Assuming an exogenously fixed investment demand is a useful simplification for understanding the role of dynamics, as well as for exploring different assumptions about saving. But a fixed investment demand is a strong assumption, and some of the results obtained so far do not hold when this assumption is relaxed. An important first step is to recognize a role for profits in determining investment demand, as Joan Robinson did in her pioneering studies of growth (1956; 1962). But this is hardly sufficient. Bhaduri and I argued that it is important to go beyond a simple dependence of investment on the rate of profit (and thus on the real wage), because the two components of the profit rate – the profit share and the rate of capacity utilization – can interact differently depending on the state of the economy. In particular, investment demand may be more responsive to the rate of profit when capacity utilization is (or is expected to be) high, but much less responsive under slack business conditions, the difference being due to business perceptions of how likely it is that they will be able to sell more goods and thus realize the prospective profits from adding to productive capacity.

In this essay I add a new dimension, distinguishing between two very different types of investment, capital widening, which is to say investment that takes place to expand capacity, and capital deepening, investment which substitutes capital for labor. Though the distinction between widening and deepening is an old one, the growth theories spawned by Harrod three-quarters of a century ago have neglected capital deepening.

The reason for introducing capital deepening is that Bhaduri and I didn't go far enough in our critique of investment theory. Yes, indeed, the responsiveness of investment demand to the rate of profit may vary pro-cyclically, but in our model it was still true that the response was always positive, or at least non-negative. But that assumption makes sense only for capital widening. By contrast, capital deepening responds *negatively* to the rate of profit. The benefits of substituting capital for labor, unlike the benefits of capital widening, vary directly with the real wage. Another reason why high wages might be good for employment.
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Moreover, there is good reason to expect that the attractiveness of capital deepening will be much less influenced by the business cycle than is capital widening. At a capacity utilization rate of 90 percent, capital widening may have little attraction at all. At a capacity utilization rate of 90 percent, capital deepening may be only 10 percent less profitable than when the economy is running at 100 percent of capacity. And the importance of this distinction is borne out empirically. Alex Field (2011) has documented the importance of capital deepening even in the depths of the Great Depression. By 1936, the US economy was back to the level of real GDP in 1929, but it could produce the same level of output with so many fewer workers that the unemployment rate was 17 percent in 1936, as against 3.2 percent in 1929.

In this section we take up capital widening and capital deepening successively.

### 4.4.1 Capital widening

We write investment demand for capacity expansion as

$$i=\mathrm{\psi}\left(\mathrm{\Omega}\left(l\right)\text{\hspace{0.17em}}\left[h(l,\mathrm{\xi})-{\left(\frac{P}{W}\right)}^{-1}l\text{}\right]-{\mathrm{\rho}}_{h}\right).$$

The expression within parentheses on the right-hand side is the quasi-rent per unit of investment taking account of the (subjective) probability that the additional capacity is utilized and the additional production is sold. ρ_{
h
}, the hurdle rate of return, is the capital cost on the assumption that capital lasts forever. $\mathrm{\Omega}\left(l\right)$ is the probability of finding a market for the additional output, which is assumed to be an increasing function of *l*, with $\mathrm{\Omega}\left(0\right)=0$ and $\mathrm{\Omega}\left(l\right)\le 1$. Finally, $\mathrm{\psi}$ reflects the level of animal spirits, the degree of optimism that determines the responsiveness of investment demand to anticipated returns.

This expression for investment demand derives from the general expression for the real return to an investment equal to $\dot{K}$, namely:

$$\frac{\mathrm{\partial}Y}{\mathrm{\partial}K}\text{}\dot{K}+\frac{\mathrm{\partial}Y}{\mathrm{\partial}L}\text{}\dot{L}-{\mathrm{\rho}}_{h}\dot{K}-{\left(\frac{P}{W}\right)}^{-1}\text{\hspace{0.17em}}\dot{L},$$

where

$\frac{\mathrm{\partial}Y}{\mathrm{\partial}K}\text{}\dot{K}$ measures the direct contribution of the additional capital and

${\left(\frac{P}{W}\right)}^{-1}\dot{L}$ measures the prospective change in labor costs associated with the investment. In the mainstream conception of equilibrium the above expression vanishes. With capital a homogeneous, undifferentiated mass, the marginal productivity of capital equals the hurdle rate of interest at equilibrium

$$\frac{\mathrm{\partial}Y}{\mathrm{\partial}K}={\mathrm{\rho}}_{h}.$$

And with employment dictated by profit maximization, the marginal productivity is equal to the real wage

$$\frac{\mathrm{\partial}Y}{\mathrm{\partial}L}={\left(\frac{P}{W}\right)}^{-1}.$$

Here, in contrast, even though capital is written as an aggregate

*K*, the capital stock is not assumed to consist of legos which can be dismantled and recombined in new shapes. I rather assume investment to involve a commitment to a particular form of capital. In the absence of lego capital there can be a persistent gap between the marginal productivity of capital and the hurdle rate of interest. Indeed it is precisely an assumed positive value of the difference

$$\frac{\mathrm{\partial}Y}{\mathrm{\partial}K}-\text{\hspace{0.17em}}{\mathrm{\rho}}_{h}$$

that in

*The General Theory* is assumed to drive capital formation. As for the labor costs associated with new capital, the difference

$$\frac{\mathrm{\partial}Y}{\mathrm{\partial}L}-{\left(\frac{P}{W}\right)}^{-1}$$

is not necessarily equal to 0 – it vanishes at a flexprice equilibrium but not under a fixprice regime.

