’s Growth Behavior**

Provided the condition for dynamic stability holds (g* > -λ), and y is quite close to 1, the behavioral assumptions ensure a tendency for actual output and total demand to converge on Y*. Conceptually, the growth process depicted in the model can be described as follows. Starting from a steady state, an exogenous increase in the growth rate of autonomous demand (g_{A}) creates a situation in which actual output and, to a lesser extent, total demand temporarily grow less rapidly than Y*. This results in excess demand (y > 1), a delayed upward revision of expectations (since g_{d} > g_{dt–θ}), and an acceleration of output growth (g) that brings with it further endogenous acceleration of demand growth (g_{d}). So long as g remains below g_{A}, it also remains below g_{d}. At this stage, y is still rising further above one.

At a certain point, g catches up to g_{d}. This occurs at the same time as g catches up to g_{A}. It then overtakes both g_{d} and g_{A}. The deviation of g above g_{A} is greater than g_{d}‘s deviation. This is partly because endogenous demand growth accounts for only a fraction of total demand growth (which is a weighted average of endogenous and exogenous growth) and partly because leakage to taxes, saving and imports occurs throughout the process.

In the convergence of y toward 1, there can be some oscillation (fluctuation) around the critical point. This is due to the lag in firms’ response to excess demand and supply, including in the adjustment of expectations. If the growth rate of autonomous demand (g_{A}) is held constant throughout the adjustment process, the oscillation is damped (declining in amplitude). This is illustrated in the first diagram below. In the second diagram, random variations in g_{A} cause more sporadic behavior. In both diagrams, the growth rate of autonomous demand is exogenously increased from 0.01 to 0.03 at time t = 5, though with random variation around these levels in the second diagram.

The next two diagrams illustrate the convergence of growth rates. Once again, the growth rate of autonomous demand is exogenously increased from 0.01 to 0.03 at time t = 5. In the first diagram, the process is entirely deterministic. In the second diagram, the growth rate of autonomous demand is subjected to random disturbance.

The way in which g interacts with g_{d} and g_{A} can be verified working from the expression for g_{d} in (9). Subtracting g from both sides of the expression gives:

This can be rewritten

In the first fraction on the right-hand side, the difference between total demand Y_{d} and induced demand (1 – α)Y is simply autonomous demand A. Modifying the expression accordingly gives:

If g is less than g_{A}, the right-hand side is positive. Since the left-hand side must then also be positive, g will clearly be less than g_{d}. Conversely, when g is greater than g_{A}, it must also be greater than g_{d}. This confirms that g deviates further from g_{A} in both directions than g_{d} deviates. The expression also shows that the growth rates of actual output and total demand coincide when equal to the growth rate of autonomous demand (g = g_{d} = g_{A}).

**Growth as Such**

Economic growth entails an ongoing expansion of output. Output, when defined in terms of National Accounting conventions, is a monetary measure. This is true whether the figure for Gross Domestic Product (or related measures) is reported in nominal terms (at market prices) or in so-called ‘real’ (meaning constant-price) terms. Growth of output, when output is defined in this way, can occur with or without environmental risk and social harm. In this sense, growth is neither inherently good nor bad.

Clearly, the desirability and sustainability of growth depends on the nature of that growth. In principle, if our measure of output is sufficiently broad, an economy geared toward learning (through education, research and development, science and technology, the arts and humanities), restoration and regeneration of the natural environment, provision of first-rate health and care services, construction of quality infrastructure, development of socially harmonious institutions and processes, as well as the fostering of creativity and play, can grow just as well as an economy geared toward military adventurism, plunder of natural resources and the proliferation of social and psychological maladies. When an economy produces more knowledge, for example, with the same expenditure on education, science and research, that should really be counted as an addition to ‘real’ output, appropriately conceived, though of course difficulties can arise in evaluation and measurement.

A key implication of Modern Monetary Theory is the capacity we have, if served by a suitably empowered and accountable currency-issuing government, to select sustainable activities and development paths.

Even so, the question might still be asked, why the need or desire to grow? Why not just satisfy ourselves with a stationary economy, or perhaps even shrink things a little?

Well, strictly speaking, nothing in a consideration of economic growth necessarily requires that we prioritize high or even positive rates of growth. Especially if we are thinking in terms of a future society with a job guarantee, full employment could be maintained with or without positive rates of growth. We can attempt to understand characteristics of an economy’s growth behavior, whether that growth is quantitatively positive or negative. But, having said that, it seems likely that most people will consider growth to be a positive, provided it is the right kind of growth. Ultimately, growth can and should be about the sustainable development of humans and other species in the fullest sense. Such growth might become less and less about the production of widgets and more and more about the attainment of knowledge, peace and understanding, social cohesion, an expanded scope for individual expression and creativity, and so on.

**Demand-Led Growth**

A motive for considering the possibility that growth is demand led is the implication that this would carry for macroeconomics in general. The central idea of demand-led growth lends credence to a key assumption underlying the Kaleckian or Keynesian view of output determination. In this view, a change in demand is assumed to induce variations in the level of production (real output) more or less independently of changes in (some index of) the price level. A critical assumption underlying this position is that there is spare capacity and spare reserves of labor-power and other resources making it possible for output to adjust to demand in the suggested way.

Demand-led growth theory makes sense of the idea that spare capacity is not merely a short-run assumption but one that is normally valid over any time frame. Arguably, it is even more applicable to the long run than to the short run. The reason – as demand-led growth theory highlights – is that output can respond to demand not only through a fuller utilization of existing capacity but, via investment, an expansion of capacity itself. Modern economies are almost always operating far below the physical maximum capacity. Average utilization rates of somewhere between 80 to 85 percent seem to be typical. In the US, for instance, the rate of utilization has fluctuated between about 67 and 90 percent since records began to be published in 1967.

How is this possible?

It is possible because long before full capacity is reached, firms will find themselves operating beyond the rates of utilization that they consider ‘normal’ – meaning beyond the utilization rates that they consider appropriate to the levels of demand that they expect to face on average – and this entices them to undertake further capacity-expanding investment.

Why do firms want to operate below full capacity?

They want to do this mostly because of uncertainty over what levels of demand they, in fact, will confront at any given time. Planned margins of spare capacity give them leeway, enabling them to respond to fluctuations in demand with variations in output rather than risk losing customers to rivals. In short, spare capacity gives flexibility.

