Advanced Macroeconomics, Fall 2023 Problem Set 2
Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit
Advanced Macroeconomics, Fall 2023
Problem Set 2
Due at 11:59 pm on Friday, October 27, 2023
1 A Simple, Two-Period Consumption Savings Problem
Consider the two-period model of consumption and saving discussed in Lecture Notes 3. The household lives for two periods and its preferences are given by:
U (c0 ) + βU (c1 )
where β 2 (0, 1) is the discount factor and U : R+ ! R is the per-period utility function. We assume U is Neoclassical, i.e. it is continuous, twice-differentiable, strictly increasing, strictly concave, and satisfies the Inada conditions. The household maximizes lifetime utility subject to its sequence of budget constraints:
c0 + a1 =w0 + (1 + r)a0 (1)
c1 =w1 + (1 + r)a1 (2)
where wt is labor income and at+1 is asset holdings. We assume the real interest rate r is constant across periods. We furthermore assume a0 ≥ 0 is given.
1. Let q0 > 0 denote the price of consumption at time 0 and let
q1 = 
> 0
Derive the household’s intertemporal budget constraint.
2. Take first-order conditions and derive the household’s intertemporal Euler equation. Provide intuition as to what this equation means.
3. For the remainder of this problem, assume that 1 + r = 1/β and that initial asset holdings are zero: a0 = 0.
Solve for the optimal consumption path of the household.
4. Derive conditions on the income path (w0 , w1 ) such that the household optimally chooses to borrow in period 0.
5. Most people earn more income in their 40s and 50s than in their 20s and 30s. That is, generally speaking, lifetime earning profiles are upward-sloping (up until retirement). What does this model imply for the optimal consumption and savings behavior of people in their 20s and 30s?
2 TheNGM with Learning-by-Doing
Consider the Neoclassical Growth Model discussed in Lecture Notes 4. In this problem we will extend it with a version of “Learning-by-Doing.”
We assume that as firms produce goods, they think of ways of new ways of improving the production process. That is, we introduce a new variable we call “human capital.” The accumulation of human capital occurs not as a result of direct investment in human capital but instead as a side effect of conventional economic activity. This type of knowledge accumulation is known as learning-by-doing.
Specifically, let t 2 f0, ..., Tg. We assume the household’s per period flow utility is CEIS:
U(c) = 
and the household has discount factor β 2 (0, 1). There is no population growth. The production function is given by:
Yt = Kt(α)(htLt)1-α
with α 2 (0, 1) and ht is our new variable called “human capital.” Written in intensive form:
y t = f(kt, ht) = kt(α)ht(1) -α .
We assume that human capital accumulates according to:
ht = bkt
where b > 0, that is, human capital accumulation is a by-product of the accumulation of physical capital. The resource constraint is given by:
ct + kt+1 = f(kt, ht) + (1 - δ)kt , At 2 f0, ..., Tg
and we have the typical non-negativity constraints on consumption and capital:
ct ≥ 0, kt+1 ≥ 0, At 2 f0, ..., Tg.
Initial capital k0 > 0 is given.
1. Write down the social planner’s problem in this economy.
2. Take the planner’s FOCs and provide the conditions that characterize the socially optimal allocation of consumption and capital, {ct , kt+1} Tt=0, in this economy.
3. How and why does the planner’s Euler equation in this economy differ from the planner’s Euler equation in the standard Neoclassical Growth Model? Provide intuition.
3 NGMwith Climate Change: The Planner’s Problem
In this problem we will consider an extension of the Neoclassical Growth Model with carbon emissions and climate change.
Time is discrete and finite, with periods denoted by t 2 f0, ..., Tg. There is no population growth. The representative household has the following lifetime utility over consumption:
βtU(ct) (3)
with discount factor β 2 (0, 1) and per-period utility function U(c). We assume that U satisfies standard regularity conditions.1
The final good is produced using labor L, “clean” capital K, and “dirty” capital X, according to the following Cobb-Douglas production function:
Yt = AtKt(α)Xt(Lt)1-α -
where At is Total Factor Productivity (TFP) at time t, α 2 (0, 1), γ 2 (0, 1), and α + γ < 1. Written in intensive form:
y t = Atkt(α)xt
where k = K/Land x = X/L.
