In the long run, a higher rate of economic growth implies a higher share of investment in income. This will necessarily correspond to a lower combined share of other forms of spending. Knowledge of this point sometimes leads to unfortunate calls to cut back government spending as a means of boosting growth. Yet, if private investment can be said, in a long-run context, largely to be induced by income – which is consistent with the evidence (see, for instance, here, here and here) – then a sustained increase in the growth rate of government spending and other components of autonomous demand could be expected, through their direct impact on income, to induce higher investment. It is not at all obvious that the investment share in income would fall as a result of this process. To the contrary, there is reason to expect the investment share to rise and the share of other spending (and quite possibly government spending) to fall in consequence of demand-led growth of this nature. The purpose of the present post is to explain this possibility.

The argument is a bit technical. It follows from analyses of the ‘Sraffian supermultiplier’ and is one point emphasized in recent papers by Girardi and Pariboni (here) and Serrano and Freitas (here). The first of these papers presents some empirical support for the argument as well as for demand-led growth in general.

In these papers, growth is considered to be driven by those components of autonomous demand that do not directly create private-sector productive capacity. In the demand-led growth literature, this kind of demand is sometimes called Z. In general, it comprises government spending, autonomous private consumption expenditure (including some expenditures classified as investment in the National Accounts, such as purchases of new residential housing) and exports. An increase in the growth rate of this type of demand, by exogenously raising the growth rate of total demand, tends to lift rates of capacity utilization as firms adjust output to demand at given prices. If the change in demand conditions is long-lasting, so that rates of capacity utilization remain high, firms will be induced to install additional plant and equipment. It is assumed that the economy is within resource limits and that there is an ample supply of labor, so that capacity can be expanded in response to persistently strong demand.

The presentation in this post follows the two papers just cited. There are a couple of minor differences. One is that the economy considered here is simpler, with the focus specifically on government spending and private investment. Another is that the model is presented in discrete rather than continuous form. This is less convenient mathematically, but may be a little easier to follow for some readers. It also helps us to see the dual nature of private investment, which has an immediate impact on demand and a delayed effect on productive capacity.

**A Basic Model of Demand-Led Growth**

Consider a closed economy in which private consumption C and investment I are both purely induced, and government spending G is the only form of exogenous demand that does not directly create private-sector capacity. Demand-led growth can be characterized by extending the familiar Keynesian income-expenditure model. For the current period t:

The exogenously given parameters c and t are the marginal propensities to consume and tax. For simplicity, they are assumed constant through time. The marginal propensity to invest, h_{t}, is determined endogenously (to be explained). Note that h_{t} = I_{t }/Y_{t} is also the investment share in income. All variables are functions of time.

Substituting (2)-(4) into (1) and solving for equilibrium income Y_{t} gives:

where α = 1 – c(1 – t) is the marginal propensity to leak from the circular flow of income to taxes and saving. The expression shows equilibrium income to be a multiple of government spending, where 1/(α – h_{t}) is referred to as the ‘supermultiplier’. For output and growth to be considered demand led, it is necessary that α > h_{t} and G_{t} > 0 (or, more generally, Z_{t} > 0).

Income in (5) is at an equilibrium level in the sense that desired leakages equal desired injections. (A previous introduction to the income-expenditure model explains this concept.) However, this equilibrium of supply and demand in product markets is not necessarily a ‘fully-adjusted position’ in which capacity has been completely reconciled to demand. Most of the time, firms in a situation described by (5) will be operating their capacity to a degree above or below the normal (or average) planned level. (For a discussion of this point, see the previous post.) Whenever this is the case, it is assumed that investment will be encouraged or discouraged according to a behavioral rule such as the following:

Here, Δh = h_{t+1} – h_{t} is the change in the investment share from time t to time t+1. It represents the behavior of the investment share through time. The coefficient γ is a positive constant. It indicates the responsiveness of the investment share to deviations in the rate of utilization u_{t} from its normal rate u_{t} = 1. When the rate of utilization is above normal (u_{t} > 1), the right-hand side of (6) is positive and the investment share in income rises (Δh is positive). Conversely, when u_{t} < 1, investment is discouraged and its share in income falls (Δh is negative). The adjustment process is considered to be gradual (meaning γ must be small).