It is not critical to this view of demand-led growth that prices be assumed constant. It is only necessary that real output respond to demand in the suggested way. In the short run, a strengthening of demand may well give rise to temporary price effects in those sectors that are operating nearer to full capacity than others. These price pressures may persist until new capacity has been installed, whether by new firms or through existing firms expanding their operations. As long as, in aggregate, the level of production responds to demand in the supposed way, there is no difficulty for demand-led growth theory, since its focus is on the behavior of real output. At the same time, the possibility of temporary price effects actually makes a working assumption of independence between the price level and output more plausible in the long run than in the short run, because the longer time frame allows for investment to have its full effects on capacity.

Surplus reserves of labor-power are also a normal feature of the system. Partly this is because expansion of capacity is achieved not only through an extension of scale at given technical efficiency but through technical progress that enables the same level of output to be produced with less labor. In addition, it is partly because capitalists can and do exploit previously untapped sources of labor-power located in non-capitalist spheres of the global economy (whether these individuals reside in so-called developed or developing economies). There is also a tendency for the size of the labor force to expand in periods of strong demand due to individual labor-supply decisions. Some individuals who become discouraged in their job search during periods of stagnation re-enter the labor force when prospects improve. As with plant and equipment, which will typically be utilized at different rates in different sectors on the basis of unbalanced growth, the supply of some types of labor-power can dry up before other types. Here, too, there can be price effects in affected sectors. As with spare capacity, the assumption of excess labor-power is actually more plausible rather than less over longer time frames. In the long run, there is time to train up new workers or innovate to reduce the need for certain types of labor-power.

The most pressing constraint is the environmental one. As already stressed, remaining inside the limits set by natural resources requires being cognizant of the type of growth and kinds of activities we prioritize. Here, currency-issuing governments have the capacity to encourage or undertake the right kinds of activity and discourage or proscribe ecologically harmful production.

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*Ugh* … I lurched toward the window. Not good.

I glared down at the crowd.

Leaning out the window it was all I could do not to spit the ghastly brew all over their unsuspecting heads.

*Damn and blast you all to hell*, I thought.

I pulled sharply away from the window, slamming it shut with the anguish of a young man in confusion and pain.

It was an intense, existential pain.

Physically I was okay, though. (Absolutely tip top.)

I glared down at the brown liquid, a dirty circular pond, withdrawing into myself, leaving my once cherished Study Mug on the emotional outer, unloved. Any sense of oneness it may once have shared with its holder and the crowd below had since crashed down upon the hardwood surface of the table, its contents not hot – nor even warm – but tepid. Unforgivably tepid.

Moments later my glare was redirected toward the moka pot, perched in all its feigned innocence on the kitchen stove. I was about to reach for something breakable to throw at it when the microwave clock came into view. It conveyed the surprising news that two full hours had elapsed since the discredited brew had come fresh off the production line. Sensations of universal oneness had apparently left me incapable of keeping track of time. Deflated, I rested against a wall to take stock, exhaling a slow, steady breath. Letting it go. Letting it all go.

Lukewarm coffee had spilled on lecture notes that awaited revision. Tactfully, courteously the notes had waited, but so far with little reward.

Fetching a cloth from the kitchen sink, I mopped up.

A stack of playing cards sat aloof on a bookshelf – mockingly, haughtily, difficult to ignore forever. Their presence oppressed me, too, like unstudied lecture notes. Procrastination only added to the burden. I approached them gingerly. Grabbed a pile in one hand and sized it up. Checked the last card in the pile (they were numbered on the back). Repeated the exercise.

Some time later I was out the door.

*****

“Dealer has six, sixteen, too many!”

Players at the table roared their approval, happy, on a winning run. The young dealer, Chintana, made the payouts. She was currently very popular with the table, a popularity that was, perhaps in part, of the “what have you done for me lately?” variety. Some say it’s the best kind of popularity.

“Shuffle time.”

I stood by a nearby table. There was a German couple, tourists, or so they said. Also a sleepy looking Indian guy pretending to be sleepy. And a twenty year old international student I recognized from class chatting in Cantonese to Enlai, the sharply dressed dealer, about how much the stockbroking game had changed over the course of his long career. I chatted to some guy playing box one, who said he was from the suburbs, while affecting disinterest in Chintana’s table, waiting for the shuffle.

I’d been in the casino for a few hours by now, pretending to drink beer, nibbling on snacks, hard at work, occasionally placing a bet, mostly just watching, mingling, joking with dealers and winning players, commiserating as losses began to mount. It was fun, really, apart from the need to wager.

At the happy table Chintana presented cards for the players’ cut. A messy-haired guy in T-shirt and jeans cut about a deck from the bottom of the eight decks. A bit more than a deck, really. About fifty-seven cards. Not more than fifty-nine, anyway. Maybe as few as fifty-five. Best guess fifty-seven.

I studied Chintana’s hands for a moment, and the stack of cards she held, rested against the baize. The bottom card faced her, away from the players, but angled somewhat to my point of view. I caught the identity of that card. It was the bottom card prior to the players’ cut, but the fifty-seventh card post cut, once Chintana had placed the cut cards to the front of the shoe, and the fifty-sixth card once she burned the first card, placing it face down in the discard tray. It could be a go.

I turned back to watch Enlai’s table, a spectator only, engaging in a bit of encouraging banter with the international student (we pretended not to know each other) and the box one suburban guy who had decided to “chip up”, pressing his luck, as if he really was from the suburbs. I cheered them on, but kept an eye on the happy table. With three boxes open, the players and dealer would typically consume about a fifth of a deck of cards per round. The sixth round was the one of possible interest. But only if it panned out as hoped. The situation could just as easily dissolve into nothingness if the card came a round earlier, but too late in the round to be useful. Or maybe another player would arrive and mess things up. Hopefully peer pressure would prevent it.

With hope still alive at the end of the fifth round. I wandered over to the happy table, lingering behind box three, a beer in one hand, a clutch of gaming chips in the other. Players eyed me anxiously, fearing that I’d open a new box and mess with The Flow. The Flow was – still is – a concept embraced by the superstitious and scientific alike. The former as true believers. The latter as a pretense, a way of discouraging new players from joining a game at inopportune moments.

They needn’t have worried. I placed a “back bet”. The players nodded their approval and relaxed, glad, or pretending to be glad, that The Flow had been preserved.

“Eight hundred dollars on box three,” said Chintana. Three hundred and twenty five of those were mine. Or, at least, used to be mine. They were up for grabs now. The remainder of the eight hundred used to belong to a happy-go-lucky corporate guy. His girlfriend turned and grinned. Said they were in marketing and due back at the office. Maybe it was true. It was not impossible.

The pit supervisor nodded approval to Chintana, then laughed at the marketers’ truancy. “More money to be made in here,” he joked.

