Econ23950 homework 2
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Econ23950 homework 2
1. (20 points) Redistribution with productivity differences
All households have the same preference, where u(c,h) = ln c + 2.6lnh. Production only requires labor; labor share is 1. One unit of labor from type 2 produces 100 units of output. Type 1 is 1.4 times as productive as type 2. Two types of households enter different labor markets and receive different wages, denoted by w1 and w2 , respectively. There are two types of firms; one hire only type 1 and the other type 2.
Y1 = 140L1 Y2 = 100L2
30% of population is type 1 and 70% is type 2. The goods market clearing condition needs to take this into consideration.
0.3c1 + 0.7c2 = 0.3 · 140L1 + 0.7 · 100L2
(a) (5 points) Define and compute the competitive equilibrium.
(b) (4 points) The alternative representation of Pareto weights is to define a relative
weight. Say the type 2 is the baseline, each type 2 household has a weight of 1. Each type 1 household has a weight of ϕ . The social planner solves the following problem.
max 0.3 · ϕ · u(c1 ,h1 ) + 0.7 · u(c2 ,h2 )
c1 ,h1 ,c2 ,h2
subject to 0.3 · c1 + 0.7 · c2 ≤ 0.3 · 140(1 − h1 ) + 0.7 · 100(1 − h2 )
All Pareto optimal allocations are traced out by varying ϕ from a value of 0 to ∞ . Find the solution to social planner’s problem for all value of ϕ .
(c) (1 point) Find the value of ϕ so that social planner’s allocation is the same as the allocation in the competitive equilibrium.
(d) (1 point) When ϕ = 1, we are in an egalitarian society. What is the Pareto optimal allocation? What is the difference between type 1’s production and its consumption? Denote this number by T.
(e) (4 points) Show that this allocation can be achieved by a market economy with
a lump-sum tax of size T from each type 1 household and a lump-sum transfer of size 
to each type 2 household.
(f) (4 points) Suppose that instead of a lump-sum tax, a flat rate tax of 15% is
levied on type 1 households. Revenue collected will be distributed equally among type 2 households via a lump-sum transfer. What is the equilibrium this time?
(g) (1 point) Tabulate GDP (the right hand side of the social planner’s constraint),
welfare of each type of household under part (a), (d), and (f).
2. (10 points) Should one prepared for old age?
John values consumption the same way every period, where u(c) = 
and γ = 2. He is impatient with β = 0.95. Something else that enters his life-time utility is the chance of surviving to future periods, denoted by κ . Say one period is 10 years. John starts to make plans when he turns 20; this is period 0. The maximum life span afterward is 6 more periods. κ 1 is John’s chance of surviving to 30 while κ6 is John’s chance of surviving to 80. John’s expected life-time utility is
6
工 βj κj u(cj )
j=0
where κ 1 = 0.987,κ2 = 0.971,κ3 = 0.940,κ4 = 0.871,κ5 = 0.743, and κ6 = 0.506.
Say John has stable income for the first 5 periods of his life and no income afterwards. That is y0 = y1 = y2 = y3 = y4 = 650, 000 and y5 = y6 = 0. The real interest rate, r , is 5%. Set up the problem with one intertemporal budget constraint as the following.
≤ 
(a) (6 points) Compute John’s optimal consumption plan.
(b) (4 points) John’s wealth at period j is denoted by aj and saving by sj . John has
no inheritance; a0 = 0. The law of motion for wealth is described as the following: sj = raj + yj − cj
aj+1 = aj + sj
Plot John’s consumption, saving, and wealth over his life time. Comment on your plots.
2022-10-28