ECON 8026 Diploma Macroeconomics 2018

Question 3 (20%) A special case of the OLG model yields an optimal growth process m+crvn c cf Torcrlrcr ccm + a] cd?" o q !<, — (1 Q)0 winqy*q 0 «<■"" r\： /■? 1 411 tatacqq

terims of pei-woikei capital stock as kt+1 — ~1+^~k , where u < a, p < 丄. Suppose

this model represents Argentina in the 18^th to the early 20^th century. So initially, ko > [(1 - a)P/(1 + P)]^1/(1-^a).

Claim: The half life of this growth process depends only on a. (Hint: The half life of an initial quantity ko is the time it takes for that quantity to be halved.)

Question 1 (30%) Overlapping generations and labor supply. Consider an overlapping generations model in which agents exist for two periods. Suppose each generation works while it is young (y) and consumes when it is old (o). Each young agent, born at date t e N, has the following lifetime preference function

^u (^ly, *+1) = _ ^(lt⁾² + ^ct+1,

for ranking the stream of work and old-age consumption outcomes (l^y,c?+i). Output at time t (which is consumed by the date-t old) is produced from labor using a linear production technology: y_t = .

A feasible allocation in this economy is a sequence {l^y,c"]"。, with given consumption for the initial old, cg, such that for all t > 0, .

1. A Pareto optimal allocation in this economy is a feasible allocation such that there is no other allocation in which one generation can be made better off without making any other worse off. Write down the steady-state Pareto planner's allocation problem precisely. (Hint: The Pareto planner maximizes a representative cohort's lifetime utility and in a steady state all the cohorts are identical.) ( 4%〉

2. Characterize (i.e., derive the conditions describing) a Pareto optimal allocation and solve for it. (Hint: You can show that the steady state allocation is such that the resulting lifetime utility of every cohort born in t > 0 is 1/4 and that the initial old will have utility of 1/2.) ( 8%〉

3. Argue that the unique competitive equilibrium for this economy without any means of storage or saving is Pareto inefficient. What would this competitive equilibrium allocation look like? Show that the resulting lifetime utility of each generation is zero. ( 3%〉

4. Suppose the initial old are now endowed with a fixed supply of money 0 < M < o, and, the initial young believe money has value in exchange. There are three parts to this question:

(a) Define, precisely, a monetary (recursive competitive) equilibrium. ( 4%〉

(b) Now, focus on a steady state monetary equilibrium. Solve it.

Prove that the existence of fiat money (an intrinsically valueless object much like a bubble asset) can potentially restore Pareto optimality in this example. (Hint: Show that the (long run) inflation rate must be zero for the steady state monetary equilibrium to be identical to the steady Pareto allocation you derived earlier.) ( 6%〉

(c) Discuss your results: Why is money welfare improving? Why do we say it is a bubble asset here? Why is inflation welfare reducing here? ( 5%〉

Question 2 (30%) Consider an example of an IS-PC-MP Keynesian model. Given exogenous process {rf}t>o and a nominal interest policy plan {臨}校。,an equilibrium is a set of bounded processes for output gap and inflation, {xt,nt}t>o, satisfying the Phillips Curve,

nt =月 Etnt₊i + KXt, (2)

and the IS curve,

1八

^xt = ^Et^xt+1 — (it — ^Etⁿt+1 — ^rt )• (3)

a

The parameters are restricted to be as follows: /3 e (0,1), a > 0, and k > 0. The notation EtZt+i is taken to denote expectations of a future variable zt+i conditional on information available at date t. (This is similar to the special case in our Dornbusch model studied in class.) Suppose the monetary policy plan is induced by a simple rule (MP):

it = ©n ⁿt + 肅, (4)

where ©n > 0. The random variable et, can be interpreted as unanticipated (or unmodelled) shifts in policy outcomes.

1. Sketch the graphs of the policy equilibrium conditions (2), (3) and (4) in (xt,nt) space. How would you name these graphs? (Hint: Label your diagrams clearly, e.g., intercepts, slopes and etc.) ( 8%〉

2. All else constant, what happens to current n_t and x_t if 財 fell suddenly? Show this effect using your graphical device from the last part. ( 7%)
3. Show what happens to your conclusion in the last part if we instead had a limiting

economy with k / +o. Provide an interpretation. ( 4%)

4. Show that you can re-write the system as:

缶=[伽 + ^K)EtE + '両^xt+1 + 心 _^)]，

and,

^xt = ^[(i _ 8加^{)Et +} ’^Etx+ ⁺ (律 f^)]-

(4%〉

5. Re-write your policy equilibrium description as a matrix system of the form yt = FEt{yt₊x} + Gzt, where y and z, respectively, contain endogenous and exogenous variables. ( 7%〉

Question 3 (30%) Consider an OLG economy where agents live for two periods. There is lump-sum tax/transfer, a_t = r_tw_t e R, where r_t < 1 for all t e N and Wt is the wage rate measured in units of consumption. The transfer to (tax on) the date-t old is Zt e R. The transfer system is such that, for each period, the budget for the system is balanced.

Define total resources at time-t, given state kt, as

f@t)= ka

where a e (0,1) and 6 e (0,1]. (We have assumed that capital stock depreciates fully every period.) Assume initial per-worker capital stock ko is given. The population of young agents, of measure Nt, grow at a constant rate n > —1. Assume No is given. All agents are prices takers—i.e. they take the real wage Wt and gross capital rental rate R as given.

Denote RJe+1 and zf+1 as agents' subjective expectation of next period capital return and transfer, respectively. Old agents own the capital stock at time t. Each young agent is endowed with 1 unit of time, and faces a lifetime sequence of budget constraints:

Ct + St < (1 — Tt )wt,

and,

^dt+1 < ^Rt+1^St + ^zt+1-

Consider agents with lifetime preference representations:

U ^(ct) + /3U (必+1),

where U(x) = ln(x), x > 0, /3 e (0,1), Ct is young-age consumption, and dt is consumption of the date-t old.

2. Set up the firm's decision problem. (Hint: denote 辟 and N?, respectively, as the firm's demand of per-worker capital and labor services at time t.) Derive exactly the firm's demand for capital and labor, given wt and R_t. ( 2%〉

Using the final goods market clearing condition, government budget constraint and consumers' budget constraints, verify that the capital market clears. (Hint: Show that kt₊i = St/(1 + n).) ( 4%〉

5. Consider a (dynamic) Modified Golden Rule Pareto optimal allocation, under a particular infinitely-lived Pareto planner that discounts each future cohort's lifetime utility using the factor y^t. Show that this Pareto allocation can be analytically solved as

c* = ( ¹ — ^aY ) (k*)^a

^Ct = 1+ 牛^(kt),

=月⁽¹ + ⁿ⁾ / ^{1 — a}Y\ _(k*_)a

從=Y U + g/Y ^(kt)' and,

k* = ^aY (k*)

^t+i (1 +仇丿(仍

(Bonus 15%〉