ECO 4933 Topics in Theory Homework 1 Answer Key

Florida International University
ECO 4933 Topics in Theory – Fall 2014
Homework #1 – Due date 09/25/14 – Answer Key
Notice: Print your homework in a letter size sheet. Make sure to include your personal information. If your homework is more than two sheets long number and staple them together. Handwritten submissions are not accepted.

1. Consider a consumer with cuasi-linear preferences represented by the utility function:

U  c, g   c  3 g where c is a private good and g is a pure public good. The price of the private good is $12 per unit and the price of the public good $1. Assume the consumer has
$1,000 to spend. Calculate the utility maximizing quantities of c* and g*.
Answer:

Max U  c, g   c  3 g
s.t. 12c  g  1000 c0 g0
L  c  3 g  0 1000  12.c  g   1c  2 g

The first order conditions are:
L
 1  0 .12  1  0
c

(1.1)

L 1  2 3
 g  0  2  0
g 3

(1.2)

0 , 1 , 2  0

(1.3)

0 1000  12.c  g   0

(1.4)

1.c  0

(1.5)

2 .g  0

(1.6)

1

We are interested in an interior solution with c, g  0 , so we assume 0  0, 1  0, 2  0 .
We rewrite (1.1) as

and (1.2) as

0 

1
12

1
33 g 2

 0 .

Combining (1.1) and (1.2) we get

1
2
1
  4g 3  1  g   
2
12
4
33 g
1

2

3

 g*  8

To obtain the optimal value of c, we replace g in the budget constraint.

12.c  g  1000
12.c  8  1000
1000  8 c  c*  82.67
12

2. Now assume that the consumer has $1,800 to spend. How will the change in income affect the optimal quantities of c and g. Why?
Answer:

g*  8 c*  149.33
The new optimal values show us that with a cuasi-linear utility function all the income affect will be absorbed by the private good.
3. Now the utility function is U  c, g   5c  g and the consumer has $1,500 to spend. Find the values of c and g that maximize his utility.
Answer:

g*  1.44 c*  124.88

2

4. FIU offers bus service between the Modesto Maidique Campus and the Biscayne Campus.
Suppose that a bus arrives at the