MicroEconomics Calculus

Solutions: Mid-term I Econ500, 10:00am (White) October 7, 2015

Question 1

max x1,x2,x3

2 √

x1 + √

x2 + 4x3 s.t.

(1) 3x1 + x2 + 2x3 ≤ m; (2) x1 ≥ 0; (3) x2 ≥ 0; (4) x3 ≥ 0.

Exogenous variables: I Endogenous: x1, x2, x3.

Question 2 maxx1,x2 x1+2x2 subject to (i) x1+2 ≤ 2x2, (ii)x1+x2 ≤ 19, (iii) 4x1+x2 ≥ 28, (iv) 2x1 + x2 ≤ 32. 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 = 3, x2 = 16 .

The value of the objective at the solution is 35 .

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Question 3 The indifference curves for utility u(x1, x2) = min{2x1 + x2,20 + x1} are depicted below. Suppose income I = 36 and prices are p1 = 3, p2 = 1. Then 6 points

optimal consumption is x1 = , x2 = . If income increases to I = 48

then consumption of good 2 decreases (circle the correct answer) by 6

units.

Finally, graph two budget lines that demonstrate that one of the goods is a Giffen

good (Clearly indicate these budgets lines in the graph). 6 points

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Question 4 A demand function is given by QD(P) = 140 − 10P. At price P = 10, the price elasticity of supply is 0.5 and total supply at P = 10 is 100 units. The supply

curve is a straight line. Then 12 points

If supply is of the form QS (P) = a + bP, then 0.5 = bP/Q = 10b/100 = 0.1b.

Thus, b = 5. Further, QS (10) = a + 50 = 100. Therefore a = 50.

The supply function is QS(P) = 50 + 5P In equi-

librium, 140 − 10P = 50 + 5P.

The equilibrium price is P = 6

The equilibrium quantity Q = 80. The price elasticity of supply is 5P/Q = 30/80 =

3/8.

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The price elasticity of supply at the equilibrium price is 3/8=0.375

Question 5 A utility function is given by u(x1, x2) = max{2x1, x2} (Note: this the max- imum instead of the minimum. Graph the indifference curves for utility levels 10,

20, and 30.

Suppose that prices are p1 = p2 = 1. Graphically determine the least costly con-

sumption bundle that provides a utility of 20. Then 12 points

Hicksean demand x1 = 10, x2 = 0 .

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Question 6 For utility u(x1, x2) = log(2x1 + x2) + x2, the MRS is given by

MRS = 2

1 + 2x1 + x2 .

(a) Suppose prices are p1 = 1, p2 = 10. Then the equation of the income offer

curve is 2

1 + 2x1 + x2 =

1

10 .

Thus, x2 = 19 − 2x1 6 points

(b) The budget line equation is x1 + 10x2 = I. This, and the income offer curve

imply 6 points

x1 = 10 − I

19 , x2 =

2I 19 − 1 .

(c) What happens to consumption when income is increased? What kind of goods

are 1 and 2? 4 points

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Question 7 Suppose that a utility function is given by u(x1, x2) = x 2 1 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 = x2 1 x2. Thus, u = (1/8)x

3 1 , i.e., x1 = 2u

1/3. 4 points

h1(1, 4, u) = 2u 1/3, h2(1, 4,u) = (1/4)u

1/3

(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.

Question 8 Suppose a consumer’s preferences are given by u(x1, x2) = x1+2x2. If prices 12 points

are p1 = p2 = 2, then the

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The equation of the income offer curve is x1 = 0

Hicksean demand is h1(2, 2, u) =, h2(2, 2, u) = u/2

Expenditures to get a utility of u at prices p2 = p2 = 2 are

e(2, 2, u) = u

Walrasian demand at income I is

x1(2, 2, I) = 0, x2(2, 2.I) = I/2

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