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Department of Economics Queen’s University
ECON 222: Macroeconomic Theory I Fall 2024
Assignment # Due: 10pm, Sunday November 3, 2024
Question 1 (10 marks): Consider the goods market equilibrium of a closed economy. Ex- plain the impact on equilibrium investment and the interest rate resulting from the following scenarios: (a) Total factor productivity rises today and remains persistently high in the future. (b) An increase in government consumption that is entirely financed by an increase in the effective tax rate on capital income.
Question 2 (15 marks): A large open economy has desired national saving given by
Sd^ = 20 + 200r
and desired national investment given by
Id^ = 30 − 200 r.
The foreign economy has desired national saving given by
SF ord = 40 + 100r
and desired national investment given by
IF ord = 75 − 400 r.
(a) Calculate the global equilibrium world interest rate rw and the implied equilibrium values of S, I, SF or, IF or and the current accounts for each country, CA and CAF or. (b) Suppose Sd^ rises by 45, so that now Sd^ = 65 + 200r. Calculate the equilibrium values of rw, S, I, SF or, IF or, CA and CAF or. (c) Suppose, with Sd^ as in part (a), that Id^ rises by 45 to Id^ = 75 − 200 r. Calculate the equilibrium values of rw, S, I, SF or, IF or, CA and CAF or.
Question 3 (25 marks): Consider the following basic version of the Solow growth model. Suppose the relationship between output and productive inputs at any point in time is represented by Y = K^0.^5 (AN )^0.^5.
Assume that population growth is n = 0.1, the savings rate is s = 0.2, and the rate of depreciation of capital is d = 0.1. (a) Suppose initially that A = 1, K = 10, 000 and N = 1000. What are the initial values of capital per worker, k 0 , and output per worker, y 0? (b) If A is constant, what are the steady state values of capital per worker and output per worker? (c) Explain, with the aid of a diagram, what happens to the growth rate of output per worker as the economy approaches the steady state.
Now suppose that productivity growth is ongoing so that A grows at the constant rate g = 0.15. (d) What are the steady state values of capital per effective worker, k e∗ , and output per effective worker, y e∗? (e) At what rate does output worker grow in the steady state in this case?
Question 4 (25 marks): Consider a variant of the Solow model augmented with endoge- nous human capital accumulation. Output per worker is given by
Y = AKαHβ^ N 1 −α−β^.
where A is a constant and α + β < 1. Here output depends on the stock of physical capital, K, the stock of human capital, H, and the number of workers, N. The workforce grows at constant rate n and both physical and human capital are assumed to depreciation at the same rate d. The stock of physical capital per worker evolves according to
∆k = sky − (n + d)k
where sk is the share of output invested in physical capital. The stock of human capital per worker evolves according to ∆h = shy − (n + d)h,
where sh is the share of output invested in human capital (e.g. through education). (a) Show that output per worker can be expressed as
y = Akαhβ
For the rest of this question, assume that A = 1, α = β = 13 , n = 0.02, d = 0.08, sk = 0. 2 and sh = 0.1. (b) In a steady-state, ∆k = ∆h = 0. Show that the steady state can be characterized by two equations each of which can be expressed in terms of k as a function of h. (c) Illustrate the curves for these two equations on a diagram with k on the vertical axis and h on the horizontal axis. Explain why you have drawn the two curves this way. (d) Based on your diagram, explain why there can only be one positive pair of steady state values (k∗, h∗) that satisfy both equations. (e) What are the steady state values of physical and human capital per worker? What is output per worker in the steady state?
