ECON 8022 - MACROECONOMIC THEORY Final Examination
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ECON 8022 - MACROECONOMIC THEORY
Final Examination
(PART - B)
Semester 2, 2022
1 Question 1: OLG model with elastic labor
We now consider an overlapping generation model model in which households value both con- sumption good and leisure.
Demographics and households. Households live for Öve periods. The preference is
given by
8j -1 ╱ 元 1(c)1-j一7(à) + (1 一 元)1(l)1-j一7(à)、 + 8j -1 ╱ 1(c)1-j一7(à)、 ;
where, lj is leisure and 元 is a parameter governing the weights for consumption and leisure. For simplicity, we also assume that when households are retired leisure is not a choice variable any more. The household period by period budgets are given by
c1乙t + s1乙t = (1 一 rt(ss) 一 rt(N))(1 一 l1 )h1wt ;
c2乙t+1 + s2乙t+1 = (1 一 r 一 r +1)(1t(N) 一 l2 )h2wt+1+ (1 + ╱ 1 一 r +1t(☆)、rt+1)s1乙t ; c3乙t+2 + s3乙t+2 = (1 一 r 一 r +2)(1t(N) 一 l3 )h3wt+2+ (1 + ╱ 1 一 r +2t(☆)、rt+2)s2乙t+1 ;
c4乙t+2 + s4乙t+3 = (1 + ╱ 1 一 r +3t(☆)、rt+3)s3乙t+3 + P4 ;t+3 ;
c5乙t+4 = (1 + ╱ 1 一 r +4t(☆)、rt+4)s4乙t+4+ P5乙 t+4:
where (1 一 lj)hjwt+j -1 is labor income, rtsj -1 is capital income, rss is social security tax, rN is labor income tax, r ☆ is capital income tax and Pj is social security beneÖt.
Production and Örm. The production sector consists of a representative Örm that has the following production technology Y = AKá H1-á where Y output, A is total factor productivity, K is capital stock, and H is e§ective labor (e.g., human capital). Capital depreciation rate is 6: The Örmís optimization problem is given by
kt(m)a乙Lt(x) ,(1 一 rt(coh)) ┌AtKt(á)Ht1-á 一 wtHt┐ 一 qtKt ; 、
where wt is the market wage rate and qt is the rental rate; and r t(coh) is the company income tax.
Government. The government runs a PAYG social security system. The social security payment for retirees is linked to the average labor income at time t as
(1 一 l1 )h1wt + (1 一 l2 )h2wt + (1 一 l3 )h3wt
where w is a replacement rate. The social security program is self-Önanced so that
Total social security payment
扌↓
5
匕 Pj乙t
j=4
Total social security tax revenue
扌↓
3
= r t(ss) 匕(1 一 lj)hjwt ;
j=1
where r t(ss) is a social security tax that adjusts endogenously to program every period.
The government also has a general government spending Gt . given by
balance the social security
The government budget is
Labor income tax revenue Capital income tax revenue
≥ 3 扌尸 ( ≥ 4(扌尸) ( Company income tax revenue
Gt = r t(l) 匕(1 一 lj)hjwt + r t(k) 匕 rtsj乙t + r(≥)t(com) [Yt 一 wtH(t]
j=1 j=1
We assume that the government just spends all tax revenues.
Assume that h1 = 1; h2 = 1:5; h3 = 1:2; 8 = 0:98; 7 = 2; 元 = 0:4; A = 1, a = 0:36 and 6 = 0:05: The government sets w = 30%; rl = 20%; rk = 20% and rcom = 25%:
1.a) Find the steady state solution, using (i) fsolve function and (ii) the value function search
with Gauss-Seidel algorithm.
1.b) Suppose that the government cuts the labor income tax rl from 20% to 15% (e.g., the recent three stage income tax cuts in Australia). Study the (long-run and short-run) macroeconomic and welfare e§ects of the personal income tax reform.
1.c) Suppose that the government cuts the company income tax rcom from 25% to 20%: Analyze the (long-run and short-run) macroeconomic and welfare e§ects of the business income tax reform:
2 Question 2: Heterogeneous agent model (HAM) with taxes
We consider a closed economy model Ölled with di§erent households, a representative Örm and a government. Time is discrete (t = 0; 1; :::; o).
Household. A typical household lives inÖnitely and has the following preference:
o
E0 匕 8tu(ct;lt) ;
t=0
where 8 is the time discount factor, ct is consumption, lt is leisure and u(ct) = 1c—t1-à(o) + 9 :
The agent is given capital k0 initially and one unit of time in each period. The agent can invest in capital market. The labor supply is nt = 1 一 l1 : The household lifetime budget constraint is
ct + kt+1 = (1 一 r )ztwtnt +t(A) !1 + (1 一 r☆ )rt] kt ;
k0 > 0; kt 2 0; 1 2 lt > 0;
where ct and it are consumption and investment; rt is interest rate; wt is wage rate; r t(☆) and r t(A) are taxes on capital income and labor income, respectively; and zt is labor productivity which follows a stochastic process that has two states: low (L) and high (H),
zt = { zz日(L):
Firm. There is a representative Örm which has access to the following CRS technology: Yt = AtF(kt;nt) = Atkt(á)nt(1) -á
where, At is the total factor productivity, kt is capital input and nt is labor input. The representative Örm rents inputs in competitive markets.
Government. The government collects tax revenue to Önance an investment subsidy program and a sequence of government purchases. The government budget constraint is given by
Gt = r t(A)wtnt + r☆ rtkt:
Note that, Gt is does not enter householdís preferences or budget constraint.
Assume the following parameter values in Table 1.
2.a) Stationary competitive equilibrium: Assume 9 = 0 and solve the mode numerically. Re- port the main results including the distribution of income and asset.
2.b) Calibration: Find 9 =? so that aggregate labor supply is n = 0:3. Report the steady state results.
Parameters |
Value |
|
Preferences Discount rate Weight on leisure |
7 = 2 8 = 0:95 9 = ? |
|
Income endowment |
z = |
|
Transition matrix Interest rate Capital income tax rate Labor income tax rate Borrowing constraint |
π LL = 0:6 and π HH Tk = 20% TA = 10% k = 0 |
= 0:7 |
Table 1: Parameter Values
2.c) Tax reforms: Start from a benchmark model in 2:b) and consider a tax reform that reduces capital tax rate from 20% to 15%. Report the results. Discuss the e§ects on output (Y), capital (K) and employment (N). Does the reform result in welfare improving? What are implications for wealth and income inequality?
2022-11-19