SIMON FRASER UNIVERSITY BURNABY. Macroeconomic Theory Core Exam Sample Questions. March 2012

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1 SIMON FRASER UNIVERSITY BURNABY BC Macroeconomic Theory Core Exam Sample Questions March 2012 Paul Klein Instructions [Please read these instructions before you commence writing the exam!] There are TWO sections in this exam. ALL questions in BOTH parts are MANDATORY. This is a closed book exam: you are not allowed to consult any notes, text, nor colleagues during the writing of this exam. If you use variables that are not defined in the question, carefully define them before you proceed. Try to be as concise and precise as possible. You have four (4) hours. Good luck!

2 Part I. True/False/Uncertain and Explain. For each question, give a coherent answer in no more that 10 lines. 1. In the Huggett (1993) model, the ratio of the variance of consumption to the variance of earnings increases as the persistence of earnings increases. 2. In the Kocherlakota (1996) model, the ratio of the variance of consumption to the variance of earnings increases as the persistence of earnings increases. 3. The First Welfare Theorem fails in overlapping generations models because agents in these economies are impatient and always save too much. 4. Consider a 2-period overlapping generations economy with capital and a population growing at some constant positive rate. Suppose the economy is at its unique steady-state competitive equilibrium which features over-accumulation of capital. A government that wants to set up a tax-transfer scheme for this economy that will yield a Pareto-superior competitive equilibrium allocation can always find at least one such scheme. 5. Arbitrage can only exist when the number of securities exceeds the number of possible states. 6. From the point of view of the standard neoclassical model, taxing capital is never a good idea. 7. In models with limited enforcement, the default constraint binds only for individuals with high endowment. 2

3 Part II. Problems Question 1. Equilibrium asset pricing Consider an environment where everyone s preferences are represented by [ ] E β t c1 α t 1 α where β > 0, α > 0. Suppose log consumption growth ln c t+1 ln c t is i.i.d. normal with mean µ and variance σ 2 and the flow of information is represented by the filtration {F t } generated by the consumption process. You may use without proof the result that if X is a normal random variable with mean µ X and variance σx 2 then E[e X ] = e µ X+ 1 2 σ2 X. (a) Derive a formula for the period t price of an asset that yields one dollar for sure in period t + 1. (b) Derive a formula for the period t price of an asset that, if purchased in period t, entitles the owner to a dividend flow starting in period t + 1 and given by c t+1, c t+2,... (c) In the following questions, X is an F t+1 -measurable random variable that is independent of c t+1 conditional on F t. All variances given are conditional on F t. (a) Derive a formula for the period t price of an asset whose period t + 1 payoff is X when the variance of ln X is σ 2 ln X and E[X F t] = 1. Is this price increasing, decreasing or constant in σln 2 X? Explain intuitively. (b) Derive a formula for the period t price of an asset whose period t+1 payoff is X when the variance of ln X is σ 2 ln X and E[ln X F t] = 0. Is this price increasing, decreasing or constant in σln 2 X? Explain intuitively. 3

4 Question 2. A Firm s Problem Consider a firm that faces a given fixed interest rate r = 0.05 and maximizes the discounted sum of present and future profits. Suppose initial capital k 0 is given and that the problem of the firm is to maximize (1 + r) t π t (1) where and π t = f (k t ) g (k t, i t ) (2) k t+1 = (1 δ) k t + i t (3) where δ = 1/5. The production function f is defined via Meanwhile, the investment cost function g is given by f (k) = k 1 2 k2. (4) g (k, i) = i + 16 (i δk) 2. (5) (a) Find the steady state capital stock k. (b) Find the optimal feedback rule, i.e. an expression for i t or k t+1 in terms of k t and parameters. Feel free to use the symbol k as shorthand in order to simplify the notation. 4

5 Question 3. OLG Consider an overlapping generations environment that exists in periods t = 1, 2,..., where people live for two periods. Denote the consumption profile of an individual born in period t by (c t (t), c t (t + 1)) and her endowment profile by (ω t (t), ω t (t + 1)). Define the aggregate endowment in period t by Y (t) = N(t 1)ω t 1 (t) + N(t)ω t (t). We will assume that there is an n > 0 such that N(t + 1) = nn(t) and Y (t + 1) = ny (t) for t = 0, 1,... Preferences for generations t = 1, 2,... are represented by the (common) utility function u(c t (t), c t (t+1)) where u is strictly increasing in each argument, bounded, differentiable and concave. Preferences of members of generation 0 are represented by the strictly increasing function ũ(c 0 (1)). (a) Define a competitive equlibrium in this environment, denoting the price of t goods in terms of t + 1 goods by r(t). (b) Give a specific example of a competitive equilibrium that is not Pareto optimal. (Make the example as concrete as possible by exactly specifying preferences and endowments. The less general, the better.) (c) Show that a competitive equilibrium satisfying r(t) = r > n is Pareto optimal. 5