“Yeah, right!” said the marketer, in good humor.

Chintana ran her hand across the baize – “no more bets” – and wished everybody luck.

Tipsy and easygoing on the outside, sober and queasy on the inside, I pushed clenched fists into jacket pockets, hoping for the best. The wager amounted to a sizable chunk of my still pitiful bankroll, a few thousand dollars plus change built up from a measly initial thousand. I knew that until that got up to perhaps a hundred thousand or so the game was always going to be a slog. A hellish, never-ending kick in the guts, placing back bets to appease other punters and avoiding costly waiting bets by resisting the comfort of a chair. Slog was right. I still sweated the turn of each card. The psychology of it all should have been conquered by this point, but had not been.

The wager as a proportion of bankroll was more or less justified. With a margin for error of five cards, my theoretical advantage on the round was about ten percent, and the standard deviation on the wager only a bit more than the bet itself. Optimal betting theory called for roughly the bet placed.

On the dealing of box three’s first card by the amazing Chintana, my spirits soared. An ace of diamonds, as hoped for! No, not as hoped for. As intended, and expected. I was a pro, damn it, or at least on the way to becoming one.

*Oh, Chintana, thank you! You are wonderful and your popularity thoroughly deserved.*

With the appearance of the ace as first card the advantage had now spiked to about fifty percent. I tried to conjure a sense of optimism to mask the gnawing in the deep dark pit of my stomach that said this was all going to end in calamity.

I longed for a picture card to go with the ace, for blackjack. Nothing but that three-to-two payoff would do now. A nice add to my small but growing – yes growing, damn it – bankroll.

Chintana showed an eight as up card. Box three, unfortunately, soon showed a seven for its second card, giving a mediocre soft eighteen. The marketers correctly stood.

I cursed the universe inwardly. *Typical*. And it *was* typical. It typically happened twenty-four times out of every four hundred and fourteen attempts that a seven would be drawn to an ace when the lovely Chintana showed an eight as up card from an eight deck shoe. Roughly one in thirteen times. The temporary fifty percent edge had eighty percent evaporated, though the situation was not terrible. There was still hope. I was reduced, once again, to hope.

Hope, that is, until Chintana revealed a seemingly heartless three of hearts for eleven and a ten of clubs – for clubbing players over the head – for three-card twenty-one, swiftly banishing eight hundred dollars, three hundred and twenty-five of them formerly mine, down the chute.

Chintana shrugged sympathetically. I gave my best impression of a good-natured roll of the eyes, commiserated with the marketers and strolled off in the direction of a bar. (The nearest high cliff was some distance away and peak hour traffic would frustrate any attempt to reach it in a hurry.)

Soon after, I headed for an exit.

*****

Back at the apartment, I brooded, waiting for the moka pot to work its magic. The lecture notes, for their part, still waited in dignified, non-judgmental silence on the table.

With Study Mug in hand, fresh and forgiven, I resumed my earlier vantage point by the window, gazing at the crowds below. Such beautiful, crazy crowds. How they wandered to and fro. Happy. Unhappy. Rich. Poor. Amazing. Wonderful. Crazy beautiful crowds.

I muttered to no one in particular, “Damn and blast you all to hell.”

The door opened.

It was Chintana. She was beautiful, eyes sparkling, changed out of her work clothes. She gave me a look. Quizzical.

I shrugged.

“It didn’t work?” she asked.

“It worked perfectly,” I assured her. “You did great. It was not obvious, but clearly visible.”

She crossed the room and nestled close. “The ace,” she said, “but it was no good.”

“Just unlucky.”

We kissed and went to the bedroom. Later, we got a ton of study done and, that semester, caned finals.

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A first step is to consider the disequilibrium behavior of an economy that, for simplicity, is taken to be stationary (non-growing) when in equilibrium. This approach is adopted in the present post. The exercise is really preparation for considering a continually growing economy – a task that is left for a possible future post.

The material in this post is somewhat technical but hopefully not difficult. Even so, the post is long (about 4000 words) and, for readers not already familiar with similar material, possibly a stretch to read all in one go. I considered separating the post into numerous shorter ones but felt that the loss of continuity would require too much repetition in setting up the discussion each time.

The post is divided into sections that provide natural stopping points for readers who wish to take breaks. The section titles are:

1. Macroeconomic Equilibrium

2. Disequilibrium Behavior

3. Adjustment Process

4. Adjustment in Simple Algebra

5. Convergence as a Power Series: λ = 1

6. Convergence as a Power Series: λ < 1

7. Adjustment of Growth Rates

Some background information relating to 1 can be found in the ‘Short & Simple’ series (mainly here). A very helpful reference for some of the material covered in later sections is Ronald Shone’s book, *An Introduction to Economic Dynamics*, Cambridge University Press. The edition I have was published in 2003. In particular, the third chapter provides more sophisticated – but still highly accessible – treatments of 3 and 4. The purpose of 7 is to lay some groundwork for a possible future post on a continually growing economy.

**1. Macroeconomic Equilibrium**

The notion of macroeconomic equilibrium relates to plans (or desires). Although, under National Accounting conventions, actual output equals actual expenditure by definition, equilibrium will only occur if actual output equals planned expenditure.

The sum of planned expenditures constitutes total demand. Actual output represents total supply. Accordingly, the equality of demand and supply requires that actual output and actual expenditure equal planned expenditure. This only occurs when unplanned expenditure is zero.

Unplanned expenditure is possible because of the way actual expenditure is defined. Included in actual expenditure is the **change in inventories**, meaning the change in unsold stocks of goods held by firms. The change in inventories is treated, in the National Accounts, as expenditure by firms to themselves.

From a theoretical standpoint, intended changes in inventories are part of planned investment. Unintended changes in inventories amount to **unplanned investment**.

Defined in this way, unplanned investment is identically equal to **excess supply**. It is the amount of output that firms intended to sell but were unable to sell within the period. In other words, unplanned investment is the excess of supply over demand.

In symbols:

Here, I_{u} is unplanned investment (the unanticipated change in inventories). Y is actual output (supply). Y_{d} is planned expenditure (demand).

Equilibrium occurs if unplanned investment is zero (I_{u} = 0) so that:

where Y* is the equilibrium level of output.

Total planned expenditure (Y_{d}) can be grouped into two broad categories. ‘Autonomous demand’ (A) is considered exogenous and independent of income. It includes the planned exogenous spending of households, firms, government and foreigners and is defined net of exogenous imports. ‘Induced demand’ is endogenous and depends on income. It is defined net of endogenous imports. In general, endogenous spending can be undertaken by households, firms or government. But to keep the present discussion as simple as possible, all expenditure other than net induced consumption (defined as induced consumption minus endogenous imports) will be considered exogenous and part of A. If the marginal propensity to leak to taxes, saving and imports is denoted by α – assumed to be a constant fraction between zero and one – there will be induced demand of (1 – α)Y.