How do we get from the generic expression for investment returns to the specific formula for capital widening? First, separate the capital cost from the quasi-rent and multiply the quasi-rent by the probability of selling the additional goods to obtain the flow of returns per unit of investment:
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$$\mathrm{\Omega}\text{\hspace{0.17em}}(l)\text{\hspace{0.17em}}\left(\frac{\mathrm{\partial}Y}{\mathrm{\partial}K}+\frac{\mathrm{\partial}Y}{\mathrm{\partial}L}\frac{\dot{L}}{\dot{K}}-{\left(\frac{P}{W}\right)}^{-1}\frac{\dot{L}}{\dot{K}}\right)-{\mathrm{\rho}}_{h}.$$

Now factor in the animal spirits of capitalists, measured by

$\mathrm{\psi}$, to obtain the following expression for investment demand:

$$i=\mathrm{\psi}\left(\mathrm{\Omega}\text{\hspace{0.17em}}(l)\text{\hspace{0.17em}}\left[\left(\frac{\mathrm{\partial}Y}{\mathrm{\partial}K}+\frac{\mathrm{\partial}Y}{\mathrm{\partial}L}\frac{\dot{L}}{\dot{K}}-{\left(\frac{P}{W}\right)}^{-1}\frac{\dot{L}}{\dot{K}}\right)-{\mathrm{\rho}}_{h}\right]\right)\text{\hspace{0.17em}}.$$

The next step is to determine the incremental labor–capital ratio

$\frac{\dot{L}}{\dot{K}}$. The simplest assumption is that when considering capacity-augmenting investment, capitalists extrapolate the current labor–capital ratio, which is to say that one additional unit of capital requires

*l* workers:

$$\frac{\dot{L}}{\dot{K}}=\frac{L}{K}=l.$$

Since

$$\left(\frac{\mathrm{\partial}Y}{\mathrm{\partial}K}K+\frac{\mathrm{\partial}Y}{\mathrm{\partial}L}L\text{}\right)\frac{1}{K}=\frac{Y}{K}=h(l,\mathrm{\xi})$$

when we substitute the equivalent expressions for

$\frac{\dot{L}}{\dot{K}}$ and

$\left(\frac{\mathrm{\partial}Y}{\mathrm{\partial}K}K+\frac{\mathrm{\partial}Y}{\mathrm{\partial}L}L\text{}\right)\frac{1}{K}$, we obtain the capital-widening formula

$$i=\mathrm{\psi}\left(\mathrm{\Omega}(l)\left[h(l,\mathrm{\xi})-{\left(\frac{P}{W}\right)}^{-1}l\text{}\right]-{\mathrm{\rho}}_{h}\right).$$

To draw the aggregate-demand schedule, plot the investment-demand schedule together with the saving schedule in the manner of

Figure 12. For simplicity let saving revert to being a constant fraction of income, and

Figure 12 becomes

Figure 17.

There is an obvious superficial resemblance between Figure 17b and Figure 12c. But the superficial resemblance conceals significant differences. In the first place, in contrast with Figure 12, there is no reason for the AD schedule generated by Figure 17a to intersect the GS schedule at its minimum point relative to the *l* axis. More important, the dynamics of Figure 17b are the opposite of the dynamics of Figure 12c. Here, between any two points generated by a given real price, say, *l*
_{2} and *l*
_{3} generated by ${\left(\frac{P}{W}\right)}_{1}$, there is an excess of investment demand over desired saving, whereas in Figure 12c the same interval is characterized by excess desired saving. For this reason stability and the comparative statics of parameter changes do not carry over from the earlier analysis.

Consider Figure 18, in which AD, GS, and LS schedules look similar to the corresponding schedules in Figure 13a. The results are very different. In the flexprice case, the stationary-price locus itself changes, and there are now three equilibria instead of one. There are two distinct regions of the stationary-price locus, one below the LS schedule along which downward pressure on prices is just balanced by downward pressure on wages. A second region, above the LS schedule, defines a set of points for which *upward* price pressure balances upward wage pressure. In the interior of these loci the real price rises because price pressure exceeds wage pressure; everywhere else wage pressure exceeds price pressure and the real price falls.

With flexprice adjustment, change is now governed by

$${\left(\frac{P}{W}\right)}^{\u2022}=\left({\mathrm{\theta}}_{1}\left\{\mathrm{\psi}\left(\mathrm{\Omega}(l)\left[h\left(l,\mathrm{\xi}\right)-{\left(\frac{P}{W}\right)}^{-1}l\text{}\right]-{\text{}\mathrm{\rho}}_{h}\right)-sh(l,\mathrm{\xi})\right\}-{\text{}\mathrm{\theta}}_{3}\left[\frac{P}{W}-{\left(\frac{P}{W}\right)}^{*}\right]\right)\text{\hspace{0.17em}}\frac{P}{W}$$

$$\dot{l}={\mathrm{\theta}}_{2}\text{\hspace{0.17em}}\left(\frac{P}{W}-{h}_{l}{(l,\mathrm{\xi})}^{-1}\right)l.$$

The Jacobian matrix is

$$\begin{array}{c}\left[\begin{array}{cc}\frac{\mathrm{\partial}{\left(\frac{P}{W}\right)}^{\u2022}}{\mathrm{\partial}\frac{P}{W}}& \frac{\mathrm{\partial}{\left(\frac{P}{W}\right)}^{\u2022}}{\mathrm{\partial}l}\\ \frac{\mathrm{\partial}\dot{l}}{\mathrm{\partial}\frac{P}{W}}& \hfill \frac{\mathrm{\partial}\dot{l}}{\mathrm{\partial}l}\hfill \end{array}\right]\\ =\left[\begin{array}{cc}{\mathrm{\theta}}_{1}\mathrm{\psi}\mathrm{\Omega}(l){\left(\frac{P}{W}\right)}^{-2}l-{\mathrm{\theta}}_{3}& {\mathrm{\theta}}_{1}\left(\mathrm{\psi}\left\{\mathrm{\Omega}\prime \left(l\right)\left[h\left(l,\text{}\mathrm{\xi}\right)-{\left(\frac{P}{W}\right)}^{-1}l\right]+\mathrm{\Omega}\left(l\right)\left[{h}_{l}-{\left(\frac{P}{W}\right)}^{-1}\right]\right\}-s{h}_{l}\right)\\ {\mathrm{\theta}}_{2}& {\mathrm{\theta}}_{2}{h}_{l}^{-2}{h}_{ll}\end{array}\right].\end{array}$$

