MicroEconomics Calculus
lolly02
sol_mid1_f2014_w1.pdf
S: M- I E500, 10:00 (W) October 8, 2014
Question 1
(a) Suppose that supplyQS (P) = 20+ 2P and that demand isQD(P) = 100− 2P. Then in equilibrium 20+ 2P = 100− 2P
P = 20, Q = 60 . 6 points
(b) Now 20+ 2P = 100− 2(P − 12). Then in equilibrium
P = 26, Q = 72 . 6 points
Question 2 A consumer’s preferences are given byu(x1, x2) = min{x1,3x2}. The per- son’s income isI = 30. The price of good 2 isp2 = 1. The price of good 1 is p1 = 3 per unit as long asx1 ≤ 5. For every unit in excess of 5, the price isp1 = 1. In the grid below, graph (1) the budget line, (2) clearly indicate the budget set by shading it, (3) the optimal consumption choice, and (4) the indifference curve through the optimal consumption choice. 12 points
The optimal consumption isx1 = 15, x2 = 5
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Question 3 Solve the following optimization problem graphically: maxx1,x2 x1+3x2 sub- ject to (1) x1 + x2 ≤ 30, (2) x2 ≥ 10+ x1, (3) x1 ≥ 4, (4) x2 ≥ 5. Indicate the feasible set by shading it. Also graph at least three lines that represent the objective. 14 points
The solutions isx1 = 4, x2 = 26 .
The value of the objective at the solution is 82.
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Question 4 Solve the following problem graphically using the grid below. Utility is u(x1, x2) = min{3x1 + 2x2,3x1 + 3x2}, and prices arep1 = 1 and p2 = 1. The person’s income isI = 20. The government imposes a tax of 3 Dollars on each unit of good 2, raising the price top2 = 4. Graph (1) the after-tax budget line, (2) the optimal after-tax consumption, (3) the indifference curve through that point, (4) 14 points
Optimal consumption after the price changex1 = 20, x2 = 0.
h1(1,1, u) = 8, h2(1,1,u) = 8, e(1,1, u) = 16
whereu is the after tax utility.
The government’s tax revenue is 0. The deadweight loss is 4.
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Question 5 A utility function is given byu(x1, x2) = 24 ln(x1 + x2) + x2. Suppose prices are p1 = 1, p2 = 2. Then
MRS= 24
24+ x1 + x2 =
1 2 .
Thus, the equation of the income offer curve is 12 points
x2 = 24− x1.
Graph the income offer curve in the grid below. And determine graphically the following
x1(1,2,30) = 18, x1(1,2,40) = 8
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Question 6 Suppose that a utility function is given byu(x1, x2) = 2 √
x1 + √
x2. Then
MRS= 2 √
x2 √
x1 =
2 3 .
Suppose that prices arep1 = 2, p2 = 3. Then 4 points
The equation of the income offer curve is x2 = 1 9 x1
The utility of (49,49) is 21. Thus, 2 √
x1+ √
x2 = 21, and therefore 2 √
x1+ 1 3 √
x1 = 21. This implies73
√ x1 = 21 and hence
√ x1 = 9. Thus,x1 = 81 andx2 = 9. 8 points
The person needs $189
Question 7 Suppose a consumer’s preferences are given byu(x1, x2) = min{4x1, x2}. The income offer curve isx2 = 4x1. The budget line equation isp1x1 + p2x2 = I. Thus,
x1 = m
p1 + 4p2 , x2 =
4m p1 + 4p2
.
Indirect utility is 6 points
u( p1, p2, I) = 4m
p1+4p2
The expenditure function is 6 points
e( p1, p2, u) = u( p1+4p2)
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Question 8 Several income offer curves are depicted below. Using them, answer the following questions. 12 points
x1(4,2,40) = 5, x2(4,2,40) = 10
Suppose the price of good 2 increases top2 = 4 but the consumer’s income is adjusted tõI so that he/she can afford the same consumption as above.
x1(4,4, Ĩ) = 10, x2(4,4, Ĩ) = 5
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Income Offer curve for p1/p2 = 2.
Income Offer curve for p1/p2 = 1.
x1
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