ECON 711 Macroeconomic Theory and Policy Assignment 3 2022
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ECON 711 Macroeconomic Theory and Policy
Assignment 3
2022
Suggested Solutions
Question 1. (3 points) Dynamic Programming. Consider the problem of an infinitely lived consumer who starts with a wealth of W0 . At each point of time she/he can consume some of the wealth and thus save the remainder. Let ct be consumption level in period t and let u(ct ) = ln ct represent the flow of utility from this consumption. Given W0 , the consumer maximizes her/his lifetime utility
o
βtu(ct )
t=0
subject to the law of motion for wealth
Wt+1 = Wt ← ct .
(a) Formulate this problem as an optimization problem subject to a single lifetime con- straint (Hint: see Ch.1 of Williamson). Use Lagrangian method to find the optimality conditions. Find the optimal decision rule for consumption, i.e the rule that tells you how much to consume for any given level of current wealth.
Solution: The consumer’s lifetime constraint is
o
ct = W0 .
t=0
The Lagrangian for the problem of maximizing
o
βtu(ct )
t=0
subject to (1) is given by
) = t βtu(ct ) ← λ ╱ t ct ← W0 \ .
Differentiating (2) with respect to ct and equating the result to zero yields
βtu\ (ct ) = λ, for t = 0, 1, ...
Notice that equation (3) implies that
βt+1u\ (ct+1) = λ, for t = 0, 1, ...
Dividing (4) by (3) yields the following intertemporal optimality condition
u\ (ct+1)
u\ (ct )
which can be rewritten in the familiar form
u\ (ct ) = βu\ (ct+1).
To obtain the decision rule for consumption, let’s substitute u(ct ) = ln ct into the first
order condition (3):
βt = λ
which simplifies to
ct = βt .
To obtain the expression for λ, substitute (6) into the budget constraint (1):
o o o
t=0 t=0 t=0
from which it follows that
= (1 ← β) W0 .
The decision rule for consumption follows from (6) and (7)
ct = (1 ← β) W0 βt .
To express the decision rule in terms of Wt , notice that
t_1
Wt = W0 ← ci .
i=0
Substituting (8) in the expression above we get
t_1
Wt = W0 ← (1 ← β) W0 βi
i=0
= W0 ← W0 e1 ← βt、
= W0 βt ,
t_1
where the second line follows from βi = (1 ← βt ) . Now, from (8) we can write
i=0
ct = (1 ← β) W0 βt = (1 ← β) Wt .
(b) Write down the functional equation for this problem. Conjecture that the solution to the functional equation takes the form of:
V (W) = A + B ln(W)
for all W . With this guess we have reduced the dimensionality of the unknown value function V (W) to two parameters, A and B . Use “guess and verify” approach to find values for A and B such that V (W) will satisfy the functional equation. What are the optimal decision rules for consumption, c = gc (W) and next-period wealth, W\ = gw (W)?
Solution: The value function is given by
V (W) = max íln(W ← W\ ) + βV (W\ )ì .
W\ ∈[0,W]
Our guess is
A + B ln(W) = max íln(W ← W\ ) + β (A + B ln W\ )ì . W\ ∈[0,W]
Notice that the constraint on W/ will neven be binding: W\ = 0 means c\ = 0 and u (c\ ) = &; W\ = W means c = 0 and u (c) = &.The first order condition is
← + βB = 0,
which we can solve for W\ to obtain the expression for the policy function
βB
1 + βB
By definition of the policy function we have
V (W) = ln(W ← gw (W)) + βV (gw (W)) .
Using the "guess" for the value function we have
V (W) = A + B ln W = ln(W ← gw (W)) + β (A + B ln (gw (W))) Using the expression (9) for gw (W) we obtain
A + B ln W = ln ╱W ← W、+ β ╱A + B ln ╱ W)、、
= ln ╱ W、+ βA + βB ln (βBW) ← βB ln (1 + βB)
= ln W ← ln (1 + βB) + βA + βB ln (βB) + βB ln (W) ← βB ln (1 + βB) = βA + βB ln (βB) ← (1 + βB) ln (1 + βB) + (1 + βB) ln (W)
from which it follows that
B = (1 + βB)
A = βA + βB ln (βB) ← (1 + βB) ln (1 + βB)
B =
W\ = gw (W) = W = W ╱ 、/ ╱ 、 = βW
To obtain the decision rule for consumption remember that c = W ← W\ c = gc (W) = W ← gw (W) = (1 ← β) W.
Question 2. (2 points) MATLAB simulations of a growth model. Consider a neoclassical growth model in descrete time discussed in Chapter 3 of Williamson (Module 2-1). Let u (c) = ln(c), f (k) = k α , and δ = 1. Then the decision rule for capital has a simple analytical form: kt+1 = αβ (kt )α . The purpose of this exercise is to
simulate time paths of the key variables along the transition towards the steady state. (a) Let α = 0.36, β = 0.95. Compute the steady-state level of capital, kss .
(b) Suppose that the initial level of the capital stock, k0 , equals 25% of its steady-state value kss . Using the decision rule for capital, simulate the economy for 30 periods. Plot the time paths of consumption, investment, output and capital and their respective steady-state levels. Make sure your plots and codes are carefully annotated (axes, variables etc) and commented.
(c) Repeat the exercise in (b) for different values of the initial capital stock, k0 . In particular, let k0 take the values: 0.5 . kss , and 1.5 . kss . Plot the time series for different parameter values on the same graph (one variable per plot).
(d) Consider the role of impatience in growth. Repeat the exercise in (b) for different values of the discount factor β . In particular, consider the economies where consumers are very impatient β = 0.8 and very patient β = 0.99. Plot the time series for different values of β on the same graph.
Solution: See livescript Assignment3sol .mlx
2022-06-15