ECON44815 ADVANCED MACROECONOMICS 2021
Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit
ECON44815
ADVANCED MACROECONOMICS
2021
SECTION A
1. Consider an overlapping generations model economy with production. The Nt people born at time t > 1 have utility function Ut = log ct(y) + β log ct(o)+1 where ct(y) is consumption when young at t, ct(o)+1 is consumption when old at t + 1, β > 0 is a parameter, and log is the natural logarithm. Each person supplies one unit of labour when young. The economy also has N0 old people at t = 1. These people collectively own the initial capital stock K1 and each has a utility function that is increasing in consumption c1(o) . A perfectly competitive profit maximizing firm produces output Yt with the aggregate production technology Yt = F (Kt , At Nt ) = Kt(α)(At Nt )1_α where Kt is the aggregate capital stock at t, At is the level of exogenous labour productivity at t, and α e (0, 1) is a parameter. Assume At+1 = aAt for all t > 1 where a > 1 is a parameter. Assume the population of successive generations evolves according to Nt+1 = nNt for all t > 0 where n > 0 is a parameter. Assume capital depreciates at the constant rate δ e (0, 1). Let k denote capital per unit of effective labour.
(a) Suppose k1 = kgr < kss where gr denotes the golden rule and ss denotes the
laissez faire steady state equilibrium. Characterize the steady state equilibrium. Along the equilibrium time path, does aggregate consumption per unit of effective labour rise or fall? Explain. (à# ≥|力kS)
(b) Now, suppose initially at t = 1 the economy is at k1 = kss and at t = 2 an
earthquake suddenly destroys half of the capital stock (with no change in popula- tion or labour productivity). What is the immediate effect of the earthquake on k2 ? Describe, using diagrams where appropriate, the time path of the economy after the earthquake causes this change in k2 . Discuss the long term economic consequences of the earthquake. (à# ≥|力kS)
2. Consider an overlapping generations model economy with production. The Nt people born at time t > 1 have utility function Ut = log ct(y) + β log ct(o)+1 where ct(y) is consumption when young at t, ct(o)+1 is consumption when old at t + 1, β > 0 is a parameter, and log is the natural logarithm. Each person supplies one unit of labour when young. The economy also has N0 old people at t = 1. These people collectively own the initial capital stock K1 and each has a utility function that is increasing in consumption c1(o) . A perfectly competitive profit maximizing firm produces output Yt with the aggregate
production technology
Yt = F (Kt , At Nt ) = θAt Nt ┌ 1 + log ╱ 、!
where Kt is the aggregate capital stock at t, At is the level of exogenous labour pro- ductivity at t, and θ > 0 is a parameter. Assume At+1 = aAt for all t > 1 where a > 1 is a parameter. Assume the population of successive generations evolves according to Nt+1 = nNt for all t > 0 where n > 0 is a parameter. Assume capital depreciates at the constant rate δ e (0, 1).
Let k = K/(AN) denote capital per unit of effective labour. Note, the production function in intensive form is f (k) = θ(1 + log k); the marginal product of capital is f′ (k) = θ/k; and the marginal product of effective labour is f (k) _ kf′ (k) = θ log k .
(a) Solve for the savings function and use it to derive (with explanation) the equilib-
rium condition that determines the time path {kt }t≥1. Use a diagram to illustrate the dynamics of the equilibrium time path. Assume θ is sufficiently large that there are two stationary, steady state, equilibria. (à# ≥|力kS)
(b) Solve for kgr , the golden rule level of capital per unit of effective labour. Hint:
Aggregate consumption is Ct = At Nt [f (kt ) _ ankt+1 + (1 _ δ)kt]. (笠# ≥|力kS)
(c) Do there exist conditions under which this model economy has a dynamically inefficient equilibrium? Explain. (§# ≥|力kS)
SECTION B
3. Consider the standard Ramsey model, where the planner chooses time paths for the representative agent’s consumption c(t) and capital k(t) to solve
maximize !0 & e_ρtu ╱c(t)、dt
subject to k˙ (t) = f ╱k(t), L、_ δk(t) _ c(t)
k(0) > 0 given, L > 0 given
where f (., .) exhibits diminishing marginal returns to capital, u(.) satisfies standard assumptions, and δ is the depreciation rate of capital. Also, assume that capital’s marginal product is increasing in labour, L: fkL > 0.
(a) Assume that the economy is initially in steady state at date 0, and the government
announces and implements a partial temporary lockdown, implying that labour is reduced to some level lower than L. Individuals expect the lockdown to be lifted at date t1 > 0. Analyze the economy’s response (regarding consumption, capital accumulation, and capital’s marginal product over time), under the assumption that the lockdown is lifted at date t1 . (←# ≥|力kS)
(b) During lockdown, but before the lockdown is expected to be lifted (before date
t1 ) the government announces that the lockdown is permanent (deviating from its previous promise). Analyze the economy’s response (regarding consumption, capital accumulation, and capital’s marginal product over time). (←# ≥|力kS)
(c) If the government planned a permanent lockdown, should it instead have an- nounced this from the beginning? Discuss. (笠# ≥|力kS)
4. Consider the decentralized Ramsey model, where the representative household gets negative utility from holding assets which are polluting. (I.e., the household is a socially responsible investor.) The disutility is proportional to the perceived pollution content, x(t), of the portfolio. Assets are allocated across physical capital (polluting) and government bonds (clean). The household chooses time paths for consumption c(t), assets a(t) and the portfolio weight ω(t) to solve
maximize !0 & e_ρt –ln ╱c(t)、_ ηx(t)!dt
subject to a˙(t) = rk (t)ω(t)k(t) + rb (t)[1 _ ω(t)]b(t) + w(t)L _ c(t)
a(0) > 0 given, L > 0 given.
