du/bbcswebdavl pld-4062304-dit-content-rid-269802 Problem Set 3 Eco 383, Spring
ID: 3109947 • Letter: D
Question
du/bbcswebdavl pld-4062304-dit-content-rid-269802 Problem Set 3 Eco 383, Spring 2017 Capital Income Taxes Consider a simple model of a consumer's intertemporal consumption problem. Paul lives for two periods, working in the first and retiring in the second. Paul earns an income of I in the first period and zero in the second. He must decide how much of his income to consume in the first period and how much to save for consumption in the second period. On any money that saves for the second period, Paul earns interest at the rate r. So, Paul's consumption. in period 2 is given by (1 +r). (I c1). Paul's utility function can be written: 1. If Paul has income equal to $100,000 in the first period, and the market interest rate is 10%, how much does Paul consume in each period? How much does Paul save? 2. If Paul must pay income tax at the rate of 25% of income on labor income in the first period, how do your answers to question (a) change? Does the ratio of second period consumption to first period consumption change? Why or why not? 3. Now assume that Paul must pay income tax at the rate of 25% of income on labor income in the first period and also on interest income in the second period, how do your answers to question (b) change? Again, does the ratio of second period consumption to first period consumption change? Why or why not? Exercises: Externalities 1. Exercise 13 from Chapter 5 in Gruber's book (page 148). 2. Exercise 15 from Chapter 5 in Gruber's book (page 148). 3. A competitive refining industry releases one unit of waste into the atmosphere for each unit of refined product. The inverse demand function for the refined product is Pa 20-Q, which represents the marginal benefit curve where Q is the quantity consumed when consumers pay price Pa. The inverse supply curve for refining is P, 2 Q. which represents the marginal private cost curve when the industry produces Q units. The marginal damage is given by MD 0,502,Explanation / Answer
a) From the utility function we have MRS = MUc1/MUc2 = (1/2c1^0.5)/(1/2c2^0.5) = (c2/c1)^0.5. At the same time we have the two period budget constraint as
c1 + c2/1.1 = 100000
1.1c1 + c2 = 110000
The slope of the budegt constraint is 1.1 and slope of the utility function (MRS) is (c2/c1)^0.5. At the optimal choice, MRS = slope of budget equation
(c2/c1)^0.5 = 1.1
c2/c1 = 1.21
c2 = 1.21c1
Substitute c2 in budget equation
1.1c1 + 1.21c1 = 110000
c1* = $47619, c2 = $56719 saving s1 = 100000 - 47619 = $52381
b) When tax rate is 25%, income becomes 100000 - 25% of 100000 = 75000
we have the two period budget constraint as
c1 + c2/1.1 = 75000
1.1c1 + c2 = 82500
The slope of the budegt constraint is 1.1 and slope of the utility function (MRS) is (c2/c1)^0.5. At the optimal choice, MRS = slope of budget equation. This gives
c2 = 1.21c1
Substitute c2 in budget equation
1.1c1 + 1.21c1 = 82500
c1* = $35714.3, c2 = $43214.3 saving s1 = 75000 - 35714.3 = $39285.7
c) Now the after tax income is 75000 but c2 = s1*(1 - 0.25) = (75000 - c1)*(1.1)*0.75 = 0.825*(75000 - c1)
c2 = 0.825*75000 - 0.825c1
0.825c1 + c2 = 61875
Using the same process we can now see that c1 = 30405.4, c2 = $36790.5
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And 3.
If there is no correction for the externality, the equilibrium will occur at the point where the marginal benefit curve,
P^D= 24-Q
intersects the marginal private cost curve,
MPC Q = +2 . This occurs at
24-Q=2+Q
Q =11
At Q =11, price is P =13.
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b) At the social optimum marginal benefit,
P^D=24- Q, will equal marginal social cost,
MSC= MPC+ MEC
This occurs where 2
24 –Q=(2+Q ) +0.5Q
Thus, the social optimum is to produce Q = 8.80 .
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c) At the uncorrected equilibrium, the marginal social cost is
MSC = 2 +1.5(11)= 18.5 .
Thus, the deadweight loss will be 0.5(11 -8.80)(18.5 -13) =6.05
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d) The emissions fee of $T should be set to shift the MPC curve so that it intersects the marginal benefit curve at Q = 8.80 , the socially optimal quantity.
At Q = 8.80 the marginal benefit is P =15.2 and the marginal private cost is MPC = + = 2 8.80 10.80 .
Therefore, the optimal tax is T =15.2- 10.8 =4.4
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