To consider the behavior of capacity utilization, start with its change over time:

The rate of utilization itself is defined as actual output Y divided by normal output Y_{n}, where normal output is the level of output at which u = 1:

Intuitively, it seems clear that since u = Y/Y_{n}, changes in u will depend upon the relative growth rates of actual output Y and normal output Y_{n}. This suggests that it will be helpful, in the first term on the right-hand side, to relate Y_{t+1} to Y_{t} and, in the second term, to relate Y_{nt} to Y_{nt+1}. We can do this by multiplying the first term by Y_{t }/Y_{t} = 1 and the second term by Y_{nt+1 }/Y_{nt+1} = 1, which leaves both terms unchanged.

By reordering the variables in each term we have:

In this expression, the first fraction on the right-hand side can be modified as follows:

Similarly,

The change in a variable divided by the initial level of that variable is its growth rate. So, here, ΔY/Y_{t} is the growth rate g_{t} of actual output over period t. Similarly, ΔY_{n }/Y_{nt} is the growth rate g_{nt} of normal output over the same period.

Substituting back into the expression for Δu gives:

If we expand this out and then factorize, a couple of terms drop out (because they sum to zero) and we are left with:

Multiplying the right-hand side by Y_{nt }/Y_{nt} = 1 (leaving it unaffected) and reordering variables gives:

The first fraction on the right-hand side (Y_{t }/Y_{nt}) is the utilization rate u_{t} of period t. The second fraction (Y_{nt }/Y_{nt+1}) is the reciprocal of one plus the growth rate of normal output. (We already showed a few steps back that Y_{nt+1 }/Y_{nt} = 1 + g_{nt}.)

Upon substituting u_{t} and 1/(1 + g_{nt}) into the expression for Δu, we arrive at the relationship we are after:

The rate of utilization increases (Δu is positive) when actual output grows faster than normal output (g_{t} > g_{nt}). Intuitively, we might have guessed Δu = u_{t} (g_{t} – g_{nt}). In fact, this is correct for the continuous case.

**Determination of the Economy’s Actual Growth Rate**

The behavioral assumptions and equilibrium conditions outlined in (1)-(7) suggest a demand-led growth process. We can see this by considering the way in which actual output changes over time.

Begin by substituting the expressions (2)-(4) for consumption, investment and non-capacity-generating autonomous expenditure into the equilibrium condition (1) (which says Y_{t} = C_{t} + I_{t} + Z_{t}):

It takes a bit of work, but we can manipulate (1′) to arrive at an expression for the growth rate of actual output.

The equivalent of (1′) for period t+1 is:

The change in actual output can now be obtained by subtracting (1′) from (1”):

Substituting for Y_{t+1} and Y_{t} gives:

Notice that Y_{t+1} – Y_{t} = ΔY and G_{t+1} – G_{t} = ΔG. We can modify the expression accordingly:

At this point, it is useful to add –h_{t}Y_{t+1} + h_{t}Y_{t+1} (= 0) to the right-hand side. This will enable us to convert more terms into changes (symbolized by Δ).

Taking Y_{t+1} and h_{t} as common divisors and factorizing suitable terms gives:

Here, h_{t+1} – h_{t} = Δh and Y_{t+1} – Y_{t} = ΔY, enabling us to modify the expression as follows:

Now we can collect all the ΔY terms on the left-hand side and factorize:

Recalling that α = 1 – c(1 – t), this becomes:

Dividing all terms by Y_{t} and multiplying the final term on the right by G_{t }/G_{t} = 1 gives:

It is stated in (5) that Y_{t} = G_{t }/(α – h_{t}). Rearranging this gives G_{t }/Y_{t } = α – h_{t }, which can be substituted into the above expression. Also, since ΔG/G_{t} and ΔY/Y_{t} are the growth rates of government spending and actual output, respectively, we have:

Finally, solving for g_{t}, we obtain the relationship we are interested in:

This says that g_{t}, the growth rate of the economy as a whole (both of actual output and total demand), depends upon the growth rate of government spending g_{Gt} plus the rate of change of the supermultiplier (which is the second, fractional term on the right-hand side). The supermultiplier varies over time as the investment share h changes (according to (6)) in response to demand and deviations of the utilization rate u from normal.

