Intermediate Macroeconomics, Sciences Po, 2014 Zsófia Bárány Answer Key to Problem Set 10 Dynamic Micro-founded Macro Model 1. Increase in future government spending in the dynamic macro model: Consider a two-period macroeconomic model with a representative household, a representative firm, and a government. The representative household makes consumption-saving and labor supply decisions. The representative firm makes labor demand and investment decisions. (a) Define the competitive equilibrium of this economy by stating (i) the optimality conditions of the representative household s and of the representative firm s optimization problems, and (ii) the market clearing conditions for equilibrium wages and real interest rate. Household optimization requires that that the marginal rate of substitution between any two goods is equal to the corresponding relative price, i.e.: * intra-temporal consumption/leisure choice MRS l,c = w and MRS l,c = w * inter-temporal consumption choice MRS C,C = 1 + r * inter-temporal leisure choice MRS l,l = (1+r)w w and that the inter-temporal budget constraint C + C = w(h l) + 1+r π T + w (h l )+π T holds. 1+r Optimization by the representative firm requires equating marginal benefit to marginal cost for each choice variable: * (N d ): MP N = w * (N d ): MP N = w * (K ): MP K d = r This implies an investment demand I d = K (1 d)k. The firm then makes profits π = Y s wn d I d in the first period and π = Y s w N d + (1 + r)k in the second period. Government spending must satisfy the government s inter-temporal budget constraint G + G = T + T. 1+r 1+r 1
The market clearing conditions require that labor supply is equal to labor demand and that output demand Y d = C d + I d + G is equal to output supply Y s = zf (K, N). The wage rate and the interest rate will adjust to clear these markets. By Walras law, then the credit market will be cleared as well. (b) The government announces that an increase in government expenditure will occur next period. Use the model to study the effects on current aggregate output and current employment. Explain your results. Figure 1: Higher Future Government Spending A future increase in government spending implies an increase in the present value of taxes, either through higher taxes today or in the future. This generates a negative wealth effect, and thus both current consumption and current leisure fall (at given prices r and w). The fall in current consumption shifts the output demand curve to the left. The fall in current leisure increases the current-period labour supply, which shifts the output supply curve to the right. The shifts of Y s and Y d imply a fall in the interest rate. The lower real interest rate generates a inter-temporal substitution effect, between current and future consumption and between cur- 2
rent and future leisure. The inter-temporal substitution of consumption induced by the lower interest rate increases current consumption (and reduces future consumption) this is incorporate in the (downward) slope of Y d, i.e. a movement along the curve. Similarly, the inter-temporal substitution of leisure caused by the reduction in the interest rate decreases current labour supply (a leftward shift of N s ), and by its implied reduction of equilibrium employment reduces output supply Y s, which is a movement along the Y s curve (recall the derivation of the Y s -curve!). Without making any further assumptions, it is ambiguous whether current output and employment increase or decrease. It is likely that the indirect effect through the interest rate change is smaller than the initial direct effect of lower lifetime wealth. Figure 1 shows this case. Under this assumption, the overall effect will be higher labour supply and an increase in employment. Higher current employment would necessarily imply higher current output, since Y = zf (K, N). Note: We can also derive the effect on other variables of interest such as current consumption and investment. For current consumption, we have already shown that the direct effect is negative while the indirect effect through real interest rate is positive. Again, assuming the direct effect dominates, current consumption is likely to fall. The only effect on investment is through the indirect effect from the lower real interest rate, which implies a lower user cost of investment. As a result, investment increases (which is incorporated in the movement along the Y d curve). 2. Productivity shocks in the dynamic macro model: Consider again the model of question 1. (a) Analyze the effects of a very short-lived temporary positive shock to total factor productivity (TFP) on output, employment, consumption, investment, the real wage and the real interest rate. Because the shock is very short-lived you should assume that its impact on lifetime wealth is negligibly small. A positive TFP shock in the current period: * Increases the current marginal product of labour, so current labour demand shifts right * Increases current output supply, directly due to technology and through its effect on employment, so the current output supply curve shifts right 3
Figure 2: Short-lived increase in TFP * There is also a positive wealth effect on consumption and a negative wealth effect on labour supply. Both are very small when the shock is very short-lived and the representative household wants to smooth consumption and leisure in response to the proceeds in lifetime wealth. For simplicity, the wealth effects are not shown in figure 2. Equilibrium effects on variables: * Real interest rate falls * Real wage rises * Output increases * Employment increases (assuming substitution effect from real wage more important than that from the real interest rate) * Consumption rises (small wealth effect + inter-temporal substitution) * Investment rises (user cost of capital lower) (b) It is often argued that shocks are persistent. A persistent shock is a shock that is strongest when it first occurs, but also has a subsequent impact as it fades out only slowly. In the context of our model, a positive persistent TFP shock increases z, but also increases z albeit by a smaller amount. What difference does an increase in the persistence of the TFP shock make to the responses of consumption, investment and the real interest rate relative to what you found in part (a)? You should assume that higher persistent implies a bigger change in future TFP. 4
Figure 3: Additonal Effect of Persistence in TFP shock If the persistence of the TFP shock increases then this means there is a larger effect on future TFP in the same direction as the effect on current TFP. Relative to part (a) this adds to the analysis the following effects: * Higher future TFP will raise future income, thus increasing the present discounted value of lifetime income. This raises desired consumption as now a larger amount of smoothed consumption is affordable, shifting the output demand curve to the right. * There is an increase in the expectation of the future marginal product of capital. This raises investment demand at a given interest rate, shifting the output demand curve further to the right. * A larger rightward shift in output demand implies a smaller fall in the real interest rate than otherwise would be the case or possibly even a rise in the real interest rate which is what is shown in figure 3. The change in the interest rate has further effects on consumption and investment, but overall, the sum of consumption and investment must rise by more than in the case of a temporary TFP shock. It could also be argued that the output supply curve will not shift to the right as much as before (or even shift to the left because of the relatively large wealth effect on leisure). This would reinforce the conclusion concerning the real interest rate, but potentially complicate the analysis of consumption and investment. 5