MicroEconomics Calculus
lolly02
sol_mid1_f2015_w1.pdf
S: M- I E500, 10:00 (W) October 7, 2015
Question 1
(a) A utility function is given by u(x1, x2) = x1 x2 + 2x2. Then the MRS is
MRS = x2
x1 + 2 .
(1,4) is given by
M RS(1, 4) = 4/3 . 6 points
(b) Suppose that for some preferences that MRS is always equal to 3. Specify a utility function that describes these preferences.
u(x1, x2) = 3x1 + x2 . 6 points
Question 2 Solve the following optimization problem graphically: maxx1,x2 x1 + 3x2, subject to (i) 8 + x1 − 2x2 ≥ 0, (ii)x1 + x2 ≤ 22, (iii) x1 ≥ 4, (iv) x2 ≥ 2. 12 points
Clearly indicate the feasible set by shading it. Also graph at least three lines rep- resenting the objective, including the one through the optimum.
The solution is x1 = 12, x2 = 10 .
The value of the objective at the solution is 42 .
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Question 3 Let u(x1, x2) = min{x1 + 2x2,3x1 + x2}. Suppose that (10,10) is an optimal consumption bundle and income I = 60. Graph a budget set (shade the set) that is yields this optimal consumption and the indifference curve through the optimum.
You can conclude that prices are p1 = 2 , p2 = 4 . 6 points
Next, suppose that prices changes to p1 = 3, p2 = 2. Determine graphically the
least costly consumption: x1 = 6 , x2 = 12 . 6 points
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Question 4 A demand function is depicted below. Suppose the demands is given by QD(P) = 30 − P. The government imposes a tax of t = 6 Dollars per unit sold on consumers, i.e., consumers pay a price P + t, while produces receive a price of
P per unit. Then the equilibrium price and quantity are P = 8, Q = 16 . The
government’s tax revenue is $128 . 12 points
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Question 5 A utility function is given by u(x1, x2) = 5x1 + 2x2. Suppose that prices are p1 = 2 and p2 = 1. The person’s income is I = 40. Then 12 points
Optimal consumption is x1 = 20, x2 = 0 .
Suppose the government introduces a tax of 2 Dollars per unit of good 1, raising the price of good 1 increases to p1 = 4. Hicksean Demand is (0,50). In order to get the same utility as before the price change,
The person’s income must be I′ = 50 .
The deadweight loss is 10; the government’s tax revenue is 0.
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Question 6 For particular preferences the MRS is given by
MRS = 2
(
x2 x1
)3
.
(a) The equation of the income offer curve is
MRS = 2
(
x2 x1
)3
= p1 p2 .
Thus, x2 = 0.5 1/3(p1/p2)
1/3. Let q be the price ratio, then the slope of the income offer curve is A(q) = 0.51/3q1/3. Thus, the elasticity of substitution is
1/3 6 points
(b) Now
MRS = 2
(
x2 x1
)3
= 2 8 .
Thus, 2x2 = x1. The budget line equation is 2x1 + 8x2 = 180. Thus optimal consumption is 5 points
x1 = 30, x2 = 15 .
(c) Goods are less substitutable than in the Cobb-Douglas case. For Cobb-Douglas the elasticity of substitution is 1, whereas is 1/3 in this case. As a conse- quence, the substitution effect is smaller than in the Cobb-Douglas case, and hence an increase in the price of good 2 should lower the demand for good 1.
5 points
Question 7 Suppose that a utility function is given by u(x1, x2) = x21 x2. Suppose that prices are p1 = 1, p2 = 4. Then 4 points
MRS = 2x2/x1 = 1/4.
The equation of the income offer curve is x2 = (1/8)x1.
In addition u = x21 x2. Thus, u = (1/8)x 3 1, i.e., x1 = 2u
1/3. 4 points
h1(1, 4, u) = 2u1/3, h2(1, 4,u) = (1/4)u1/3
(Recall: u is the utility level that we want to obtain, which is kept here is a variable)
The minimum expenditures to get utility u at these prices is: 4 points
e(1, 4, u) = 3u1/3.
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Question 8 Suppose a consumer’s preferences are given by u(x1, x2) = x1 + log(x2).
Then MRS = x2 = p1/p2. 2 points
The equation of the income offer curve is x2 = p1/p2
The budget line equation is p1 x1+p2 x2 = I. Thus, p1 x1+p1 = I, i.e., x1 = I/p1−1.
(Walrasian) demand is therefore 5 points
x1( p1, p2, I) = (I/p1) − 1, x2( p1, p2, I) = p1/p2
We must have x2 = p1/p2 and u = x1 + log(x2). Thus, u = x1 + log(p1) − log(p2).
Hicksean demand is therefore 5 points
h1( p1, p2, u) = u + log( p2) − log( p1), h2( p1, p2, u) = p1/p2
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