Over short or medium time frames, Δh will usually differ from zero (because h_{t+1} will not equal h_{t}). When this is the case, the last term in (8) will be non-zero and the bracketed part of the first term on the right-hand side will differ from one. As a result, there will be no logical necessity for a higher g_{Gt} to correspond to a higher g_{t}. However, in the long run, if g_{Gt} were held constant and firms, hypothetically, were given enough time to adjust capacity to demand, the rate of utilization would eventually settle down to normal (u_{t} = 1). At that point, there would be no tendency for the share of private investment in GDP to change. Referring back to (6), in which Δh = h_{t}γ (u_{t} – 1), Δh would then be zero, due to u_{t} being equal to 1. The economy’s growth rate in (8) would then match the growth rate of government spending, with the bracketed part of the first term on the right-hand side equal to one and the second term equal to zero.

This suggests that the economy’s growth rate *tends* to increase with the growth rate of government spending. This result holds, within the model, as a long-run tendency. It is not something that holds at every point in time.

**The Growth Rates of Private Investment and the Capital Stock**

A glance at (6) or (8) gives the (correct) impression that private investment will grow at a different rate than the economy as a whole whenever the rate of utilization differs from normal. We can confirm this, starting from (6):

Dividing both sides by h_{t} gives:

We can write the left-hand side of (6′) as:

Substituting into (6′) gives:

Or, adding 1 to both sides,

Working with the left-hand side again and recalling that h = I/Y we have:

where g_{It} is the growth rate of private investment over period t. Substituting back into the left-hand side of the previous expression gives:

Or, multiplying all terms by 1 + g_{t},

Simplifying shows how the growth rate of private investment relates to the growth rate of the economy as a whole:

When utilization is above normal (u_{t} > 1), private investment grows faster than actual output (and total demand). When utilization is below normal (u_{t} < 1), private investment grows slower than actual output. Only in a fully-adjusted position, in which u_{t} = 1, does private investment grow at the same rate as the economy as a whole.

The behavior of private investment has implications for the growth rate of the capital stock K. Assuming that the ratio of normal output to capital θ (= Y_{n }/K) is constant, these implications will also apply to normal output Y_{n}.

The change through time of the capital stock K is equal to gross investment I minus depreciation δK, where δ is the rate of depreciation and assumed positive and constant:

This says that net investment (gross investment minus depreciation) results in an increment to the capital stock. The effect occurs with a delay. Here, the capacity effect of investment is assumed to lag behind the demand effect by one period.

Dividing both sides of the expression by K_{t} and noting that ΔK/K_{t} is the growth rate of the capital stock g_{Kt+1} enables us to write:

Multiplying the first term on the right-hand side by (Y_{t }/Y_{t})(Y_{nt }/Y_{nt}) = 1, we can rewrite this expression as:

Here, I_{t }/Y_{t} is the investment share in income h_{t}, Y_{t }/Y_{nt} is the utilization rate u_{t}, and Y_{nt }/K_{t} is the normal output-to-capital ratio θ. We can therefore write:

Since the normal output-to-capital ratio is assumed constant, we can also write:

According to these expressions, the capital stock and normal output grow more rapidly when the investment share h_{t} and utilization rate u_{t} are high. The growth of both lags behind the growth of demand and actual output.

**Fully-Adjusted ‘Long-Period’ Positions**

Reference has already been made to so-called fully-adjusted positions. These are the positions that an economy would eventually reach if the growth rate of government spending (or, more generally, the growth rate of non-capacity-creating autonomous demand Z) remained constant for long enough that firms had time to adjust capacity completely to demand.

