ECON44815 ADVANCED MACROECONOMICS
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ECON44815
ADVANCED MACROECONOMICS
SECTION A
1. Consider an overlapping generations model economy with two-period lives and 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 labor 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 labor productivity at t, and α e (0, 1) is a parameter. Assume the following: At+1 = aAt for all t > 1 where a > 1 is a parameter; the population of successive generations evolves according to Nt+1 = nNt for all t > 0 where n > 0 is a parameter; capital depreciates at the constant rate δ e [0, 1]; and the parameters satisfy an > 1.
(a) Let kt := Kt /(At Nt ) be the level of capital per unit of effective labor. Derive the dynamic equation that determines the evolution of kt in a laissez faire perfectly competitive equilibrium, illustrate it in a diagram, and solve for the steady state value kss > 0. Show that in the steady state Ut+1 - Ut is constant, and explain which property (or properties) of the utility function generate this outcome.
(50 marks)
(b) Consider two economies. Economy 0 has δ = 0 while economy 1 has δ = 1;
otherwise they are identical, including their initial conditions. Compare the time paths of {kt , yt , rt , wt , ct(y), ct(o)}t>1 for the two economies where yt is output per unit of effective labor, rt is the rate of return on capital, and wt is the wage per unit of effective labor. Explain which economy is more likely to be dynamically inefficient. (50 marks)
2. Consider an overlapping generations model economy with two-period lives and 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 labor 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 ) = ρAt Nt + Kt(α)(At Nt )1-α where Kt is the aggregate capital stock at t; At is the level of exogenous labor productivity at t; and ρ > 0 and α e (0, 1) are parameters. Assume the following: At+1 = aAt for all t > 1 where a > 1 is a parameter; the population of successive generations evolves according to Nt+1 = nNt for all t > 0 where n > 0 is a parameter; capital depreciates at the constant rate δ e (0, 1); and the parameters satisfy an + δ > 1.
(a) Let kt := Kt /(At Nt ) denote capital per unit of effective labor. Derive the dynamic equation that determines the evolution of kt in a laissez faire perfectly competitive equilibrium and illustrate the steady state kss > 0. Explain your work. (35 marks)
(b) The equilibrium allocation of consumption between old and young at t is captured by the ratio ct(o)/ct(y). Derive an expression for the steady state value of this ratio and discuss how it is affected by the parameter ρ. You may use the following: wt-1 /wt equals one in steady state, where wt is the wage per unit of effective labor. (30 marks)
(c) The possibility of dynamic inefficiency depends on the model’s parameter values. Discuss how ρ affects this possibility. (20 marks)
(d) Based on parts (b) and (c), discuss the following: is the steady state equilibrium more likely or less likely to be dynamically inefficient if the old get a larger share of consumption? (15 marks)
3. Consider an overlapping generations model economy with two-period lives and 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 labor 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 labor productivity at t, and α e (0, 1) is a parameter. Assume the following: At+1 =
aAt for all t > 1 where a > 1 is a parameter; the population of successive generations evolves according to Nt+1 = nNt for all t > 0 where n > 0 is a parameter; and capital depreciates at the constant rate δ e [0, 1].
The government operates a tax/transfer scheme. At time t > 1, each young person pays a tax of Tty and each old person pays a tax of Tto , both in units of the consumption good, and where a negative value indicates a transfer. The scheme has the pay as you go property: NtTty + Nt-1Tto = 0 for all t > 1.
Let Bt denote the stock of government debt in units of the consumption good at the end of time period t, with initial value B0 = 0. Let kt := Kt /(At Nt ) denote capital per unit of effective labor. Also, let τt(y) := Tty /At and τto := Tto /At .
(a) Derive, with explanation, the dynamic equation that determines how the endoge- nous variable kt+1 is determined in equilibrium, influenced by its lagged value kt and also influenced by the exogenous policy variables τt(y) and τ. [You can eliminate the τ o variables since the pay as you go property yields τto = -nτt(y) for all t > 1.] (30 marks)
(b) Suppose τt(y) = > 0 for all t > 1. Suppose also that the economy has reached a steady state equilibrium in which kt = kt+1 = > 0. Substitute these values into the dynamic equation from part (a) to get an equation that implicitly defines the endogenous steady state capital variable in terms of the exogenous steady state policy variable . Demonstrate that the pay as you go scheme reduces the steady state level of below where it would have been in the absence of the scheme. I.e., show that >0 < =0 . (25 marks)
(c) Use the dynamic equation from part (a) to demonstrate that, in principle, the government can choose {τt(y)}t>1 to achieve the golden rule level of capital, kt = kgr for all t > 2, starting from the initial condition k1 > 0. (25 marks)
(d) Since the policy settings in part (c) achieve the golden rule, and since the golden rule maximizes aggregate consumption across all feasible steady states, one might suspect that the equilibrium with these policy settings is Pareto superior to the laissez faire equilibrium. (Laissez faire is τt(y) = 0 for all t > 1.) Is this suspicion correct? An informal discussion is sufficient. (20 marks)
SECTION B
4. 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:
o
mPx θ k Et ,Qt,t+k ┌ Pt* Yt+klt - Ψt+k(Yt+klt)┐、
t k=0
subject to the sequence of demand constraints
Yt+klt = ╱ 、-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+klt 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 i is a demand parameter.
(a) 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. (50 marks)
(b) Log-linearize this price setting rule around a zero inflation steady state and care-
fully interpret your results. (50 marks)
5. Consider the following non-stochastic maximization problem facing a representative
competitive household:
maximize
subject to
o
βt U(ct )
t=0
ct + kt+1 = wt + (1 + rt )kt
where the subscript t stands for time, ct is consumption, kt is the capital stock, wt is the wage rate, rt is the rental price of capital, β is the subjective discount factor, and U(ct )
is instantaneous utility. Labour is inelastically supplied at unity. Let the aggregate production function be Yt = At Kt(α) where α is capital’s share and At is non-stochastic total factor productivity (TFP). 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, and (iii) how the household transfers consumption from date t to t + 1. (33 marks)
(b) Derive the household’s first order conditions and carefully interpret them.
(33 marks)
(c) Let U(ct ) = ln ct and δ = 1. Derive the optimal consumption and investment rules. Interpret your results. (34 marks)
6. (a) Consider the following social planning problem:
maximize
subject to the resource constraint
and given the forcing process
Et o β s
Ct + Kt+1 - (1 - δ)Kt = At(1) -α Kt(α)
ln At+1 = ρ ln At + ln it+1
where Ct and Kt are the planner’s consumption and capital stock at date t, At is labour productivity, {ln it+1} is serially uncorrelated, 0 < β < 1, 0 < δ < 1, 0 < ρ < 1, and σ is the relative risk aversion parameter. Labour is inelastically supplied at unity. Derive the log-linearized first order condition and resource constraint. Show all your steps. (50 marks)
(b) Using the basic new Keynesian model, set up the household’s consumption, saving,
and labour supply problem explaining the household’s optimization problem very carefully. Then derive the consumption Euler equation and interpret your results. (50 marks)
2022-04-22