Since

${h}_{u}$ is negative, a sufficient condition for a negative trace is that the expression

$${\mathrm{\theta}}_{1}\mathrm{\psi}\mathrm{\Omega}(l\text{}){\left(\frac{P}{W}\right)}^{-2}l-{\mathrm{\theta}}_{3}\text{}$$

be negative. This condition holds at

*E*″ and

*E*, as we can see from the expression for the slope of the stationary-price locus. This slope is given by

$${\left(\frac{d\left(\frac{P}{W}\right)}{dl}\right)}_{{\left(\frac{P}{W}\right)}^{\u2022}=0}=-\frac{\mathrm{\psi}\left\{\mathrm{\Omega}\prime \left(l\right)\left[h\left(l,\mathrm{\xi}\right)-{\left(\frac{P}{W}\right)}^{-1}l\text{}\right]+\mathrm{\Omega}\left(l\right)\left[{h}_{l}-{\left(\frac{P}{W}\right)}^{-1}\right]\right\}-s{h}_{l}}{\mathrm{\psi}\mathrm{\Omega}\text{}{\left(\frac{P}{W}\right)}^{-2}\text{}l-\frac{{\text{}\mathrm{\theta}}_{3}}{{\text{}\mathrm{\theta}}_{1}}}$$

and is negative at

*E*′ and positive at

*E* and

*E*″. The numerator is positive at

*E* and

*E*′ because increasing

*l* from its value at either point moves the economy from the boundary of the stationary-price locus, where

${\left(\frac{P}{W}\right)}^{\u2022}=0,$ into the interior, where

${\left(\frac{P}{W}\right)}^{\u2022}>0$. In other words at both

*E* and

*E*″

$$\frac{\mathrm{\partial}{\left(\frac{P}{W}\right)}^{\u2022}}{\mathrm{\partial}l}=\mathrm{\psi}\left\{\mathrm{\Omega}\prime \left(l\right)\left[h\left(l,\mathrm{\xi}\right)-{\left(\frac{P}{W}\right)}^{-1}l\text{}\right]+\mathrm{\Omega}\left(l\right)\left[{h}_{l}-{\left(\frac{P}{W}\right)}^{-1}\right]\right\}-s{h}_{l}$$

is positive. Since

${\left(\frac{d\left(\frac{P}{W}\right)}{dl}\right)}_{{\left(\frac{P}{W}\right)}^{\u2022}=0}$ is positive at

*E* and negative at

*E*′, it follows that the denominator must be negative at

*E* and positive at

*E*′. At

*E*″ the numerator is negative,so the denominator is also negative. Consequently the trace is necessarily negative at

*E*′′ and

*E*, but sign indefinite at

*E*′. The determinant, however, is negative at

*E*′. This establishes that there are two stable equilibria,

*E*″ and

*E*; the middle one, at

*E*′, is unstable.

In Figure 18 the stability of the equilibrium at *E* is unaffected if the intersection of the GS schedule and the stationary-price locus takes place to the right of the maximum of the stationary-price locus. Because the GS schedule is more steep than the stationary-price locus relative to the *l*-axis, the determinant of the Jacobian is unaffected. Since both the slope of the stationary-price locus and the numerator change sign relative to the case we just considered, the denominator continues to be negative. Likewise, the equilibrium at *E*′ continues to be unstable if the intersection is to the right of the minimum point on the stationary-price locus.

As we have done with earlier models, we ask what happens when one or another of the parameters shifts. The results are presented in the three panels of Figure 19. We focus on the stable equilibrium at *E*. (For the most part the stable equilibrium at *E*″ responds similarly.)

Start with the AD schedule. An increase in investment demand – an improvement either in animal spirits or in the confidence of being able to utilize additional capacity – shifts the AD schedule downward. As does a fall in the propensity to save. Note the contrast with the case where profits drive saving and investment is fixed. The AD schedule has the same U-shape in the two cases; but in the present case the downward-sloping portion of the AD schedule reflects a situation where, as the economy approaches the AD schedule from the left, investment catches up to saving, and the upward-sloping segment reflects a situation where investment falls to the level of saving. It takes *less* output to generate the requisite amount of saving when the investment propensity rises or the saving propensity falls, so the downward-sloping segment is displaced to the left. By the same logic, the upward-sloping segment is displaced to the right. In the case of the Cambridge saving equation, in which investment is fixed and saving is proportionate to profits, the downward-sloping portion is associated with saving catching up to investment and the upward-sloping segment with saving falling to investment. So an increase in investment demand or a fall in the saving propensity – see Figure 14 – *increases* the amount of output necessary for saving to catch up with investment and thus displaces the downward-sloping segment to the right. Again the same logic means the upward-sloping piece is displaced to the left.

Now the new equilibrium involves greater employment per unit of capital and a higher real price, which is to say a lower equilibrium real wage. The new rate of inflation at *F*, which can be measured by the distance from the equilibrium to the (unchanging) conventional-wage schedule, is higher than the old rate.

Supply-side shocks affect equilibrium similarly to the previous flexprice models. In Figure 19b, an increase in the price of energy reduces the marginal productivity of labor, which in turn reduces output (net of energy costs) and reduces profitability. The reduction in the marginal productivity of labor directly shifts the GS schedule to the left and, because the effect on profitability is stronger than the effect on output, shifts the AD schedule inward. The result is a downward shift in equilibrium employment per unit of capital and in the equilibrium real price. Inflation is reduced.

Figure 19c shows the effect of an increase in the conventional wage. Observe that the stationary-price locus shrinks. Why? The reason is that a decrease in ${\left(\frac{P}{W}\right)}^{*}$ raises wage pressure, which, with price pressure unchanged, has the effect of reducing the real price. To restore the balance between price and wage pressure, it is necessary to increase *l*. For as *l* increases, the difference between investment demand and desired saving also increases (since the derivative

$$\mathrm{\psi}\left\{\mathrm{\Omega}\prime \left(l\right)\left[h\left(l,\mathrm{\xi}\right)-{\left(\frac{P}{W}\right)}^{-1}l\text{}\right]+\mathrm{\Omega}\left(l\right)\left[{h}_{l}-{\left(\frac{P}{W}\right)}^{-1}\right]\right\}-s{h}_{l}$$

is positive at both

*E* and

*E*′), and this increases price pressure while wage pressure remains constant. As

*l* increases, wage pressure and price pressure once again become equal, on the new (dashed) stationary-price locus. At

*E*″ the above derivative is negative, so it would be necessary to dial back employment per unit of capital in order to increase price pressure sufficiently to balance the changed level of wage pressure. The shift in the equilibrium from

*E* to

*F* reflects the contraction of the stationary-price locus and the consequent fall in the real wage which makes employment less profitable at the margin and leads capitalists to curtail the level of employment.

