Dynamic Optimization
Problem Set #2
INTERMEDIATE MACROECONOMICS
Fall 2022
Colorado State University
Benjamin Basow*
1 Instructions
Carefully read all aspects of the prompt before attempting each academic exercise. You will be
graded on your ability to co
ectly and completely respond to each prompt. The term co
ectly
is defined as a students ability to make precise and accurate statements. A precise and accurate
statement may incorporate both mathematical computation and written economic interpretation.
The term completely is defined as a students ability to respond to all aspects of a specified prompt
of interest. All mathematical computations for specific derivations need be shown within your final
manuscript - i.e. show your work. A physical copy is due 10/16/22 in class.
2 Generalized Overlapping Generations Model
Consider a world where there exists two types of agents that live for two periods. In the first
period agents are young and in the second period agents are old. Within this economy there exists
households and firms. The households lifetime utility function is defined as:
U = u(Cy,t) + βu(Co,t+1) (1)
where Cy,t is consumption of the young in period t, Co,t+1 is consumption of the old in period t+1,
and β ∈ (0, 1) is an impatience parameter. The household faces two budget constraints. The first
udget constraint is associated with the young and is defined as:
Cy,t + st = wt (2)
where Cy,t is consumption of the young in period t, st is savings at time t, and wt is real wage at
time t. The second budget constraint is associated with the old and is defined as:
Co,t+1 = [Rt+1 + (1− δ)]st (3)
*Department of Economics, Colorado State University, A12B Clark Building, Fort Collins, Colorado XXXXXXXXXX,
XXXXXXXXXX
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ECON 304 INTERMEDIATE MACROECONOMICS FALL 2022
where Co,t+1 is consumption of the old in period t+ 1, Rt+1 is the rental rate of capital in period
t+ 1, δ ∈ (0, 1) is a capital depreciation, and st is savings in period t. In our economy there exists
a single representative firm. The production function for the representative firm is defined as:
Yt = AF (Kt, Nt) (4)
where Yt is aggregate output at time t, A is a time invariant technology variable, Kt is aggregate
capital owned by the old, Nt is aggregate labor supplied by the young.
2.1 Prompt 1
Set up the dynamic optimization problem using the households lifetime utility function, the budget
constraint associated with the young, and the budget constraint associated with the old. Addition-
ally, define the choice variable of interest. The “set up” should be of the form:
max
{choice variable}
objective funtion (5)
s.t. budget constraint young
udget constraint old
2.2 Prompt 2
First, isolate Cy,t on the left hand side of the young budget constraint. Co,t+1 is already isolated
on the left hand side of the old budget constraint by construction. Substitute the modified young
udget constraint and the unmodifed old budget constraint into the objective function and redefine
your optimization problem. The redefined optimization problem should have the form:
max
{choice variable}
objective function (6)
2.3 Prompt 3
Solve the optimization problem by taking the partial derivative of the objective function defined
in prompt 2 with respect to your defined choice variable, set that partial to zero, and create an
equality expression.
2.4 Prompt 4
Derive the intertemporal euler equation using the expression defined in prompt 3, the modified
young budget constraint, and the unmodified old budget constraint. Explain the intuition of the
euler equation.
2.5 Prompt 5
Set up the firm optimization problem. The objective of the firm is to maximize profits by choosing
optimal levels of captial Kt and labor Nt.
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ECON 304 INTERMEDIATE MACROECONOMICS FALL 2022
2.6 Prompt 6
Derive the firm optimality conditions that equate the marginal products of the firm to their respec-
tive factor prices. The firm optimality condition are derived by taking the partial derivative of the
objective function with respect to each choice variable and setting that expression to zero.
3 Overlapping Generations Model with Logarithmic Utility
Using the same overalapping generation framework, now consider a lifetime utility function with a
logarithmic functional form defined as:
U = ln(cy,t) + β ln(co,t+1) (7)
The household faces identical budget constraints to the previous prompts and are defined as:
Cy,t + st = wt (8)
Co,t+1 = [Rt+1 + (1− δ)]st (9)
All variable and parameter definitions are identical to the previous prompts.
3.1 Prompt 1
First substitute equations (8) and (9) into equation (7) and then set up the dynamic optimization
problem.
3.2 Prompt 2
Conduct the optimization of the objective function and solve for the savings rule. In other words,
solve for savings st as a function of parameters β and real wage wt after you have conducted the
optimization.
3.3 Prompt 3
For the firm, now consider a production function that displays constant elasticity of substitution.
The production function is defined as:
F (Kt, Nt) = A · [αKρt + (1− α)N
ρ
t ]
1
ρ (10)
where the parameters α ∈ (0, 1) and ρ ∈ (0, 1) are elasticities associated with each factor input.
All variables and parameter definitions are identical to the previous prompts. Set up the profit
maximization problem and conduct the optimization.
4 Overlapping Generations Model Application
Study page 166, section “8.3.1 Government Intervention” in our secondary textbook. Study how to
derive the savings rule from the household optimization problem. Pay particular attention to how
the budget constraints change and how this interacts with aspects of this model. You do not need
to write anything down for this prompt; just study the material.
3
Instructions
Generalized Overlapping Generations Model
Prompt 1
Prompt 2
Prompt 3
Prompt 4
Prompt 5
Prompt 6
Overlapping Generations Model with Logarithmic Utility
Prompt 1
Prompt 2
Prompt 3
Overlapping Generations Model Application