Total demand, then, is the sum of the induced and autonomous components of planned expenditure:

Actual output, which is identically equal to actual expenditure, includes not only planned expenditure but also unplanned investment:

When unplanned investment is zero, actual output (Y) equals planned expenditure (Y_{d}) and is at the equilibrium level (Y*). Substituting the expression for Y_{d} in (3) into equilibrium condition (2) gives:

Solving for Y:

Since both autonomous demand (A) and the marginal propensity to leak (α) are taken as given, equilibrium output (Y*) is stable unless there is an exogenous change in A or α.

Although, for given A and α, Y* is stable, actual output (Y) and planned expenditure (Y_{d}) only equal Y* in equilibrium. Our present concern is with what happens out of equilibrium.

**2. Disequilibrium Behavior**

In the income-expenditure model, it is assumed that firms respond to unexpected changes in inventories (excess demand or supply) by varying levels of production. The model is valid provided there is sufficient spare capacity, underutilized labor-power and resources for firms to respond in this way.

In a situation of excess supply, unwanted inventories mount (I_{u} > 0) and firms are assumed to cut back production. This initiates a multiplier process that causes actual output (and to a lesser extent demand, through the effects on induced demand) to decline toward the equilibrium level.

Conversely, when there is excess demand, firms are assumed to step up production in an attempt to replenish dwindling inventories. This sets off a multiplier process in which actual output (and to a lesser extent demand) rise toward the equilibrium level.

So, in a situation of excess demand:

An expansion of production will induce consumption and cause both actual output and planned expenditure to rise toward the equilibrium level of output.

In a situation of excess supply:

and the reverse process occurs.

**3. Adjustment Process**

Suppose the economy is initially out of equilibrium. We can consider the process of adjustment.

Throughout the exercise, both autonomous demand (A) and the marginal propensity to leak (α) will be held constant. This means that the equilibrium level of output (Y*) will also remain constant throughout the adjustment process as actual output and planned expenditure converge on Y* = A_{ }/_{ }α.

A well known diagram is often used to illustrate the situation. It is shown below. In the diagram, expenditure (E) is measured along the vertical axis. Output and income (Y) are measured along the horizontal axis.

The level of planned expenditure associated with each level of output is indicated by the Y_{d} schedule. At points off this schedule, actual expenditure differs from planned expenditure by an amount corresponding to unplanned investment.

The 45-degree line shows all the points at which actual output equals actual expenditure; that is, all the points at which Y = E.

As has already been established, equilibrium requires that planned expenditure (Y_{d}) equals actual output (Y). This occurs at the point of intersection between the Y_{d} schedule and the 45-degree line. Since actual output equals actual expenditure by definition, the point of intersection also represents a situation in which planned expenditure equals actual expenditure.

All points on the Y_{d} schedule to the left of the 45-degree line show situations of excess demand (Y_{d} > Y), whereas points on the Y_{d} schedule to the right of the 45-degree line indicate excess supply (Y > Y_{d}).

Suppose initially that actual output is at the level Y_{0}. This is a situation of excess demand. Planned expenditure E_{0} exceeds actual output Y_{0}.

The difference between demand and supply, which defines unplanned investment, is represented by the horizontal distance from output Y_{0} to output Y_{1}. It is also measured by the vertical distance between the Y_{d} schedule and the 45-degree line at output Y_{0}.

In the present situation, unplanned investment is negative because demand exceeds supply (Y_{d} > Y).

Although only output of Y_{0} is actually produced in period zero (denoted t = 0), firms are able to satisfy demand of E_{0} by running down inventories.

The unexpected depletion of inventories acts as a signal for firms to expand production in the next period (t = 1) to the level of output Y_{1}.

The diagram is drawn on a simplifying assumption that firms make the full adjustment from Y_{0} to Y_{1} in a single period. In general, they might only make part of this adjustment within the period. (They might even make more than the full adjustment, but this possibility will be ignored in the present discussion.) The basic logic of the model is not affected by the strength of reaction to excess demand or supply, but the convergence process will take a bit longer in the case of partial adjustment.

Even the full adjustment from Y_{0} to Y_{1} does not immediately result in equilibrium, because the additional income (ΔY = Y_{1} – Y_{0}) induces extra private consumption. This is represented as a movement up along the Y_{d} schedule.

So, at output Y_{1}, planned expenditure is still greater than actual output. This can be seen by the fact that, at output Y_{1}, the Y_{d} schedule is still above the 45-degree line.

Importantly, though, there is less excess demand at time t = 1 than at time t = 0. The reason for this is that the slope of the Y_{d} schedule is flatter than the 45-degree line. Demand rises more gradually than supply as both increase toward the equilibrium level of output.

Specifically, the Y_{d} schedule’s slope is less than one. (Its slope is 1 – α, which is less than 1 because α > 0.) In contrast, the slope of the 45-degree line is equal to 1. The difference in slopes makes it possible for supply to catch up to demand as firms attempt to adjust actual output to demand.

Within the model, the relative flatness of the Y_{d} schedule is actually one of two conditions that, when satisfied, ensure convergence of actual to equilibrium output, or what is called ‘dynamic stability’.

Both conditions for dynamic stability relate to the Y_{d} schedule and are depicted in the diagram. They are that:

(i) there is a positive level of autonomous demand (A > 0), which means that the Y_{d} schedule must have a positive vertical intercept; and

(ii) the marginal propensity to leak is positive (α > 0), which means that the slope of the Y_{d} schedule (1 – α) must be less than one.

The first condition – a positive vertical intercept – ensures that if the Y_{d} schedule crosses the 45-degree line at an equilibrium point, it will do so at a positive level of output. This is important because only positive levels of output make economic sense.

The second condition – a positive marginal propensity to leak – ensures that the two schedules will cross, because it means the Y_{d} schedule is flatter than the 45-degree line.

Provided the conditions for dynamic stability are satisfied, the convergence process will eventually cause output to reach (asymptotically) the level Y*, where supply equals demand (Y = Y_{d} = Y*).

The process is indicated by the arrows. Each time firms respond to a deficiency of inventories by stepping up production, this induces extra demand, but not as much demand as the additional income that has been created, because of a positive marginal propensity to leak from the circular flow of income.