In the situation described by Figure 19c, wage stimulus loses its luster. An increase in the conventional wage raises the equilibrium real wage but it leads to a decrease in employment and growth, and a very high conventional wage might make it impossible for the economy to grow at all. Similarly real-wage resistance, the ability of the working class to defend itself against erosion of the conventional wage, may defeat the possibility for a growth equilibrium to exist at all.

There is a clear pattern in these displacements of equilibrium. Both a shift of the AD schedule and a shift of the LS schedule are movements along a stationary GS schedule. Along this schedule the real price moves in the same direction as employment, Figure 19a is evidently consistent with a Phillips curve since the conventional wage remains fixed and inflation is proportional to the distance between the equilibrium $\frac{P}{W}$ and ${\left(\frac{P}{W}\right)}^{*}$. This is true in Figure 19b as well. In Figure 19b the displacement of the GS schedule adds to the effect of the shift along this schedule. In Figure 19c, the increase in the conventional wage moves the LS schedule and the equilibrium real price in the same direction, so to determine the impact of a change in the conventional wage on inflation we need to calculate the derivative $\frac{d\left(\frac{P}{W}\right)\text{}}{d{\left(\frac{P}{W}\right)}^{*}}\text{}$, where $\frac{P}{W}$ is the equilibrium price. We have

$$\frac{d\left(\frac{P}{W}\right)\text{}}{d{\left(\frac{P}{W}\right)}^{*}}=\text{}\frac{\frac{{\mathrm{\theta}}_{3}}{{\mathrm{\theta}}_{1}}}{\frac{{\left(\frac{P}{W}\right)}^{-2}}{{h}_{ll}}\left(\mathrm{\psi}\mathrm{\Omega}\prime \left(l\right)\left[h\left(l,\mathrm{\xi}\right)-{\left(\frac{P}{W}\right)}^{-1}l\text{}\right]-s{h}_{l}\right)-\mathrm{\psi}\mathrm{\Omega}l{\left(\frac{P}{W}\right)}^{-2}+\frac{{\mathrm{\theta}}_{3}}{{\mathrm{\theta}}_{1}}}.$$

The denominator is positive at a stable equilibrium because a condition of stability is that the determinant of the Jacobian is positive, and the denominator is the determinant after dividing through by θ

_{2} θ

_{1}. For an upward shift in the conventional wage to be inflationary the derivative

$\frac{d\left(\frac{P}{W}\right)\text{}}{d{\left(\frac{P}{W}\right)}^{*}}\text{}$ must be less than unity. Since

${h}_{ll}<0$, a necessary condition for this inequality is that

$$\mathrm{\psi}\mathrm{\Omega}\prime \left(l\right)\left[h\left(l,\mathrm{\xi}\right)-{\left(\frac{P}{W}\right)}^{-1}l\text{}\right]-s{h}_{l}$$

be positive, which is to say that at equilibrium investment must be more responsive to additional output and employment than is saving.

Consider now the alternative, fixprice, adjustment mechanism. In the fixprice case, the picture is qualitatively the same as for the Cambridge saving equation, except for the fact that the GS schedule does not go through the minimum point of the AD schedule. That is, Figure 20 looks like Figure 13b. But the disequilibrium relationship of investment and saving is reversed, and as a result stable and unstable equilibria change places: *E*′ is now unstable, while *E* is stable.

The equations describing the dynamics of a fixprice regime are

$${\left(\frac{P}{W}\right)}^{\u2022}=\left\{{\mathrm{\theta}}_{2}\left[l-\text{GS}(\frac{P}{W},\mathrm{\xi})\right]-{\mathrm{\theta}}_{3}\left[\frac{P}{W}-{\left(\frac{P}{W}\right)}^{*}\right]\right\}\text{}\frac{P}{W}$$

$$\dot{l}={\mathrm{\theta}}_{1}\left\{\mathrm{\psi}(\mathrm{\Omega}(l)\left[h\left(l,\mathrm{\xi}\right)-{\left(\frac{P}{W}\right)}^{-1}l\text{}\right]-{\text{}\mathrm{\rho}}_{h})-sh(l,\mathrm{\xi})\right\}l.$$

The Jacobian matrix is

$$\left[\begin{array}{cc}\frac{\mathrm{\partial}\text{\hspace{0.17em}}{\left(\frac{P}{W}\right)}^{\u2022}}{\mathrm{\partial}\frac{P}{W}}& \frac{\mathrm{\partial}{\left(\frac{P}{W}\right)}^{\u2022}}{\mathrm{\partial}l}\\ \frac{\mathrm{\partial}\dot{l}}{\mathrm{\partial}\frac{P}{W}}& \hfill \frac{\mathrm{\partial}\dot{l}}{\mathrm{\partial}l}\hfill \end{array}\right]=\left[\begin{array}{cc}{\mathrm{\theta}}_{2}\frac{{h}_{l}^{2}}{{h}_{ll}}-{\mathrm{\theta}}_{3}& {\mathrm{\theta}}_{2}\\ {\mathrm{\theta}}_{1}\mathrm{\psi}\mathrm{\Omega}\text{}{\left(\frac{P}{W}\right)}^{-2}l& {\mathrm{\theta}}_{1}\left(\mathrm{\psi}\left\{\mathrm{\Omega}\prime \left(l\right)\left[h\left(l,\mathrm{\xi}\right)-{\left(\frac{P}{W}\right)}^{-1}l\right]+\mathrm{\Omega}\left(l\right)\left[{h}_{l}-{\left(\frac{P}{W}\right)}^{-1}\right]\right\}-s{h}_{l}\right)\end{array}\right]$$

$$\text{sgn}\left[\begin{array}{cc}\frac{\mathrm{\partial}{\left(\frac{P}{W}\right)}^{\u2022}}{\mathrm{\partial}\frac{P}{W}}& \frac{\mathrm{\partial}{\left(\frac{P}{W}\right)}^{\u2022}}{\mathrm{\partial}l}\\ \frac{\mathrm{\partial}\dot{i}}{\mathrm{\partial}\frac{P}{W}}& \hfill \frac{\mathrm{\partial}\dot{l}}{\mathrm{\partial}l}\hfill \end{array}\right]=\left[\begin{array}{cc}-& +\\ +& -\end{array}\right].$$

