## ECONOMICS

Chapters 5 & 6 Homework #3 Due: September 26th, 2017

ECON 361 – Intermediate Microeconomics Page 1 of 1

1. Thomas has a quasi-linear utility function of the form

U (x; y) = x3 + 2y:

a) What are Thomas’s marginal utility functions, MUx

and MUy?

b) Suppose I = 30, Px = 12, and Py = 2. What is utility

maximizing basket?

c) Derive Thomas’s demand curve for x (in terms of ex-

ogenous variables I, Px and Py).

d) Derive Thomas’s demand curve for y (in terms of ex-

ogenous variables I, Px and Py).

2. Frank’s preferences over oranges (x) and other goods (y)

are given by U (x; y) =

p

xy, and he has an income of

I = $120. (Hint: Might need calculator)

a) What are his marginal utility functionsMUx &MUy?

b) Calculate the optimal basket when Px = 6 & Py = 3.

c) Suppose Px decreases to Px = 1, nd the new optimal

basket.

d) Calculate the income and substitution eects when Px

decreases to Px = 1.

e) What is the compensating variation of the price

change.

f) What is the equivalent variation of the price change.

3. Suppose the market for cars has two segments, businesses

and home users. The demand curve for cars by businesses

is P = 120 40Qb, where Qb is the quantity of cars

demanded by businesses with the price is P. The demand

curve for cars by home users is P = 40 10Qh, where

Qh is the quantity of cars demanded by home users when

the price is P. Both businesses and home users will never

demand negative amounts of cars, so for suciently high

prices, the demand will be 0.

a) Graph the demand curves for each segment, and draw

the market demand curve for cars.

b) Write the equation for the demand curve for all prices

P 0 (make sure that it matches with part (a)).

c) When the price is P = $20, what is the consumer

surplus for businesses, home users, and the market as

a whole.

4. A rm uses the inputs of fertilizer, labor, and hothouses

to produce roses. Suppose that when the quantity of

labor and hothouses is xed, the relationship between the

quantity of fertilizer and the number of roses produced

is given by the following table:

Tons Fertalizer 0 1 2 3 4 5 6 7 8

1000’s of Roses 0 0:5 1 1:7 2:2 2:5 2:6 2:5 2

a) What is the average product of fertilizer when 4 tons

are used?

b) What is the marginal product of the sixth ton of fer-

tilizer?

c) Does this total product function exhibit diminishing

marginal returns? If so, over what quantities of fer-

tilizer do they occur?

d) Does this total product function exhibit diminishing

total returns? If so, over what quantities of fertilizer

do they occur?

5. A rm is required to produce 50 units of output using

quantities of labor and capital (L;K) = (3; 8). For each

of the following production functions, state whether it is

possible to produce the required output with the given

input combination. If it is possible, state whether pro-

ducing Q = 50 with input combination is technically ef-

cient or inecient.

a) Q = 8L + 4K

b) Q = 10

p

KL

c) Q = min f17L; 7Kg

d) Q = 2KL + L 6.

6. Consider the production function Q = KL2 L3.

a) Sketch a graph of at least 4 isoquants from this pro-

duction function.

b) Does this production function have an uneconomic

region? If so, describe the region algebraically (Hint:

your answer will be an inequality like this: K < L)?

7. A rm’s production function is Q = 3L1=3K2=3.

a) Does this production function have constant, increas-

ing, or decreasing returns to scale?

b) Determine MRTSL;K for this production function.

c) What is the elasticity of substitution for this produc-

tion function? (Hint: what type of production func-

tion is this?)

8. A rm’s production function is initially Q1 = K

1

3L

2

3 .

Over time, the production function changes to Q2 =

KL

2

3 .

a) Does this change represent technological progress?

b) Is this change labor saving, capital saving, or neutral?