A similar process would occur from a position of excess supply at which the point on the Y_{d} schedule happened to be to the right of the 45-degree line. There would be a multiplied decline in actual output as each contraction in the level of production shrunk induced demand, but not by as much as the reduction in income.

**4. Adjustment in Simple Algebra**

The adjustment process just described can be expressed in simple algebra.

Recall, from (1), that unplanned investment is identically equal to the amount of excess supply in the economy:

It has been assumed that, out of equilibrium, firms attempt to eliminate part or all of the unplanned investment in the next period. This suggests that, in the next period, firms will adjust production by the amount

where λ is a positive fraction. The minus sign in (7) indicates that firms adjust actual output in the opposite direction to the change in unplanned investment. For example, if demand is 990 and actual output only 980, implying unplanned investment of -10, it is assumed that output in the next period will be increased by some fraction (λ) of 10.

The expression for unplanned investment in (1) can be substituted into (7):

This says that the change in actual output (ΔY) can be expressed as a fraction (λ) of excess demand (Y_{d} – Y).

From (3), we know that Y_{d} = (1 – α)Y + A. Substituting for Y_{d} in (8) and rearranging gives:

This expression makes it possible to work out actual output for the next period as the sum of current-period output and next period’s change in output (Y + ΔY). To do so, we can add Y to both sides of (9) and rearrange:

If we are considering periods t and t+1, this can instead be written:

This is a recursive equation. We can repeatedly substitute the value of actual output in one period (Y_{t}) to obtain actual output of the following period (Y_{t+1}).

For example, suppose in period t = 0 that autonomous demand is 500, with the marginal propensity to leak a constant 0.5. The equilibrium level of output will be Y* = A_{ }/_{ }α = 1000. But perhaps firms have only produced actual output of 980 in the period.

According to the behavioral assumption, firms will increase production each period until equilibrium is restored. In the simplest case of full adjustments (λ = 1), actual output in successive periods will be:

The squiggly equals sign in the final row says that Y_{n} is ‘asymptotically equal’ to the equilibrium value. That is, actual output approaches the equilibrium level asymptotically.

The logic of the adjustment process just outlined can also be considered in a slightly different way by assuming, as an alternative exercise, that the economy is initially in equilibrium and then working through the effect of a one-off exogenous change in autonomous demand.

Suppose A = 490, α = 0.5, and the economy is initially in equilibrium with both actual output and total demand equal to 980. If autonomous demand then increases by 10 to 500 (giving ΔA = 10), this will result in excess demand and call for an expansion of production by firms. If we take the change in autonomous demand to have occurred in “period 0”, then the disequilibrium adjustment process will be an exact replica of the one shown above.

If λ happens to be less than 1, the adjustment will be more gradual. The following table, which continues with the same example, shows the sequence of actual output and total demand for various values of λ. Total demand is calculated each step by substituting the most recent value for actual output into (3), which says that Y_{d} = (1 – α)Y + A.

In the table, autonomous demand is held constant at A = 500, the marginal propensity to leak is α = 0.5 and a situation of excess demand is assumed to exist in period 0.

Here are a couple of cases represented in graphical form:

**5. Convergence as a Power Series: λ = 1**

We have observed that actual output and demand both tend to converge to a level that is a multiple of autonomous demand. In equilibrium, Y = Y_{d} = Y* = A_{ }/_{ }α.

As is well known, the expenditure multiplier (1_{ }/_{ }α) can be interpreted as the sum of a power series:

When α takes a value between zero and one, which is always the case under the assumptions of the income-expenditure model, the above series asymptotically approaches 1_{ }/_{ }α as n becomes large.

The power series can be interpreted as tracing out each step of the adjustment process in which actual output adjusts to its equilibrium level.

Starting from a position of equilibrium, reconsider the consequences of a one-off exogenous change in autonomous demand, denoted ΔA.

The change in autonomous demand will cause a multiplied change in equilibrium output of:

Here, the sum of the power series given in (11) has been substituted for the multiplier 1_{ }/_{ }α.

Continuing with the example, if actual and equilibrium output were initially 980, the marginal propensity to leak 0.5 and the change in autonomous demand 10, then equilibrium output would eventually increase by 20. Applying (12), with α = 0.5 and ΔA = 10:

If it is assumed that firms make a full adjustment each period (meaning λ = 1), then each step in the convergence process toward equilibrium will correspond to one of the terms in the infinite series shown above.

The initial change in autonomous demand (ΔA = 10) causes a situation of excess demand. Actual output is 980, whereas total planned expenditure has suddenly increased to 990. Firms meet the higher demand by running down inventories by 10. This represents unplanned investment of -10.

By assumption, firms respond by trying to eliminate the unplanned investment. In step 0 of the adjustment (t = 0), firms expand production by 10. This is equal to the first term in the power series: 10 x 1 = 10(1 – 0.5)^{0}.

The increase in actual output of 10 induces extra expenditure of 5, equal to (1 – α)ΔY. As a result, there is still excess demand, but now only of 5 rather than 10.

In step 1 (t = 1), firms expand production again, this time by 5. This corresponds to the second term in the power series: 10(0.5) = 10(1 – 0.5)^{1}.

More generally, the adjustment that firms make in step t of the convergence process will be given by the *(t+1)*th term in the series: ΔA(1 – α)^{t} = 10(1 – 0.5)^{t}. For instance, in step 3, firms adjust output by 10(1 – 0.5)^{3} = 1.25.

The sum of the adjustments made over n steps asymptotically approaches 20, with both actual output and planned expenditure converging on 1000.

Visually, this adjustment has already been illustrated in the graph provided earlier for the case of λ = 1.

**6. Convergence as a Power Series: λ < 1**

We have seen that if λ takes a value less than 1, the adjustment process will be more gradual. The power series needs to be modified to:

This is a more general version of (12). In the special case of λ = 1, the above formula reduces to the earlier one. The effect of λ < 1 is to spread the adjustment process over more steps. The adjustment in the initial step is smaller, but the size of the adjustment each period shrinks less rapidly than when λ = 1.

The value of λ affects the speed of adjustment but, within the model, not the level to which actual output and demand converge. Since Y* = A_{ }/_{ }α, the point of convergence depends only on autonomous demand (A) and the marginal propensity to leak (α).

The rest of this section can be skipped without missing much economic content. But for readers who may be curious, it is pretty easy to arrive at the formula given in (12′). Each step of the adjustment process can be described algebraically, beginning with t = 0, until a clear pattern emerges. Since this kind of scenario arises quite frequently in economics, perhaps it is worth spelling out the working.