6 Question 4. A Linear Economy Consider an environment where a representative agent has preferences represented by The resource constraint is given by β t [ln c t 1 2 h2 t ]. c t + g t = h t ; t = 0, 1,... where {g t } is an exogenously given sequence. Additional constraints are c t 0 and h t 0. The government can raise revenue in two ways only: it can issue one-period bonds or it can tax labour income proportionally at rate τ t. Initial government debt b 0 is zero. (a) Show that any allocation satisfying the resource constraint and the equation β t h 2 t = 1 1 β is part of a competitive equilibrium profile. Hint: First state what is meant by this precisely. (b) Suppose you are given a competitive equilibrium allocation. How would you go about finding the sequence of government debt issues b t, t = 1, 2,...? (c) Let p t denote the relative price of consumption in period t in terms of consumption in period 0. Show that in any competitive equilibrium, p t b t 0 as t. Hint: You may invoke without proof the converse of what you showed in (a). 6

7 Question 5. Bellman Let X be a non-empty set. Let f : X X R be a bounded function, and let β [0, 1). Let Γ : X X be a correspondence. Let a X be fixed. We will call a sequence x = {x t } admissible if x 0 = a and x t+1 Γ(x t ) for each t = 0, 1,... Let v : X R be a bounded function that satisfies for all x X. Show that for all admissible x. v(x) = sup {f(x, y) + βv(y)} (6) y Γ(x) v(a) β t f(x t, x t+1 ) Hint: start by proving, by mathematical induction, that, for any admissible sequence x and any T = 0, 1, 2,... T 1 v(a) β t f(x t, x t+1 ) + β T v(x T ). 7

8 Question 6. Suppose we want to maximize where subject to β t u(c t ) u(c) = lim σ σ c 1 σ 1 1 σ and a t+1 = Ra t c t lim t R t a t 0 where a 0 > 0 is a given constant and where σ 0, R > 0 and β 0 are given parameters. Suppose βr 1 σ < 1. If you find the algebra too messy, assume σ = 1. (a) Write down the optimal c t and a t as a function of t and parameters. (b) Derive a feedback representation of a solution, i.e. functions g and h such that a t+1 = h(a t ) and c t = g(a t ) define an optimal solution. 8

9 Question 7. Suppose you want to maximize subject to β t u(c t ) a t+1 = (1 + r)a t c t c t 0 a t+1 0 where r > 0 and a 0 > 0 are given parameters. The function u : R + R is (weakly) concave, bounded and once differentiable. Suppose β = (1 + r) 1. Show that c t = c 0 = ra 0 is an optimal solution by going through the following steps. (a) Derive the value function implied by the above consumption plan. (b) Show that the value function you derived in (a) satisfies Bellman s functional equation as it applies to this problem. Question 8. Consider an environment where a representative consumer/worker has preferences represented by β t u(c t ) where 0 β < 1 and u : R + R is a strictly increasing, differentiable and strictly concave function. Production opportunities are defined by the following resource constraint c t + k t+1 f(k t, l t ) where f is increasing in both arguments, differentiable and strictly concave. Initial capital k 0 is given. The economy is also endowed with one unit of human capital which generates one unit of labour services in each period (l t = 1). Consumption c t and the capital stock k t cannot be negative. 9

10 (a) Define what is meant by a feasible consumption allocation {c t }. (b) Suppose the consumer owns the physical and the human capital and rents them out to the firm at rates r t and w t respectively. (Consumers have no claim on any profits of the firm.) Write down the utility maximization problem of the consumer. Hint: You may want to define p t = p 0 (c) Write down the profit maximization problem of the firm. Denote total output (which is sold to the consumer as either consumption or investment) by y t. Note t k=1 r 1 k that it is static because the firm owns no capital. (d) Write down the market clearing conditions. Verify that they imply y t = r t k t + w t. Question 9. Consider an overlapping generations environment where agents live for two periods. Each generation t has infinitely many small members of total mass N(t). Population growth is constant and given by n. More explicitly, N(t + 1) = nn(t). Preferences are represented by u h t = ln c h t (t) + ln c h t (t + 1). Endowments are ωt h = [2, 1]. (a) Find the competitive equilibrium. Notice that it is independent of n. (b) Suppose n = 1. Show that the competitive equilibrium allocation is not Pareto optimal. (c) Suppose n = 1/3. Is the competitive equilibrium Pareto optimal? 10

11 Question 10. Suppose you want to maximize where 0 < β < 1, subject to β t c t qa t+1 + c t = a t, a t+1 0 and c t 0, where q > 0 and a 0 > 0 is given. (a) Suppose q = β. (a) Verify that the function v(a) a satisfies the functional equation v(a) sup {a qa + βv(a )}. a [0, a q ] (b) Verify that the sequences c t = 0, ã t = q t a 0 are feasible and that, with v(a) = a, for all t = 0, 1,... v(ã t ) = c t + βv(ã t+1 ) (c) Show that the sequence described in (ii) is not optimal. Why does Bellman s principle of optimality not apply? (b) Suppose β q. Find a candidate solution and show, by invoking an appropriate theorem, that it is a solution. 11