Describe and graph how each of the following affect the k = 0 and C = 0 locus. How does c and k react immediately after change and what is the new steady state? Note: k and c will denote per intensive unit of labor (same as k and c in the lecture slides).

First note that we have the dynamics of capital stock and consumption(per intensive unit of labor) as follows:

k = f (k) — (g + n + d)k — c

^c _ ^{f/(k) —} 3 ^— p ^— 0g

C = 6

(a) [5pt] Permanent rise in productivity growth rate g.

Answer: From k = 0, we have

c = ^{f (k) — (}g + n + ^{3)k (1)}

A rise in g makes the level of c consistent with k = 0 lower for any k. Then k = 0 shifts down. From C = 0, we have

f/(k) = 3 + p + 旳 (2)

A rise in g increases f/(k) and thus decreases k. Then C = 0 shifts left. When the shock happens, k keeps unchanged(predetermined by the economy) and the level of c goes down(when the new saddle path is lower), and then k and c move to the new steady state where c大 and k大 are both lower than before.

(b) [5pt] Rise in the preference for today's consumption 6

Answer: There is no preference parameter showing in k, thus k = 0 does not change. Let's look at equation (2). A rise in 0 implies a higher f/(k) for = 0. Due to the concavity of f (), the higher level of f/(k) would happen at a lower level of capital, so the curve shifts to the left.

When the shock happens, consumption will increase a little bit to be on the new saddle path, at the end, k大 and c大 will be lower than before.

(c) [5pt] Proportional downward shift of production function f (k).

Answer: A downward shift of f (k) leads to lower output and f/(k) for any level of k. The k = 0 curve will shift down. The C = 0 curve will move left since f/(k) is lower now, k needs to decrease to keep the C = 0 relationship.

The immediate effect on c is unclear(need more information to pin down the exact saddle path), and as usual, k does not move immediately. As time pass by, the new steady state level will be lower for both c大 and k大.

(d) [5pt] Decrease in the depreciation rate of capital 6

Answer: A decrease in 6 will raise the k = 0 curve. In addition, f/(k) will decrease, C = 0 curve will shift right.

The immediate effect on c is unclear and k does not move immediately. As time pass by, the new steady state level will be higher for both c大 and k大.

Question 2

Consider an economy with infinitely lived representative households which provide labor services in exchange for wages, receive interest income on assets, purchase goods for consumption and save by accumulating additional assets. Thus, the representative household maximizes its lifetime welfare:

产 产(c )1-9

u(c"e^-(^o-n)以= e^-(^p-n)tdt

Jo Jo ¹ — ⁰

subject to its flow budget constraint and the No-Ponzi-Game condition, where n is the rate of population growth, 0 > 0, p > 0 and p > n. Assume further that the government purchases per capita are gt = Gt/Lt, which are financed by a constant tax on consumption 1 > Tc > 0, and the government budget is balanced. The productive sector of the economy has competitive firms which produce goods, pay wages for labor input and make rental
payments for capital inputs. The firms have neoclassical production function, expressed in per capita terms y芒=Akf where 0 < a < 1 and capital depreciates at the rate 6 > 0.

(a) [5pt] Specify the household's dynamic optimization problem

Answer:

Let's derive the budget constraint first. In this case, total assets satisfy A芒= L_tw_t + L貞芒a芒— c芒L芒 — T_cCtLt, let at = At, we have

-^dAt ¹ . ^dLt _₂

Lt ^{— At}!T^Lt

^At ^Lt

^—匸L

—nat

今 dt = wt + rtat — Ct — TC — nat

The Household wants to do the following optimization:

s.t. equation (3), ao given

(b) [10pt] Derive the first order conditions of the household's optimization problem.

Answer:

A present-value Hamiltonian is:

(c ) 1_9

^H —— ^e_(°_^ra)t + Rt^(wt + ^(rt ^{— n)a}t ^— (1 + ^Tc)^Ct)

(c) [5pt] Obtain the Euler equation.

Answer:

From (4), we have

成=[项(q)_⁹_i(C"e_i _ (p - n)e_S(q)_9]

thus

—=-9(c"T(潟-⁽p - ⁿ⁾ (6)

〃芒

Combine (6) & (5), we have

^u = n — r 芒=—Q — — p + n

ut 芒 ct ¹

今 Q— = r芒 — p (EE)

以

(d) [5pt] Write down government's budget constraint, the government spending per capita, and g.

Answer: Government's budget constraint is:

^Tc^ct^Lt = ^Gt

今 ^Tc^ct = ^gt

今 9t = ^C^ct

(e) [3pt] How does the tax affect the consumption choice?

Answer: By equation (EE), we have

幺Q = rt - p (7)

ct

From (7) we can see that the consumption tax does not affect the consumption choice.

(f) [5pt] Write down and solve the problem of a profit-maximizing representative firm. Using the results above specify the competitive market equilibrium.

Answer:

The representative firm wants to

max Ht = Lt[Ak^a — (r + 6)kt — wt]

kt,Lt

taking first order conditions w.r.t k, we have

LtaAkf^-1 = L芒(r芒 + 5)今 r芒=aAk^^-1 — 5

taking first order conditions w.r.t L, we have

dk+

^wt = /^(k芒)+ ^Lt//^(k芒)dL = ^Ak? ^{— aAk}t ^±kt = ^{(1 — a)Aka}

In the equilibrium, d_t = k芒,combining prices and (3), we have

k 芒=(1 — a)Ak£ + aAk£^-1k 芒—(5 + n)k 芒—(1 + r_c)ct

今 k芒=Akf — (5 + n)k芒—(1 + r_c)ct

When combining the prices and (7), we have

Q = ¹⁽顽尸^{—5 — P)}

(g) [5pt] Derive the conditions for the steady-state level of capital and consumption per capita and draw the phase diagram.

Answer: Using (8) and (9), at steady-state, we must have k = 0 and = 0, which means:

Aka — (5 + n)kt — (1 + r_c)ct = 0

aAk?T = 5 + p

Then we can solve for steady-state capital and consumption per capita using the above two equations. The phase diagram see the picture at the end.

(h) [2pt] Find the value of k大 for a = 0.5, A = 4, 5 = 0.4, and p = 0.6. Answer: From (g), we have

Plugging in the numbers, we will end up with k大 = 4.

(i) Assume that the economy is initially at a steady state with k大 and c大 > 0. What are the effects of a temporary increase in government purchases on the paths of consumption, capital and interest rate (draw their behavior over time).

Answer: When there is a temporary increase in government purchases, r_c must increase because of balanced budget. From (8) and (9), we can see that & locus doesn't change and k芒 locus shifts down.

Since government increases the spending, the consumption must go down. However, it cannot go as much as the increase of government spending (if they do, there will be a jump in consumption(and hence marginal utility) when government spending back to normal, since the return of government spending is anticipated, the discontinuity in marginal utility will not be optimal).

Thus, at the time the government spending increases, consumption goes down and then gradually increases. While the capital per capita goes down for a while. Once the government purchases return to previous level, it will be on the old saddle path and increase to the old steady state. The interest rate will move oppositely with capital. It will go up and then decrease to the previous steady-state level.

See the associated graphs below.

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