The sequence of events begins with a one-off change in autonomous demand, or ΔA. This creates excess demand of ED_{0} = ΔA. In response, firms begin to adjust actual output to demand by trying to eliminate unplanned investment.

Let Δ_{t}Y be the change in actual output at step t of the adjustment process. The change in actual output (and income) will induce additional private consumption (net of endogenous imports) of Δ_{t}C. Specifically, a fraction (1 – α) of the extra income will go to induced demand. This constitutes excess demand, but less than was created in earlier steps.

The excess demand of period t (ED_{t}) will be reduced in period t+1 to the extent that extra output is produced in the new period, but will be increased to the extent that extra demand is induced. That is, ED_{t+1} = ED_{t} – Δ_{t}Y + Δ_{t}C.

Putting all this together, a pattern emerges within a few steps:

The adjustment of actual output in step t will be equal to λΔA(1 – αλ)^{t}. When t = 0, this reduces to λΔA. When t = 1, the expression reduces to λΔA(1 – αλ).

The total change in actual output over the entire adjustment process will be the sum of all these step adjustments, which is what (12′) says.

**7. Adjustment of Growth Rates**

It is also possible to represent the adjustment process in terms of growth rates. This will be useful if considering an economy with a continual tendency to grow over time (a task left for another time).

At the moment, we are considering a simpler economy that only grows (or shrinks) in the process of adjustment to equilibrium. So, in the present context, the growth rate will settle at zero once equilibrium is reached.

Recall that unplanned investment, as defined earlier in (1), is identically equal to the amount of excess supply in the economy:

We have expressed the change in actual output as a function of excess demand. Combining (7) and (8) from earlier:

If, conceptually, we break down the steps of the adjustment process into tiny enough increments of time, we can think of the adjustment process as a continuous one and recast (8) as:

Here, Y dot is the derivative of output with respect to time (dY_{ }/_{ }dt).

The growth rate of actual output (g) is found by dividing Y dot by Y:

This growth rate diverges from zero whenever the economy is out of equilibrium. In situations of excess demand, Y_{d }/_{ }Y is greater than one and the actual growth rate rises above zero. The growth rate will remain positive until equilibrium (Y_{d }/_{ }Y = 1) is attained. Conversely, the actual growth rate is negative in situations of excess supply, and remains that way until convergence is complete.

Under present assumptions, the equilibrium growth rate (g*) is determined by the exogenously given growth rate of autonomous demand. This can be seen by differentiating the expression for equilibrium output (Y* = A_{ }/_{ }α) with respect to time and dividing by Y*:

Since autonomous demand is being held constant throughout the adjustment process, g_{A} and g* are both zero over the relevant time horizon.

In more elaborate versions of the model, the behavior of the equilibrium growth rate can be more complicated. The simplicity, in the present context, is due to: (i) all demand other than net induced consumption being treated as exogenous; and (ii) the marginal propensity to leak being assumed strictly constant. Introducing more endogenous elements to demand will complicate the behavior of g*. (Examples involving endogenous investment and an endogenous component to government spending are discussed in a previous post.)

The convergence of the actual growth rate (g) to the equilibrium growth rate g* (and g_{A}) is illustrated below for a couple of values of λ.

The growth behavior of total demand is also illustrated in these diagrams. Like actual output, demand grows at a rate other than zero outside equilibrium. Its growth rate (g_{d}) does not diverge as far from the equilibrium growth rate as g does, because the required adjustment of demand is less than the required adjustment of output. By the logic of the model, whenever there is disequilibrium, production needs to “catch up” to demand. To do so, it needs to grow (or shrink) faster than demand during the process of adjustment.

The growth behavior of demand is implicit in (3), which defines planned expenditure as the sum of induced and autonomous demand:

Differentiating this expression with respect to time gives:

Dividing through by Y_{d} and multiplying Y dot by Y_{ }/_{ }Y and A dot by A_{ }/_{ }A gives:

The expression for g in (13) can now be substituted into the growth equation for demand. Upon rearrangement:

Since the level of autonomous demand is held constant throughout the adjustment process, the growth rate of autonomous demand (g_{A}) is zero, making the second term on the right-hand side of (14) also zero:

The sign of g_{d} simply depends on whether the fraction Y_{ }/_{ }Y_{d} is greater than, equal to, or less than one. When there is excess demand, Y_{ }/_{ }Y_{d} is less than one and g_{d} is positive. This is consistent with the behavior depicted in the diagrams above. In the reverse case of excess supply, Y_{ }/_{ }Y_{d} is greater than one and g_{d} is negative. In equilibrium, the term on the right-hand side of (14′) vanishes and g_{d} equals zero, as expected.

It has been stated that, out of equilibrium, the divergence of g from the equilibrium growth rate (g* = g_{A} = 0) is greater than the divergence of g_{d}. This can be verified working from the expressions for g and g_{d} provided by (13) and (14′), respectively:

The growth rate of actual output (g) will be greater in absolute value than the growth rate of total demand (g_{d}) provided:

From (4), Y = (1 – α)Y + A + I_{u}, which implies that I_{u} = αY – A. This can be substituted for I_{u} in the above inequality to obtain the following condition:

This is one of the two conditions for dynamic stability already discussed. Since, within the model, there is always a positive amount of autonomous demand, the condition A > 0 automatically holds. This verifies that, out of equilibrium, the absolute value of g will exceed the absolute value of g_{d}. In other words, g will take more extreme values than g_{d} during convergence to equilibrium.

It is possible to say more about the growth behavior of actual output, total demand and equilibrium output, but this is perhaps better left for a discussion of a continually growing economy.

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**The Basic Framework**

The level of ‘real’ economic activity – as measured in the National Accounts as Gross Domestic Product at constant prices – will be considered demand determined along the lines of Keynes and Kalecki. It is assumed that normally there are reserves of labor-power and other resources as well as spare productive capacity in the form of underutilized plant and equipment. So long as this assumption holds, it is possible for output to be adjusted to demand through variations in the level of activity.

Inside resource and capacity limits, the level of output is regarded as being determined more or less independently of prices. In exceptional circumstances, when demand outstrips the capacity of the economy to respond in real terms, there would be demand-side price pressures and/or non-price rationing. But, for the most part, the economy is considered to be demand constrained, inside ultimate supply-side limits.

This is regarded as true over any time frame. In a period so short that capacity is treated as fixed, output is adjusted to demand through variations in the rate of capacity utilization. In response to stronger demand, firms utilize their given facilities more fully. Over longer time frames, in which capacity should be considered variable, output can be adjusted to demand not only through variations in the utilization rate but by altering capacity itself, investment being the means for doing this.

