Economics 312/702 - Macroeconomics: Problem Set 2
Q1. Consider a variation on the Solow model where the savings rate is variable instead of constant. Suppose that as usual output is produced competitively via a Cobb-Douglas production function: Y = KαN1-α, and the population grows at the constant rate n > 0. Suppose that there is no depreciation or productivity growth (δ = g = 0), and that total savings is given by St = s(r)Yt, where:
s(r) = s¯rφ
where we introduce the constants s¯ > 0 and φ > 0. Thus s(r) is increasing in r, so that higher real interest rates induce higher savings rates.
(a) Solve for the steady state equilibrium per-worker quantities of capital, output, and consumption (as a function of α, φ, s¯, n) if φ < α/1-α.
(b) What are the effects of an increase in the interest elasticity of savings φ on the per-worker quantities of capital, output, and consumption? Consider both the short-run and long-run, continuing to assume φ < α/1-α even after the increase.
(c) Now suppose φ = α/1-α. How does this change the model? What are the effects of an increase in the saving fraction ¯s on capital and output now?
Q2. This question uses the optimal growth model to analyze the effects of immigration. For simplicity suppose that there is no productivity growth, that the population grows at a constant rate n, and depreciation is the constant δ. Suppose that the economy is initially in the steady state.
(a) Now suppose that there is a one-time increase in the labor force from immigration (N' > N), but the population growth rate n remains constant. Analyze the short-run and long-run effects of this change for the levels of per-capita capital, consumption and output, and the growth rates of (total) output and per-capita output.
(b) Now suppose instead that there is an increase in immigration as a continuing process, so that n increases to a higher value n'. Analyze the short-run and long-run effects of this change for the levels of per-capita capital, consumption and output, and the growth rates of (total) output and per-capita output.
(c) If wages are equal to the marginal product of labor (as they would be in a competitive equilibrium), how are they affected in by immigration? Do your answers differ if the increase in labor is a one-time change or an ongoing process?
Q3. An individual lives for three periods, and has preferences given by:
u(c1) + βu(c2) + βδu(c3),
where both β, δ are constants between zero and one. The interpretation is that she discounts one period into the future at β and two periods forward at βδ. She has a given total lifetime wealth with present value y and so must meet the budget constraint:
c1 + c2/(1 + r) + c3/(1 + r)2 = y.
(a) Using the first order conditions of her utility maximization problem, find an (Euler) equation relating u'(c1) to u'(c2) and another equation relating u'(c2) to u'(c3).
(b) In period 2, the individual rethinks her choice of consumption and savings in period 2 (and hence her choice of consumption in period 3). She now maximizes
u(c2) + βu(c3)
subject to the present value budget constraint. Note that she treats c1 as fixed, since it is already determined. Again find a relationship between u'(c2) and u'(c3), and compare this with your answer above. When will they be the same?
Q4. Consider an infinite horizon model of consumption and savings where consumers have "habits", meaning that they care about consumption relative to their own past consumption. Thus preferences are given by:
t=0∑∞βtu(ct - ct-1)
Suppose the consumers face a constant interest rate and so face the flow constraint:
ct + at+1 = xt + (1 + r)at
where at are assets and xt is income as of date t
(a) Write down the Lagrangian for the consumer's optimization problem, and find the first order conditions for the choice of consumption at arbitrary dates t and t + 1.
(b) How does the Euler equation with habits compare to the case (as in class) without habits?
(c) As in class, suppose that income and hence consumption are now random, so introduce expectations appropriately. Further, suppose β(1 + r) = 1 and u(ct - ct-1) = ct - ct-1 - a/2(ct - ct-1)2.
What does the Euler equation look like now? Now what information at date t helps predict consumption at t + 1?