We assume that the use of dirty capital in the production process leads to carbon emissions while the use of clean capital does not. Carbon emissions, in turn, brings about climate change, and climate change lowers TFP. We represent this relationship in a simple way as follows:
At =A(-)xt(-)η
with A(-) > 0 and η 2 (0, γ). That is, TFP is strictly decreasing in the use of dirty capital. The resource constraint for the economy is given by:
ct + kt+1 + xt+1 = Atkt(α)xt + (1 - δk )kt + (1 - δx )xt , (4)
where δk 2 (0, 1) and δx 2 (0, 1) denote the depreciation rates of clean and dirty capital, respectively. Finally, we have the typical non-negativity constraints:
ct ≥ 0, kt+1 ≥ 0, xt+1 ≥ 0, At 2 f0, ..., Tg.
Initial clean and dirty capital, k0 > 0 and x0 > 0, are exogenously given.
1. The social planner chooses an allocation,
{ct , kt+1, xt+1} Tt=0,
consisting of consumption, clean capital, and dirty capital in every period. Write down the social planner’s problem in this economy.
2. Take the planner’s FOCs and provide the conditions that characterize the socially optimal allocation in this economy. Note that there will be two Euler equations, one corresponding to investment in clean capital and the other corresponding to investment in dirty capital.
3. Provide intuition for:
(a) the planner’s Euler equation corresponding to clean capital
(b) the planner’s Euler equation corresponding to dirty capital
4 NGMwith Climate Change: Decentralization
We now consider the competitive equilibrium of the economy described in the previous problem.
The representative household supplies labor inelastically, `t = 1, and has preferences over consumption given by (3). The household owns clean and dirty capital and rents it out to the firms.
Let wt denote the wage rate, rt(k) the rental rate on clean capital, andrt(x) the rental rate on clean
capital. The household faces a sequence of period budget constraints given by:
ct + kt+1 + xt+1 + bt+1 = (1 + rt(k) - δk )kt + (1 + rt(x) - δx )kt + (1 + Rt)bt + wt + Mt ,
where Mt is a lump-sum transfer from the government. The household takes all prices, including wage rates and rental rates, as given. As in the lecture notes, you may assume an ad hoc borrowing constraint on bond holdings:
bt+1 ≥ b, At 2 f0, ..., T - 1g
with b < 0, as well as the terminal condition on debt:
bT+1 ≥ 0.
Initial clean and dirty capital holdings, k0 > 0 and x0 > 0, as well as initial bond holdings, b0 = 0, are exogenously given.
There exists a government that has the power to tax capital. In particular, we assume that the representative firm is taxed at rate τk when it rents clean capital and is taxed at rate τx when it rents dirty capital. This implies that the firm’s after tax rental rates on clean and dirty capital are given by:
(1 + τk )rk and (1 + τx )rx ,
respectively. With the tax revenue it collects from these two taxes, the government distributes the revenue to the household via the lump sum transfer, Mt. Finally, bonds are in zero net supply in every period.
1. Write down the government’s period t budget constraint.
2. Set up the representative firm’s problem and solve for the equilibrium rental rates and the equilibrium wage rate. Note: the representative firm takes TFP, A t, as given.2
3. Set up the representative household’s problem and derive its optimality conditions. What are the no arbitrage conditions in this problem?
4. Show that the household’s budget constraint, combined with the government budget constraint, market clearing, and equilibrium prices imply the resource constraint (4).
5. Provide the conditions that characterize the competitive equilibrium allocation and prices in this economy given tax rates τk and τx.
6. Solve for the tax rates, τk and τx, that implement the socially optimal allocation. We call these the optimal tax rates.
(a) Is the optimal tax rate on dirty capital, τx, strictly positive, strictly negative, or equal to zero? Provide intuition.
(b) Is the optimal tax rate on clean capital, τk , strictly positive, strictly negative, or equal to zero? Provide intuition.
2023-10-26