Both the level of activity and the growth rate of the economy are in this way considered to be demand led.

This perspective on output and growth is intentionally left open in terms of other key economic questions, such as value, price and distribution. This leaves space for competing theories on these matters. If distribution, for instance, is thought to impact systematically on the level of output, it can be introduced through explicit assumptions about the determinants of variables or parameters that, in the present discussion, are simply taken as exogenously given. A well known example is Kalecki’s adoption of the classical assumption that workers, in aggregate, consume all their wages, which makes the size of the multiplier dependent on the distribution between wage and profit income.

The perspective presented here is also open when it comes to the roles of money and finance, although the natural fit with endogenous money and state money theories will probably be evident.

Cumulative developments in the sectoral balances that are thought to impact on private consumption could be worked into an explanation of that part of consumption that here is simply treated as autonomous. Within the framework to be discussed, this would also have an impact on private investment.

**Keynesian Output Determination**

We can start with the standard Keynesian model of output and income determination (although Kalecki’s framework would serve just as well). Total output (Y) is equal to the sum of expenditures. In the simplest versions of the model, all expenditures other than the greater part of private consumption are assumed to be exogenous. Induced consumption is taken to equal c(1 – τ)Y, where c and τ are the marginal propensities to consume and tax, respectively. For simplicity, both are assumed constant. Supposing a closed economy, also for simplicity, exogenous demand includes autonomous private consumption, private investment and government spending. These categories of demand can be grouped together as autonomous expenditure (A). Output will equal the sum of induced consumption and autonomous expenditure:

All variables are taken to be functions of time. Unless there is risk of ambiguity, time subscripts will be omitted. If we let α = 1 – c(1 – τ) represent the marginal propensity to leak to taxes and saving, which implies that c(1 – τ) = 1 – α, the above expression can instead be written as:

This notational choice will simplify some of the algebra later.

If (1) is viewed in terms of actual output, it is merely an identity. The identity holds because of the National Accounting convention of treating changes in inventories, whether anticipated or not, as part of investment. Over time, however, unanticipated variations in inventories can be expected to induce an output response. These inventory changes, to the extent that they are unexpected, amount to unplanned investment. A key assumption of the Keynesian income-expenditure model is that firms expand or cut back production in an attempt to eliminate unplanned investment or disinvestment. Granted this output response to the emergence of excess supply or demand, and given the level of exogenous variables and parameters, there will be a tendency for actual output Y to converge on output level Y* (obtained by solving (1) for Y):

This notional level of output is usually referred to as ‘equilibrium output’. In what follows, it will be the focus of attention. It is possible to delve into the relationship between Y and Y*, but this exercise is left for another day.

The term ‘equilibrium’ is often taken to have quite negative connotations in heterodox circles. With that in mind it is perhaps worth stressing that the requirements of equilibrium, as defined by (2), are far less onerous than those pertaining to some other notions of equilibrium. In particular, there is no implication that all markets must clear. Equilibrium, in the present sense, simply refers to a situation in which planned leakages (in a closed economy, taxes plus saving) happen to equal planned injections (government spending plus private investment) so that the sum of all monetary demand (planned expenditures) equals the sum of output supplied (also measured in monetary terms). There may be mismatches of supply and demand in many markets, but if the excess demands and supplies happen to sum to zero, actual output will be at Y*.

If capacity is taken as fixed – as is often the case in short-run models – Y* will be stable until there is a change in one of the exogenous variables or parameters (summarized as A and α). As will be apparent by the end of the post, over longer time frames, in which capacity is free to vary, neither Y* nor the growth rate of Y* will necessarily be stable other than in the simplest versions of the model that take all expenditure other than induced consumption as exogenous. Over longer time frames, a stable growth rate requires more than an equality of planned leakages and injections. In particular, capacity would need to be fully adjusted to demand.

Importantly, the model does exhibit ‘dynamic stability’ provided that: (i) there is a positive level of autonomous demand (A > 0); and (ii) the marginal propensity to leak is greater than zero (α > 0). The property of dynamic stability is a consequence of the key behavioral assumption that firms seek to eliminate unanticipated changes in inventories. Dynamic stability ensures that there is a continual tendency for Y to converge toward Y*.

The model’s property of dynamic stability perhaps provides some justification for focusing, at least in certain analytical contexts, on the behavior of Y*, rather than explicitly dealing with the distinction between actual and equilibrium output. In what follows, the asterisk in Y* will be dropped unless ambiguity seems likely, with output Y* simply referred to as Y. But it should be kept in mind that discussion pertains to the behavior of notional output Y* rather than actual output Y.

**A Growing Economy**

The growth behavior of this notional level of output, which we are simply denoting Y, can be considered by differentiating (1) with respect to time:

Here, Y dot is the derivative of output with respect to time (dY/dt). Similarly, A dot = dA/dt. Collecting terms involving Y dot to the left-hand side gives:

Dividing both sides by αY and multiplying the right-hand side by A/A (= 1) modifies this to:

A time derivative of a variable divided by the variable itself gives the variable’s instantaneous growth rate. In our case, Y dot/Y is the instantaneous growth rate of output Y, which can be denoted g. Likewise, A dot/A is the instantaneous growth rate of autonomous spending A, which can be denoted g_{A}:

Recalling from (2) that Y = Y* = A / α, the fraction on the right-hand side reduces to one. So, in this simple version of the model, equilibrium output grows at the same rate as autonomous demand:

The growth rate of equilibrium output remains constant so long as the growth rate of autonomous demand remains constant (which, of course, may not be for long at all).

**A First Complication: Endogenous Private Investment**

In terms of growth behavior, the model as presented so far is as simple as it gets. The reason for this is that: (i) all expenditure other than induced private consumption has been assumed exogenous and part of autonomous expenditure; and (ii) the only endogenous component of spending (induced private consumption) has been treated as a constant fraction of income, because of the simplifying assumption that the marginal propensities to consume and tax are both constant. The behavior of equilibrium output becomes a little more complicated once we allow for the idea that private investment is induced by growing output and income. The basic rationale here is that higher rates of capacity utilization, which occur when output is expanded relative to capacity to meet rising demand, will encourage additional investment as a means for firms to adjust capacity to demand.

Following the supermultiplier theory, this situation can be represented by a simple modification of expression (1). Output will now equal the sum of induced consumption, endogenous private investment and autonomous demand:

The second term on the right-hand side, hY, is endogenous planned investment (I = hY). Unlike the marginal propensity to leak, the ‘marginal propensity to invest’ h, while also a fraction between zero and one, is not considered to be a constant. Instead, h is assumed to rise when the rate of capacity utilization is above ‘normal’ and to fall when the utilization rate is below normal. In the present context, the marginal propensity to invest can also be thought of as the investment share in income, because I = hY implies h = I/Y. It is supposed that investment rises and falls as a proportion of income in response to variations in the rate of utilization.

