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Macroeconomics/Macroeconomics III

Winter Quarter 2021

Final Examination

1. The social planner maximizes welfare of the representative household’s utility:

where β e (0, 1) under the resource constraint of

The production function in this economy is Leontief-type such that

The initial capital stock is K0 > 0. The population, Lt, grows at the rate of n. We assume

(a) Derive the steady state levels of capital per capita and consumption per capita. Explain how this economy converges to the steady state.

(b) Solve for ct and kt on the transition dynamics toward the steady state for a given initial per- capita capital, k0 , and the optimal initial consumption per capita, c0 .

(c) The economy is initially at the steady state. Suppose that productivity of capital, a, unan- ticipatedly increases permanently. Explain the change of steady state levels of capital and consumption per capita. Illustrate the dynamics of the economy in response to the shock.

2. Suppose that each household lives for 2 periods: young and old. They work, consume and save when young; and they just consume when old. There is no bequest so that the initial asset for each household is zero. Assuming log utility, the optimization problem for a young household born in t is

Firms have the following production function:

where kt and et are ﬁrm-level capital and labor, respectively. Kt(y) represents the externality eﬀect such that productivity of individual ﬁrms positively depends on aggregate capital. We assume perfect competition and ﬁrms are symmetric. Moreover, we assume that the total number of ﬁrms is 1; no population growth so that L1t = L2t = 1 Vt; the depreciation rate is δ e (0, 1).

(a) Taking wt and rt+1 as given, ﬁnd the optimal saving for the generation born at the beginning of t, or at+1 .

(b) Derive wt and rt+1 in equilibrium.

(d) Find the conditions for the economy to have a balanced growth path with a strictly positive growth rate in output per worker.

3. The ﬁnal-good ﬁrm produces output Yt by choosing a combination of intermediate inputs xt(i) (i = 1, 2, . . . , n) so as to maximize proﬁt:

using the CES-form aggregation technology:

Here, σ represents the elasticity of substitution, ε is a parameter to indicate love for variety, and the price of product i in period t is denoted by pt(i) .

(a) Solve the quantity demanded xt(j) (j = 1, 2, . . . , n) using pt(j), Yt , and Pt .

(b) Log-linearize xt(j) (denote by t(j)) and express it using log-linearized pt(j), Yt , and Pt .

(d) Given product symmetry (xt(i) = xt and pt(i) = pt for all i), write Pt using pt . (e) Calculate ε so that the eﬀect of n on Pt disappears.

4. Suppose that inﬂation rate πt and output xt are given by the backward-looking Phillips curve and the IS curve, respectively:

where εt and µt are independent and identically distributed shocks, which are uncorrelated each other

and means are zero (i.e., Et[εt+i] = Et[µt+i] = Et[µt+iεt+j] = 0 for i, j = 1, 2, . . . ).

(a) Answer whether each of πt and xt is a forward- or backward-looking variable.

(b) Suppose κ = 0. Then, answer the condition for πt to be unique and stationary (determinate). Note: no need to answer for xt .

(c) Express the equations as in equation (1) in the lecture note of “Solving Rational Expectations Equilibrium,” where the left-hand side of the equation is the vector of (πt, Etxt+1)/ .

(d) Derive the characteristic equation for the eigenvalues (denote by λ) of the matrix (B in the lecture note) that determines determinacy.

(e) Suppose σκ 0. Solve the condition for equilibrium to be unique and stationary (determinate).