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Computational Economics, University of California Irvine,
Instructor: Oliko Vardishvili
Spring 2023
Take Home Exam 1 1
(Upload a zip of code & pdf files in Canvas) Note that for the final grade, I
will ask you questions during lecture from your own solutions.
Problem 0 Warm up: Elastic Labor Supply
Now we will depart from the Aiyagari model with capital accumulation
studied in class by introducing elastic labor supply. Now the preferences
take the following form: (
cν(1− l)1−ν
1− µ
Take the following parameters: ν = 0.374, µ = 4.2 Use the following pa-
ameters: β = 0.96, α = 0.36, δ = 0.08, ρ = 0.6, σ2ϵ = 0.16, such that
var(log(zt)) = 0.32. Besides, set m = 2.45 to discretize shocks.
1. Set up the dynamic programming problem and derive the first-orde
2. Compute the steady state of the model using discrete value function
Problem 1 Steady States
Now we again consider inelastic labor supply. We will depart from the
Aiyagari model with capital accumulation studied in class by introducing
taxes in two very simple ways.
Use the following parameters: γ = 0.5, β = 0.96, α = 0.36, δ = 0.08,
ρ = 0.6, σ2ϵ = 0.16, such that var(zt) = 0.04. Besides, set m = 2.45 to
discretize shocks.
Utility: u = c
1−γ , production function: Y = AK
1This take home is prepared for and adopted from Arpad A
aham’s lecture series.
2Note that the values imply a relative risk aversion is equal to 2, a standard value in
macro literature.
3[prob, logs, invdist] = markovappr(ρ, σϵ, 2.45, N) in Matlab.
ˆ Reform 1: Introducing a 15% consumption tax. The tax revenues are
ebated equally across all agents of the economy.
ˆ Reform 2: Introducing a capital income tax which generates the same
evenues as the previous tax schedule (tax on rk, not on the principal).
The tax revenues are rebated equally across all agents of the economy
again. [Hint: τkrk = Trc]
1. Define the recursive competitive equili
ium in these cases. Note that
you have to add a government in the definition. Derive the Eule
equations analytically and compare them with each other and with
the basic model (without taxes).
2. Solve using some adjustments of the programs (with 3 solution meth-
ods: value function iteration, policy function iteration, and endoge-
nous grid point methods provided in the class) for the steady state
level of aggregate capital and the stationary decision rules and distri-
ution of agents for the two tax reforms. [Hint: Note that tax revenues
and consequently the tax rebate is a function of aggregate capital (o
the interest rate), so you have to make only small modifications.]
3. Plot asset policy, consumption policy, and distributions and compare
the smoothness of the functions across these solution methods. Ex-
plain which solution method approximates better the policy functions.
4. Check how aggregate capital accumulation changes as a result of the
two tax reforms. Provide intuition in terms of insurance and output
5. Check how the distribution of agents across consumption levels and
asset levels changes due to the two reforms. You may want to use
oth graphical representation and some statistics such as the Gini
coefficient or coefficient of variation.
6. Calculate the welfare effect of these tax reforms. Do it in two ways: (i)
use the aggregate social welfare and compare it across the three cases
(benchmark and two reforms); (ii) check also who benefits and who
loses due to these reforms. You can use the consumption equivalent
measures for these welfare comparisons.
7. Comment on what sense, these welfare comparisons across steady
states are meaningful or misleading.
8. Find an interesting quantitative question which you can answer using
this model and answer it.
Problem 2 Transitional dynamics
Now assume that at t=0 the tax schedule is in the steady state with
consumption taxes (which you solved above. i.e., steady-state econ-
omy under Reform 1). At the beginning of period 1, the government
makes a surprise announcement that it abolishes consumption taxes
and switches entirely to taxing a capital income (the second steady
state you solved above).
1. Define the recursive competitive equili
ium with transitions.
2. Compute the transition path of the economy using the algorithm pro-
vided in handout and the matlab code provided in tutorial. [try
T=200]. Plot the transition paths of interest rate, wage, capital and
welfare. Comment on the results you obtain.
3. a. Answer the following question: ”How much do we need to change
consumption of the agent in every state in the stationary equili
so that he’d be indifferent between living through the tax reform and
living in the pre-reform economy?” Decompose welfare increase due to
increased consumption level and due to reduced uncertainty.
. Now discuss which tax system is welfare improving taking into
account the transitional dynamics (as opposed to steady state com-
parisons.). Also discuss which taxes are more distortionary.
4. What fraction of the overall population would support the reform?
Compute and plot the measure of consumption equivalent variation.
5. Use your results to analyze which tax schedule is better in terms of
efficiency and distribution and why.
6. Find an interesting quantitative question which you can answer using
this model and answer it.
Answered 4 days After May 30, 2023


Pradeep Kumar answered on Jun 04 2023
3 Votes

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