Of course, every time the growth rate of Z is perceived to change in a persistent way, the economy will be set off on a different notional growth path. The economy will never actually reach a fully-adjusted position. These positions are a theoretical construct intended to shed light on growth tendencies. A fully-adjusted position describes, at any point in time, the growth rate to which the economy is notionally tending given the current behavior of Z.

In the model, a tendency for the economy to converge on a fully-adjusted position is assured provided the conditions for demand-led growth are met. These conditions, mentioned earlier, are that α > h_{t} and G_{t} > 0 (or, more generally, Z_{t} > 0). For the first of these conditions to hold, the speed of adjustment cannot be too rapid (that is, γ cannot be too large). When the two conditions are satisfied, the growth process is said to be ‘dynamically stable’. A formal stability analysis can be found in the paper by Serrano and Freitas (pp. 9-12) linked to at the beginning of this post.

In a fully-adjusted position, there is no tendency for either the investment share h_{t} or utilization rate u_{t} to change. In other words, Δh and Δu will both be zero.

The first condition (6”) is satisfied when the rate of utilization is normal (u_{t} = 1). Substituting this equilibrium value for u_{t} into (7′) gives a growth rate for the economy as a whole, g_{t}, equal to the growth rate of normal output, g_{nt} (and of the capital stock g_{Kt}). By (8) and (9), government spending and private investment will also grow at this same rate. That is, in a fully-adjusted position, g_{Gt} = g_{t} = g_{It} = g_{Kt} = g_{nt}. At this point, capacity and normal output will have finally caught up with demand and actual output, and so will be growing at the same rate as these other variables until something happens to disturb the growth rate of government spending (or Z).

Substituting u_{t} = 1 into (10′) (which says g_{nt} = h_{t}u_{t}θ – δ) and solving for h_{t} gives the investment share in income h* associated with the fully-adjusted position:

Since, in a fully-adjusted position, the growth rate of normal output g_{nt} is the same as the growth rate of the economy as a whole, which in turn is the same as the growth rate of government spending (or Z), we can also express this as:

This is the investment share in income that tends to be brought about by the demand-led growth process. Noting that the rate of depreciation δ and the normal output-to-capital ratio θ have both been assumed constant, the expression shows that the investment share h_{t} tends to rise with the growth rate of the economy g_{t} and the growth rate of government spending g_{Gt}.

**The Growth Rate and Share in Income of Government Spending**

We are now, at last, in a position to relate the growth rate of government spending to the shares of private investment and government spending in income.

Substituting the equilibrium investment share h* into (5) we have:

Noting, again, that in a fully-adjusted position g_{nt} = g_{Gt}, this can be written:

Upon rearrangement we obtain:

All elements on the right-hand side of (12) are constant. As anticipated, an exogenous increase in the growth rate of government spending g_{Gt} is associated with a smaller share of government spending in income (G_{t }/Y_{t}).

The causation suggested by the model runs from government spending to actual output (and total demand) to induced investment. An increase in g_{Gt} lifts the rate of output growth g_{t} and pushes utilization beyond normal, accelerating the growth rate of investment g_{It}.

In this simplified version of the model, the economy tends toward a situation in which private investment, the capital stock, normal output and actual output all grow at a rate that is determined by the growth rate of government spending.

More generally, when we include other components of non-capacity-creating autonomous demand Z in the model, the economy tends toward a growth rate determined by the combined behavior of all components of Z. Inside resource limits, the economy grows faster when Z grows faster. When this occurs, the share of private investment in total income will rise and the share of Z in total income will fall.

There is more leeway for the share of government spending in income once we include all components of Z. It is then possible that G_{t }/Y_{t} could rise or fall with g_{Gt}, depending on the behavior of other components of Z.

But the basic point remains; namely, there is no contradiction in a higher growth rate of government spending translating into more rapid economic growth alongside a higher investment share and lower government-spending share in income.