The trace of the Jacobian is negative but the determinant is sign-indefinite. The determinant is positive at

*E*, where the AD schedule cuts the stationary-price locus from below, and negative at

*E*′, where the AD schedule cuts the stationary-price locus from above.

The comparative statics of shifting the underlying schedules are shown in Figure 21. With one exception, these comparative-statics exercises yield results that are qualitatively the same as in a flexprice regime. A decrease in desired saving or an increase in investment demand, pictured in panel (a), increases both employment and inflation. An increase in the conventional wage, as in panel (c), reduces employment but has ambiguous effects on inflation.
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This, like the corresponding flexprice result, supports the conclusion that Bhaduri and I reached to the effect that investment responsiveness to profitability provides a channel through which higher wages depress employment.

The only exception to the previous pattern is a change in the price of energy. As in the flexprice case, an increase in the price of energy shifts the goods-supply schedule to the left and the aggregate-demand schedule inwards. But, unlike the flexprice case, the effect on the real price, inflation, and the level of economic activity is ambiguous. In the fixprice case, as before,

$${\text{GS}}_{\mathrm{\xi}}={\left(\frac{dl}{d\mathrm{\xi}}\right)}_{\text{GS}}=-\frac{{h}_{l\mathrm{\xi}}}{{h}_{ll}}<0$$

$${\text{GS}}_{\frac{P}{W}}={\left(\frac{dl}{d\left(\frac{P}{W}\right)}\right)}_{\text{GS}}=-\frac{{h}_{l}^{2}}{{h}_{ll}}>0$$

and, for capital-widening investment,

$${\text{AD}}_{\frac{P}{W}}={\left(\frac{dl}{d\left(\frac{P}{W}\right)}\right)}_{\text{AD}}=\frac{-\mathrm{\psi}\mathrm{\Omega}\text{}{\left(\frac{P}{W}\right)}^{-2}l}{\text{}\mathrm{\psi}\left\{\mathrm{\Omega}\prime \left(l\right)\left[h\left(l,\mathrm{\xi}\right)-{\left(\frac{P}{W}\right)}^{-1}l\text{}\right]+\mathrm{\Omega}\left(l\right)\left[{h}_{l}-{\left(\frac{P}{W}\right)}^{-1}\right]\right\}-s{h}_{l}\text{}}$$

$${\text{AD}}_{\mathrm{\xi}}={\left(\frac{dl}{d\mathrm{\xi}}\right)}_{\text{AD}}=\frac{-\left(\mathrm{\psi}\mathrm{\Omega}-s\right){h}_{\mathrm{\xi}}}{\mathrm{\psi}\left\{\mathrm{\Omega}\prime \left(l\right)\left[h\left(l,\mathrm{\xi}\right)-{\left(\frac{P}{W}\right)}^{-1}l\text{}\right]+\mathrm{\Omega}\left(l\right)\left[{h}_{l}-{\left(\frac{P}{W}\right)}^{-1}\right]\right\}-s{h}_{l}}.$$

We have

$${\left(\frac{P}{W}\right)}_{\mathrm{\xi}}=\frac{-{\text{GS}}_{\mathrm{\xi}}+{\text{AD}}_{\mathrm{\xi}}}{{\text{GS}}_{\frac{P}{W}}+\frac{{\mathrm{\theta}}_{3}}{{\mathrm{\theta}}_{2}}-{\text{AD}}_{\frac{P}{W}}}.$$

The term

${\text{GS}}_{\mathrm{\xi}}$ measures the direct effect of an increase in the price of energy on the goods-supply schedule. This effect is positive: an increase in the price of energy increases the real price associated with any given level of employment. But there is a second effect on

${\left(\frac{P}{W}\right)}_{\mathrm{\xi}}$ via aggregate demand, reflected in the second term of the numerator. This effect is negative since at equilibrium the denominator in the formula for

${\text{AD}}_{\mathrm{\xi}}$
$$\mathrm{\psi}\left\{\mathrm{\Omega}\prime \left(l\right)\left[h\left(l,\mathrm{\xi}\right)-{\left(\frac{P}{W}\right)}^{-1}l\text{}\right]+\mathrm{\Omega}\left(l\right)\left[{h}_{l}-{\left(\frac{P}{W}\right)}^{-1}\right]\right\}-s{h}_{l}\text{\hspace{0.17em}}={I}_{l}-{S}_{l}$$

is negative, and the numerator is positive.

Figure 21b shows the balance of these two effects as positive – the denominator of

${\left(\frac{P}{W}\right)}_{\mathrm{\xi}}$,

${\text{GS}}_{\frac{P}{W}}+\frac{{\mathrm{\theta}}_{3}}{{\mathrm{\theta}}_{2}}-{\text{AD}}_{\frac{P}{W}}$, is necessarily positive as a condition of equilibrium.

As in previous fixprice exercises, however, the sign of the numerator is not guaranteed. Whether the real price goes up or down depends rather on how much the GS schedule shifts relative to how much the AD schedule shifts. An increase in the price of energy initially adds to inflationary pressures on prices and causes $\frac{P}{W}$ to rise. When the marginal propensity to invest (as a function of the labor–capital ratio), ${I}_{l}$, is much larger than the marginal propensity to save, ${S}_{l}$, this is pretty much the whole story. At a higher real price profits are higher, and as a result there is greater investment demand.