The consumer takes the time paths of the prices rk (t), rb (t), and w(t) as given.
(a) Show that the household’s budget constraint can be written as
a˙(t) = rb (t)a(t) + [rk (t) _ rb (t)]ω(t)k(t) + w(t)L _ c(t). (à ≥|力kS)
(b) Total pollution is proportional to production: X(t) = eAf ╱k(t), L(t)、, where e
is a pollution parameter and A a technology parameter. Assume the perceived pollution content of the portfolio is x(t) = ω(t)eAf ╱k(t), L(t)、. Derive the Euler equation and the portfolio optimality condition for the household. (笠# ≥|力kS)
(c) The producing firm hires capital and labour on the spot market, taking the wage rate as given, but realizes it affects its cost of capital, via pollution. The producing firm seeks to solve
maximize Af ╱k(t), L(t)、_ rk (t)k(t) _ w(t)L(t) subject to X(t) = eAf ╱k(t), L(t)、
and the household’s optimal portfolio rule
where f(., .) exhibits constant returns to scale. Find the firm’s first order condi- tions. (」# ≥|力kS)
(d) Using your result in (c) and the consumption Euler equation, draw the c˙(t) = 0 line in a phase diagram. Also draw the k˙ (t) = 0 line. (笠# ≥|力kS)
(e) Assuming the economy is at a steady state, suppose there is a positive permanent
technology shock (A increases). Draw the trajectory and the time paths for consumption and capital. (笠à ≥|力kS)
(f) In the scenario in (e), is the pollution premium, rk _ rb , higher or lower in the new
steady state? Does it look as if the household becomes more or less concerned with the environment? Give an interpretation. (笠# ≥|力kS)
SECTION C
5. Consider the following maximization problem facing a representative competitive house- hold:
&
maximize Et L βs U(ct+s , 1 _ lt+s)
s=0
subject to ct+s + kt+s+1 = wt+slt+s + (1 + rt+s)kt+s
where time starts at t, the subscript t +s stands for date t +s, ct+s is consumption, lt+s is labour supply, kt+s is the capital stock, wt+s is the wage rate, rt+s is the rental price of capital, β is the subjective discount factor, U(ct+s , 1 _ lt+s) is the instantaneous utility function, and Et is the date t expectations operator. Let the aggregate production function at any date t be Yt = Zt Kt(α)Lt(1) _α where α is capital’s share and Zt is a productivity shock. Note the difference between the productivity shock Zt from the
way the productivity shock At was covered in lectures. Let the investment technology be Kt+1 = (1 _ δ)Kt + It where δ is the physical rate of depreciation and It is physical investment. Uppercase letters stand for aggregate variables and lowercase for individual variables.
(a) Write in simple terms the environment facing the household. In particular, explain
(i) the timing of decisions of the household from date t to t+1, (ii) all the markets in which the household transacts at each date t, (iii) how the household transfers consumption from date t to t + 1, and (iv) how the productivity shock Zt differs from the productivity shock that we covered in the lectures. (笠à ≥|力kS)
(b) Derive the household’s first order conditions and carefully interpret them.
(笠à ≥|力kS)
(c) Let U(ct+s , 1 _ lt+s) = ln ct+s + π ln(1 _ lt+s) with π > 0, and let δ = 1. Derive the optimal consumption, investment and labour supply rules. Interpret your results.
(笠à ≥|力kS)
(d) Let the productivity shock follow the process Zt = Zt_(ρ)1 εt where 0 < ρ < 1 and εt is a white noise. Derive the reduced form process for output and carefully interpret the results. (笠à ≥|力kS)
6. Consider the following optimization problem for a price setting firm. The firm, opti- mizing in period t, chooses the price Pt* that maximizes the current market value of the profits generated while that price remains effective with probability θ . Formally, it solves the problem:
&
mPa*x L θk Et –Qt,t+k ,Pt* Yt+k|t _ Ψt+k(Yt+k|t)、!
t k=0
subject to the sequence of demand constraints
Yt+k|t = ╱ 、_e Ct+k, k = 0, 1, 2, . . .
where Et is the expectations operator at date t, Qt,t+k is the sequence of stochastic discount factors for nominal payoffs, Ψt+k(.) is the cost function, Yt+k|t is the date t + k output of the firm that last reset its price at date t, Pt+k is the general price level at date t + k , Ct+k is the aggregate demand at date t + k, and e is a demand parameter.
(a) What kind of economic environment will give rise to this price setting problem?
Carefully develop this price setting problem of the firm from first principles by writing the cash flow of the firm each period. (§# ≥|力kS)
(b) Derive the first order condition associated with this price setting firm showing all
your work. Carefully interpret this first order condition. What happens to this price setting rule when θ = 0? Explain. (§# ≥|力kS)
(c) Log-linearize this price setting rule around a zero inflation steady state and care- fully interpret your results. (←# ≥|力kS)
2022-05-12