**A Graphical Representation**

We have seen, in our simplified model, that a faster growth rate of government spending g_{Gt} results in a higher investment share h_{t} and lower government-spending share (G_{t }/Y_{t}).

We can gain an understanding of why this is the case by reconsidering the following expressions:

Expression (8) says that when the investment share is rising (Δh > 0), actual output and total demand grow faster than government spending (g_{t} > g_{Gt}). Now, from (6), we know that the investment share rises (Δh > 0) when utilization is above normal (u_{t} > 1). When this occurs, investment, according to (9), will grow faster than actual output (g_{It} > g_{t}). Putting all this together, when the rate of utilization exceeds normal (u_{t} > 1), the investment share rises (Δh > 0) and investment grows faster than actual output, which in turn grows faster than government spending (g_{It} > g_{t} > g_{Gt}).

In other words, when there is an exogenous increase in the growth rate of government spending, this causes investment to grow faster than income so that the *level* of investment rises relative to the *level* of income. As a consequence, the investment share (h_{t} = I_{t }/Y_{t}) rises. At the same time, the demand effects of the acceleration in the growth rate of investment cause income to grow faster than government spending. This causes the *level* of government spending to fall relative to the *level* of income. Accordingly, the share of government spending in income (G_{t }/Y_{t}) falls.

By way of illustration, assume the following initial values and parameters:

Suppose that for 20 periods (defined as years) the economy is stationary, with output, demand and the capital stock constant rather than growing. Then, at time t = 20, there is an exogenous increase in the growth rate of government spending to 1 percent per annum (g_{Gt} = 0.01). Government spending continues to grow at this rate for 40 periods. Then, at time t = 60, there is another exogenous increase in the growth rate of government spending, this time to 3 percent per annum (g_{Gt} = 0.03).

Two cases will be shown. They differ only in the speed-of-adjustment parameter, γ. In case 1, γ = 0.02. In case 2, γ = 0.1.

**Case 1: γ = 0.02.** The figure below shows the behavior of the growth rates of investment, actual output and the capital stock in response to the behavior of government spending. The capital stock (and also normal output) adjust with a lag due to the demand effect of investment being felt before the capacity effect. The adjustment takes place without noticeable cyclical fluctuations in the growth rates because γ is very small.

The next figures show the behavior of the investment share h_{t}, the government-spending share (G_{t }/Y_{t}) and the rate of utilization u_{t}. Note that the vertical axis does not start at zero. This is to highlight the changes in the variables. The government-spending share drops from 30 percent to about 28.3 percent. The investment share rises from 20 percent to about 21.5 percent. As discussed, exogenous increases in g_{Gt} cause the investment share to rise and the government-spending share to fall. The response of investment is due to the behavior of the utilization rate, which on average is above normal for the duration of the transition period, before eventually settling back to normal.

**Case 2: γ = 0.1.** When γ is larger (but not too large), the growth rates (other than g_{Gt}) cycle around the growth rate of government spending in a ‘damped’ manner (meaning the cycles get smaller in amplitude over time) until converging on g_{Gt}.

Likewise, the investment and government-spending shares also cycle toward their new levels. The utilization rate fluctuates around an average during the transition period that is above normal before settling back to normal once capacity has been fully adjusted to demand. You can get an idea that the average utilization rate is above normal during the transitional phase by noting the peaks (high points) and troughs (low points) in the successive cycles. In particular, the peak of the utilization rate in each cycle is further above 1 than the troughs are below 1. This causes the share of investment to rise as firms attempt to adjust capacity to demand.

**Further Reading on Demand-Led Growth Theory**

A good place to start is Serrano, 1995, The Sraffian Supermultiplier. Similar conclusions to those discussed in the present post can be reached from a Kaleckian starting point once the role of non-capacity-creating autonomous demand is taken into account. See, in particular, papers by Allain (here) and Lavoie (here). A helpful survey of the demand-led growth literature is provided by Cesaratto (here).