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Fall 2022

Econ 501: Macroeconomic Analysis and Policy

Final Exam

Question 1 Solow-Swan Model (25 pts)

Consider the production function F(K,L) = K^αL^1-α (where 0<α<1 and technology is normalized at 1 and does not grow). Suppose that the savings rate is exogenous and fixed at 0.3, the depreciation rate, δ = 0.06, and population growth, n = 0.04.

a) Show that the production function exhibits constant returns to scale. 3pts

b) Derive the per-worker production function y = f(k) 4pts

c) Solve for the steady-state level of capital per worker for α=0.5. 4pts

d) Solve for the steady-state output per worker. 3pts

e) Solve for the steady-state consumption and investment levels. 3pts

Now assume that the savings rate is not fixed anymore.

f) Solve for the Golden-Rule level of capital per worker and the associated consumption level. 4pts

g) Solve for the savings rate that allows for the golden rule level of capital in the steady state. Provide intuition about how the government can intervene to increase consumption. 4pts

Question 2 Ramsey-Cass-Koopmans Model (15 pts)

Consider an economy that follows the assumptions of the Ramsey-Cass-Koopmans model.

a) Write down the k˙ = 0 and c˙ = 0 equations and graph the phase diagram. 5pts

b) How does c and k react immediately after a global pandemic permanently decreases the productivity growth rate g? Explain and graph on the phase diagram from a). 5pts

c) How does c and k react immediately after a proportional upward shift of the production function f(k)? Explain and graph on the phase diagram from a. 5pts

Question 3 Endogenous Growth (20 pts)

The production function for a representative firm i is Y_i = K_i^α(AL_i)^1−α. A represents the level of knowledge in the economy and is perfectly linked to the level of capital in the economy, K. Markets are competitive and there is no depreciation of capital. Households maximize their utility, resulting in the following consumption growth: c˙/c = 1/ θ[αr − ρ] and accumulate assets according to their budget constraint: a˙ = w + ra – c.

a) Write down the representative firm’s profit maximization problem rewriting the production function in terms of the level of capital in the economy K and in per-capita terms. Determine r and w. 5pts

b) Assuming that all the firms are the same, what is the equilibrium K? Rewrite to show that this is an AK model and describe any implications on growth rates. 5pts

c) Obtain the equilibrium r. What would be the equilibrium consumption growth rate and the capital law of motion? 5pts

d) Intuitively, is this consumption growth optimal? Explain why is it or why not 5pts

Question 4 Consumption (20 points)

a) Using the permanent income hypothesis in a two-period setting, derive the consumption for a representative agent that pays Social Security (SS) in the first period and will receive the amount (SS) in the second period as a government transfer to protect those retired. 5pts

b) Will this social security structure increase or decrease consumption (in comparison with no social security administration in the economy)? 5pts

c) How will the consumption choice change if the social security payment is paid with interests ((1+r) SS )? 5pts

d) As a policymaker, which of the two alternatives (with or without SS) would you choose in order to stimulate the economy? Explain. 5pts

Question 5 Unemployment (20 points)

a) According to the Mortensen-Pissarides Model and assuming that bargaining power is equal between employers and workers and that Vv=0 (meaning that V_E − V_U = V_F), write down the value functions for the worker and the firm and find the equilibrium wage as a function of the parameters a, b, r and α, and output. 8 pts

b) Who (employer or worker) keeps a higher share of the output when a<α? Explain. 5 pts

c) Now assume that after Covid, workers realize that they have more options and twice more bargaining power than firms. How does the model change and what are the implications for the wage determination? 7 pts