The final term in (1′) is also somewhat altered compared with (1). Z, like A, refers to autonomous expenditure. But this expenditure now also has the characteristic that it does not directly add to private-sector productive capacity. In the literature, this category of expenditure is sometimes referred to as ‘non-capacity-enhancing autonomous demand’. In a closed economy, Z includes autonomous private consumption and government spending. In an open economy, it would also include exports.

Solving (1′) for Y enables us to express equilibrium output as a multiple of autonomous demand Z:

Dynamic stability now requires Z > 0 and α > h. The fraction 1/(α – h) is known as the ‘supermultiplier’. It is larger than the regular multiplier because of the inclusion, in its denominator, of the marginal propensity to invest. (At the same time, compared with (2), a term for investment has been removed from the numerator, so the supermultiplier operates on a level of autonomous expenditure that is smaller than in the simpler version of the model.) For example, if α (the marginal propensity to leak to taxes and saving) is assumed to be 1/2 and, at a given point in time, the marginal propensity to invest happens to be 1/6, the regular expenditure multiplier will be 2 whereas the supermultiplier will be 3. An exogenous increase in Z will cause an expansion of output and income that induces further private consumption and investment.

Differentiating (1′) with respect to time, we have:

Here, the product rule [d(uv)/dt = u’v + uv’, where u’ and v’ are the time derivatives of u and v, respectively] has been applied to the term hY because both h and Y are functions of time. Collecting all the terms involving Y dot to the left-hand side and factorizing gives:

Following the same basic procedure as earlier, we can divide both sides by (α – h)Y and multiply Z dot by Z/Z to obtain:

Referring back to (2′) we can see that Z/(α – h) = Y, which makes the last fraction in the above expression one. After reordering terms and noting that g = Y dot/Y and g_{Z} = Z dot/Z, we have:

Growth of equilibrium output now depends not only on the instantaneous growth rate of autonomous demand but also on the instantaneous growth rate of the supermultiplier, which is the second term on the right-hand side of (3′). This last point can be seen by differentiating the supermultiplier 1/(α – h) with respect to time [applying the quotient rule, which says d(u/v)/dt = (u’v – uv’)/v^{2}] and dividing this derivative by the supermultiplier itself. The result is the second term on the right-hand side of (3′).

It is possible to say more about this growth behavior by fleshing out the notion of induced investment, its relationship to the behavior of the utilization rate, and the ongoing attempts by firms to adapt capacity to demand. For now, suffice to note that unless capacity is fully adjusted to demand, h dot (the instantaneous change in the investment share) will be nonzero (other than possibly momentarily when switching sign). As a consequence, equilibrium output will grow more or less rapidly than autonomous demand, according to whether the investment share is rising or falling (h dot is positive or negative). In contrast to the simplest version of the model, the growth rate of equilibrium output is variable rather than constant – even though it pertains to a situation in which supply continually equals demand – unless the investment share has stabilized.

For readers who may be interested, a few earlier posts (for example, here and here) discuss aspects of growth under the assumption of endogenous private investment.

**Another Complication: A Job Guarantee**

It is possible to endogenize other components of spending. Doing so will further modify the growth behavior of equilibrium output. An example that has been touched on in a couple of earlier posts concerns a job guarantee. Under this program, a part of government spending would respond endogenously to variations in output and employment in the broader economy (by which is meant the non-job-guarantee sector). Strictly speaking, this spending would adjust in response to fluctuations in the composition of employment between the two sectors. But it is possible to consider the spending as endogenous with respect to output and income as well, because of the close connection between output and employment.

Modifying expression (1) once more, output for the economy as a whole is now the sum of induced private consumption, endogenous private investment, endogenous job-guarantee spending and autonomous demand:

Here, jY represents government spending on a job guarantee (G_{j}), where j is the proportion of income spent on the program. Like h, j is variable rather than constant. But unlike h, j varies countercyclically rather than procyclically. It automatically rises in economic downturns and falls during upturns. A couple of earlier posts (here and here) discuss the behavior of job-guarantee spending. In those posts, job-guarantee spending was denoted G_{g}. From now on, I intend to opt for G_{j} instead (j for ‘job guarantee sector’) so that g can be reserved for growth rates. Since G_{j} = jY, it follows that the job-guarantee spending share in income is j = G_{j}/Y. An expression for j can be derived from the expressions for G_{j} and Y that are presented in the earlier posts. This is left for a possible future post.

Solving for equilibrium output gives:

Dynamic stability now requires Z > 0 and α – h – j > 0. The supermultiplier has become 1/(α – h – j). An exogenous increase in autonomous demand Z will boost output as well as employment in the broader economy. One consequence is that h will tend to become bigger due to a strengthening of the inducement to invest. Viewed in isolation, this will tend to make the supermultiplier larger. But another consequence of stronger autonomous demand is that j will tend to shrink, because rising output implies a stronger employment outcome in the broader economy, reducing the need for job-guarantee spending. This, viewed in isolation, tends to make the supermultiplier smaller. The overall impact on the supermultiplier will depend on the combined impact of stronger autonomous demand on the investment share (h) and job-guarantee spending share (j).

Adopting the same procedure as we have already adopted twice in earlier sections of the post enables the derivation of an expression for the growth rate of equilibrium output:

Here, time derivatives are denoted with the prime (‘) symbol rather than dots, because j already has a dot. Expression (3”) says that the growth rate of equilibrium output deviates from the growth rate of autonomous demand to the extent that there are variations in the investment and/or the job-guarantee spending shares in income. As before, the growth rate of equilibrium output will equal the sum of the growth rates of autonomous demand and the supermultiplier.

**Concluding Remark**

Even though we have abstracted from the distinction between actual output Y and equilibrium output Y*, which amounts to assuming that the economy is continuously operating at a level of output that equates aggregate supply with aggregate demand (or planned leakages with planned injections), this does not imply a constant rate of growth (let alone a constant level of output) other than in the simplest versions of the model. In the presence of endogenous investment and/or a job guarantee, stability in the equilibrium growth rate would further require that capacity had fully adjusted to demand (stabilizing the investment share h) and/or the composition of employment had settled at a particular sectoral configuration (stabilizing the job-guarantee-spending share j). Needless to say, there is little reason to expect that such stability in the equilibrium growth rate would ever be maintained for long, or even be attained at all.

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