When the difference between the two propensities is small, the initial inflationary effect is countered by the reduction in income caused by the increase in the price of energy. As income falls, so does saving. At the same time, a fall in income leads to a reduction in profit and therefore lower investment demand. With investment demand and saving closely balanced at the margin but with ${I}_{l}-{S}_{l}$ being negative, it takes a substantial reduction in output to bring saving and investment back into line with each other. Once again the idea that exogenous price increases cause stagflation – greater inflation and lower output – turns out to be overly simple. This is the outcome in Figure 21b, but a higher energy price can lead also to Phillips-type outcomes in which the real price and inflation, as well as output, move downward together.

A sufficient condition for the change in employment to be negative is that the real price responds negatively to a change in $\mathrm{\xi}$. The change in employment is given by

$${l}_{\mathrm{\xi}}={\text{AD}}_{\mathrm{\xi}}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{\text{AD}}_{\frac{P}{W}}{\left(\frac{P}{W}\right)}_{\mathrm{\xi}}={\text{GS}}_{\mathrm{\xi}\text{}}+{\left(\frac{P}{W}\right)}_{\mathrm{\xi}}\left({\text{GS}}_{\frac{P}{W}}+\frac{{\mathrm{\theta}}_{3}}{{\mathrm{\theta}}_{2}}\right)$$

so that if

${\left(\frac{P}{W}\right)}_{\mathrm{\xi}}$ is negative, so is

${l}_{\mathrm{\xi}}$. Observe that the converse does not hold.

${l}_{\mathrm{\xi}}$ can be negative even if

${\left(\frac{P}{W}\right)}_{\mathrm{\xi}}$ is positive, provided that

${\text{GS}}_{\mathrm{\xi}\text{}}$, which is necessarily negative, is sufficiently large in absolute value.

### 4.4.2 Capital deepening

Capital deepening – cost-cutting investment – is an entirely different story. Again we start from the generic formula

$$\frac{\mathrm{\partial}Y}{\mathrm{\partial}K}\text{}\dot{K}+\frac{\mathrm{\partial}Y}{\mathrm{\partial}L}\text{}\dot{L}-{\mathrm{\rho}}_{h}\dot{K}-{\left(\frac{P}{W}\right)}^{-1}\text{\hspace{0.17em}}\dot{L},$$

but now we assume that output is fixed so

$$\frac{\mathrm{\partial}Y}{\mathrm{\partial}K}\text{}\dot{K}+\frac{\mathrm{\partial}Y}{\mathrm{\partial}L}\text{}\dot{L}\text{\hspace{0.17em}}=0.$$

Hence the prospective return to a single unit of investment is

$${\left(\frac{P}{W}\right)}^{-1}\text{\hspace{0.17em}}\frac{\frac{\mathrm{\partial}Y}{\mathrm{\partial}K}}{\frac{\mathrm{\partial}Y}{\mathrm{\partial}L}}\text{\hspace{0.17em}}-{\mathrm{\rho}}_{h}={\left(\frac{P}{W}\right)}^{-1}\text{\hspace{0.17em}}\frac{h-{h}_{l}l}{{h}_{l}}\text{\hspace{0.17em}}-{\mathrm{\rho}}_{h}$$

and investment demand becomes

$$i=\mathrm{\psi}\text{\hspace{0.17em}}\left({\left(\frac{P}{W}\right)}^{-1}\text{\hspace{0.17em}}\frac{h-{h}_{l}l}{{h}_{l}}-{\mathrm{\rho}}_{h}\right),$$

with ψ, as before, reflecting animal spirits.

Assuming that saving is a constant fraction of income, the aggregate-demand schedule can be derived in the same manner as we derived Figures 12 and 17. The picture is in Figure 22.

There is a new wrinkle. Unlike capacity-augmenting investment, which is assumed to maintain the labor–capital ratio unchanged, cost-cutting investment is specifically intended to reduce the labor–capital ratio. We have

$$\dot{l}=\left(\frac{\dot{L}}{L}-\frac{\dot{K}}{K}\right)\frac{L}{K}$$

$$\frac{\dot{L}}{L}\text{\hspace{0.17em}}=-\frac{h-{h}_{l}l}{{h}_{l}}\text{\hspace{0.17em}}\frac{\dot{K}}{K}\text{\hspace{0.17em}}\frac{K}{L}=-\frac{h-{h}_{l}l}{{h}_{l}}\text{\hspace{0.17em}}\mathrm{\psi}\left({\left(\frac{P}{W}\right)}^{-1}\text{\hspace{0.17em}}\frac{h-{h}_{l}l}{{h}_{l}}-{\mathrm{\rho}}_{h}\right){l}^{-1}$$

$$\frac{\dot{K}}{K}\text{\hspace{0.17em}}=\mathrm{\psi}({\left(\frac{P}{W}\right)}^{-1}\frac{h-{h}_{l}l}{{h}_{l}}-{\mathrm{\rho}}_{h})$$

so that

$$\dot{l}=-\frac{h}{{h}_{l}}\mathrm{\psi}\left({\left(\frac{P}{W}\right)}^{-1}\frac{h-{h}_{l}l}{{h}_{l}}-{\mathrm{\rho}}_{h}\right).$$

Therefore in a flexprice regime the goods-supply schedule by itself no longer determines

$\dot{l}$ and is no longer the stationary

*l* locus. Instead we have

$$\dot{l}={\mathrm{\theta}}_{2}({h}_{l}-{\left(\frac{P}{W}\right)}^{-1})\text{}l-\frac{h}{{h}_{l}}\left(\mathrm{\psi}{\left(\frac{P}{W}\right)}^{-1}\frac{h-{h}_{l}l}{{h}_{l}}-{\mathrm{\rho}}_{h}\right).$$

In a fixprice regime the stationary-

*l* locus is no longer the aggregate-demand schedule. Instead we have

$$\begin{array}{c}\dot{l}={\mathrm{\theta}}_{1}\mathrm{\psi}\left({\left(\frac{P}{W}\right)}^{-1}\frac{h-{h}_{l}l}{{h}_{l}}-{\mathrm{\rho}}_{h})-sh\right]\text{\hspace{0.17em}}l-\frac{h}{{h}_{l}}(\mathrm{\psi}{\left(\frac{P}{W}\right)}^{-1}\text{}\frac{h-{h}_{l}l}{{h}_{l}}-{\text{}\mathrm{\rho}}_{h})\text{\hspace{0.17em}}\\ =\left({\mathrm{\theta}}_{1}-\frac{h}{{h}_{l}l}\right)\text{\hspace{0.17em}}\mathrm{\psi}\left({\left(\frac{P}{W}\right)}^{-1}\frac{h-{h}_{l}l}{{h}_{l}}-{\mathrm{\rho}}_{h}\text{\hspace{0.17em}}\right)l-{\mathrm{\theta}}_{1}shl.\end{array}$$

The flexprice and fixprice equlibria are given in panels (b) and (c) of

Figure 23.

In a flexprice regime the stationary-*l* locus simply shifts to the left, so that it lies above the GS schedule. The shift is more complicated in a fixprice regime: the shift of the stationary-*l* locus is of ambiguous sign near the origin but unequivocally negative as *l* increases without bound. So as *l* increases, the intersection of investment demand and desired saving is no longer single valued for a given value of the real price. Figure 24a shows the displacement of investment demand as a function of *l*, and Figure 24b reflects this displacement in $\frac{P}{W}\text{}\text{\hspace{0.17em}}\text{x}\text{\hspace{0.17em}}l$ space. (The stationary*-l* locus approaches the aggregate-demand schedule as $l$ goes to zero. The difference between the two schedules becomes infinite as $l$ increases without bound.)

The equilibria in Figures 23b and 23c are strange in that they imply that the boss's right hand does not know what the left hand is doing, and vice versa.
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In both a flexprice regime and a fixprice regime, the right hand is substituting capital for labor, presumably because it is cheaper to produce with more capital and less labor. At the same time, in a flexprice regime, the left hand is hiring additional workers because the price of goods exceeds their marginal cost. In a fixprice the left hand is hiring more workers because goods are flying off the shelves. The right and left hands offset each other with the result that at *E* (and at *E*′) the labor–capital ratio remains stationary over time even as both employment and the capital stock increase.

Examination of the Jacobian matrices reveals what by now has become a familiar pattern. In the flexprice case we have

$$\begin{array}{l}J=\left[\begin{array}{cc}\frac{\mathrm{\partial}{\left(\frac{P}{W}\right)}^{\u2022}}{\mathrm{\partial}\frac{P}{W}}& \frac{\mathrm{\partial}{\left(\frac{P}{W}\right)}^{\u2022}}{\mathrm{\partial}l}\\ \frac{\mathrm{\partial}\dot{l}}{\mathrm{\partial}\frac{P}{W}}& \frac{\mathrm{\partial}\dot{l}}{\mathrm{\partial}l}\end{array}\right]\text{\hspace{0.17em}}\\ \text{\hspace{1em}}=\left[\begin{array}{cc}-{\mathrm{\theta}}_{1}\mathrm{\psi}{\left(\frac{P}{W}\right)}^{-2}\frac{h-{h}_{l}l}{{h}_{l}}-{\mathrm{\theta}}_{3}& -{\mathrm{\theta}}_{1}\left[\mathrm{\psi}{\left(\frac{P}{W}\right)}^{-1}\text{}\frac{{h}_{ll}}{{h}_{l}^{2}}+s{h}_{l}\right]\\ {\mathrm{\theta}}_{2}{\left(\frac{P}{W}\right)}^{-2}+\frac{h}{{h}_{l}}\frac{h-{h}_{l}l}{{h}_{l}}\mathrm{\psi}{\left(\frac{P}{W}\right)}^{-2}& {\mathrm{\theta}}_{2}{h}_{ll}-\frac{{h}_{l}^{2}-{h}_{ll}h}{{h}_{l}^{2}}\mathrm{\psi}\left[{\left(\frac{P}{W}\right)}^{-1}\text{}\frac{h-{h}_{l}l}{{h}_{l}}-{\mathrm{\rho}}_{h}\right]+\frac{h}{{h}_{l}}\mathrm{\psi}{\left(\frac{P}{W}\right)}^{-1}\frac{{h}_{ll}h}{{h}_{l}^{2}}\end{array}\right]\\ \text{\hspace{1em}}=\left[\begin{array}{cc}-& +\\ +& -\end{array}\right]\end{array}$$

with the result that in

Figure 23b
*E* is stable and

*E*′ is unstable.

In the fixprice case

$$\begin{array}{l}J=\text{\hspace{0.17em}}\left[\begin{array}{cc}\frac{\mathrm{\partial}{\left(\frac{P}{W}\right)}^{\u2022}}{\mathrm{\partial}\frac{P}{W}}& \frac{\mathrm{\partial}{\left(\frac{P}{W}\right)}^{\u2022}}{\mathrm{\partial}l}\\ \frac{\mathrm{\partial}\dot{l}}{\mathrm{\partial}\frac{P}{W}}& \frac{\mathrm{\partial}\dot{l}}{\mathrm{\partial}l}\end{array}\right]\\ \text{\hspace{1em}}=\left[\begin{array}{cc}{\mathrm{\theta}}_{2}\frac{{h}_{l}^{2}}{{h}_{ll}}-{\mathrm{\theta}}_{3}& \text{}{\mathrm{\theta}}_{2}\\ -\left({\mathrm{\theta}}_{1}-\frac{h}{{h}_{l}}\right)\mathrm{\psi}{\left(\frac{P}{W}\right)}^{-2}\frac{h-{h}_{l}l}{{h}_{l}}& -\left({\mathrm{\theta}}_{1}-\frac{h}{{h}_{l}}\right)\mathrm{\psi}{\left(\frac{P}{W}\right)}^{-1}\frac{{h}_{ll}}{{h}_{l}}-{\mathrm{\theta}}_{1}s{h}_{l}+\left(\frac{{h}_{ll}h}{{h}_{l}^{2}}-1\right)\mathrm{\psi}\left[{\left(\frac{P}{W}\right)}^{-1}\frac{h-{h}_{l}l}{{h}_{l}}-{\mathrm{\rho}}_{h}\right]\end{array}\right]\\ =\left[\begin{array}{cc}-& +\\ -& +\end{array}\right].\end{array}$$

In Figure 23c
*E*′ is once again unstable whereas *E* is stable only if the speed of adjustment of prices and wages, reflected in the parameters ${\mathrm{\theta}}_{2}$ and ${\mathrm{\theta}}_{3}$, is sufficiently rapid relative to the speed of adjustment of employment, ${\mathrm{\theta}}_{1}$, that the trace is negative.

The economics behind the new wrinkle deserve some attention. The equilibria in Figures 23b and 23c reflect both the continuing downward pressure on employment caused by the substitution of capital for labor and the continuing incentive to create new jobs resulting from either the excess of price over marginal cost (in a flexprice regime) or the excess of expenditure over income (in a fixprice regime). At equilibrium the pressures to destroy and create jobs are equipoised so that the labor–capital ratio remains stationary.

The comparative statics of parameter changes are similar in most respects to what has already been encountered. Figures 25–27 show, respectively, the effect of a higher level of investment demand (or a lower level of desired saving); the effect of a higher energy price; and, finally, the effect of a higher conventional wage.

As has been the case right along, an increase in investment demand or a decrease in desired saving, pictured in Figure 25, expands employment and output and raises inflation in both flexprice and fixprice regimes. The effect of a higher price of energy also parallels earlier results. In the flexprice case, the outcome is unambiguous, higher energy prices invariably leading to diminished economic activity and a lower rate of inflation, as in Figure 26b. By contrast, in the fixprice regime, both the real price and employment can go up or down. Indeed, $\frac{P}{W}$ and $l$ can move in opposite directions, as in Figure 26c, or they can move together in the same direction.

A change in the conventional wage, pictured in Figure 27, demonstrates the importance of whether investment is capacity-augmenting or cost-cutting. In a flexprice regime, as in Figure 27b, a higher conventional wage is contractionary with respect to employment and of ambiguous sign with regard to inflation.
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This is the same as the earlier cases. The surprise is that under a fixprice regime, an increase in the conventional wage is associated with a *higher* level of economic activity. The reason is that a higher conventional wage drives the equilibrium real wage up, and a higher real wage means more investment demand since the higher the real wage the more profitable it is to substitute capital for labor. This is of course the opposite of capital widening, in which investment demand falls as the conventional wage rises and the profit rate falls.

Observe that as Figure 27c is drawn, a higher conventional wage is not only expansionary, it is also inflationary, that is, the new equilibrium *F* is associated with a higher level of inflation than the original equilibrium *E*. But this is a consequence of assuming that the original equilibrium is on the rising portion of the stationary-*l* locus. In case the equilibrium lies on the falling portion, a higher conventional wage continues to be expansionary but is now associated with a lower rate of inflation. We have

$$\frac{d\left(\frac{P}{W}\right)\text{}}{d{\left(\frac{P}{W}\right)}^{*}}=\frac{\left({\mathrm{\theta}}_{1}-\frac{h}{{h}_{l}}\right)\mathrm{\psi}{\left(\frac{P}{W}\right)}^{-1}\text{}\frac{{h}_{ll}}{{h}_{l}}+{\mathrm{\theta}}_{1}s{h}_{l}-\left(\frac{{h}_{ll}h}{{h}_{l}^{2}}-1\right)\mathrm{\psi}\left[{\left(\frac{P}{W}\right)}^{-1}\text{}\frac{h-{h}_{l}l}{{h}_{l}}-{\mathrm{\rho}}_{h}\right]{\mathrm{\theta}}_{3}}{\mathit{\text{det}}J}$$

so

$$\begin{array}{c}\frac{d\left(\frac{P}{W}\right)\text{}}{d{\left(\frac{P}{W}\right)}^{*}}-1=\\ \text{}\frac{{\mathrm{\theta}}_{2}}{\mathit{\text{det}}J}\left\{\left({\mathrm{\theta}}_{1}-\frac{h}{{h}_{l}}\right)\mathrm{\psi}{\left(\frac{P}{W}\right)}^{-1}\text{}\frac{{h}_{ll}}{{h}_{l}}+{\mathrm{\theta}}_{1}s{h}_{l}-\left(\frac{{h}_{ll}h}{{h}_{l}^{2}}-1\right)\mathrm{\psi}\left[{\left(\frac{P}{W}\right)}^{-1}\text{}\frac{h-{h}_{l}l}{{h}_{l}}-{\mathrm{\rho}}_{h}\right]\frac{{h}_{l}^{2}}{{h}_{ll}}-\left({\mathrm{\theta}}_{1}-\frac{h}{{h}_{l}}\right)\mathrm{\psi}{\left(\frac{P}{W}\right)}^{-2}\frac{h-{h}_{l}l}{{h}_{l}}\text{}\right\},\end{array}$$

which is necessarily negative since

$$\mathit{\text{det}}J>0$$

$$-\frac{{h}_{l}^{2}}{{h}_{ll}}\text{\hspace{0.17em}}>0$$

and, as the slope of the stationary-

$l$ locus relative to the

*l* axis, the ratio

$$\frac{-\left({\mathrm{\theta}}_{1}-\frac{h}{{h}_{l}}\right)\mathrm{\psi}{\left(\frac{P}{W}\right)}^{-1}\text{}\frac{{h}_{ll}}{{h}_{l}}-{\text{}\mathrm{\theta}}_{1}s{h}_{l}+\left(\frac{{h}_{ll}h}{{h}_{l}^{2}}-1\right)\mathrm{\psi}\left[{\left(\frac{P}{W}\right)}^{-1}\text{}\frac{h-{h}_{l}l}{{h}_{l}}-{\mathrm{\rho}}_{h}\right]}{\left({\mathrm{\theta}}_{1}-\frac{h}{{h}_{l}}\right)\mathrm{\psi}{\left(\frac{P}{W}\right)}^{-2}\frac{h-{h}_{l}l}{{h}_{l}}}\text{}$$

is negative along the falling portion.