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```					                                                                                 Econ 3070
Prof. Barham
Problem Set 3 Questions
(All book questions on this homework are from Edition 2)

Reminder:
Maximization question must be solved using method shown in class not
by using the tangency condition used in the text.

1. Ch5, problem 5.5 (show how you get the marginal utilities that are given in the question)

2. Aunt Joyce purchases two goods, perfume and lipstick. Her preferences are represented
by the utility function
U (P , L ) = PL ,
where P denotes the ounces of perfume used and L denotes the quantity of lipsticks
used. Let PP denote the price of perfume, PL denote the price of lipstick, and I denote
Aunt Joyce’s income.

a. Derive her demand for perfume. Your answer should be an equation that gives P as
a function of PP , PL , and I.

b. Derive her demand for lipstick. Your answer should be an equation that gives L as a
function of PP , PL , and I.

c. Is lipstick a normal good? Draw her demand curve for lipstick when I = 200. Label
the demand curve D1. Draw her demand curve for lipstick when I = 300 and label
this demand curve D2.

d. What can be said about her cross-price elasticity of demand of perfume with respect
to the price of lipstick?

3. Uncle Bob purchases two goods, tweed sport coats and bow ties. His preferences are
represented by the utility function
U (B, C ) = B 0.25 C 0.75 ,
where B denotes the number of bow ties purchased and C denotes the number of sport
coats purchased. Let \$25 be the price of bow ties and \$60 be the price of sport coats.
And finally, let I denote Uncle Bob’s income.

a. Derive Uncle Bob’s Engel curve for bow ties. Your answer should be an equation
that gives B as a function of I.

b. Draw Uncle Bob’s Engel curve for bow ties on a graph with B on the horizontal axis
and I on the vertical axis.
Econ 3070
Prof. Barham
c. Are bow ties a normal good? What can be said about Uncle Bob’s income elasticity
of demand for bow ties?

4. Ch 5, problem 5.8
5. Ch 5, problem 5.10
6. Ch 5, problem 5.18

7. Suppose that the production function for lava lamps is given by
Q = KL2 − L3 ,
where Q is the number of lamps produced per year, K is the machine-hours of capital,
and L is the man-hours of labor.

Suppose K = 600.

a. Draw a graph of the production function over the range L = 0 to L = 500, putting L
on the horizontal axis and Q on the vertical axis. Over what range of L does the
production function exhibit increasing marginal returns? Diminishing marginal
returns? Diminishing total returns?

b. Derive the equation for average product of labor and graph the average product of
labor curve. At what level of labor does the average product curve reach its
maximum?

c. Derive the equation for marginal product of labor. On the same graph you drew for
part b, sketch the graph of the marginal product of labor curve. At what level of
labor does the marginal product curve appear to reach its maximum? At what level
does the marginal product equal zero?

8. Consider again the production function for lava lamps: Q = KL2 − L3 .

a. Sketch a graph of the isoquants for this production function.

b. Does this production function have an uneconomic region? Why or why not?

9. For each of the following production functions, graph a typical isoquant and determine
whether the marginal rate of technical substitution of labor for capital ( MRTS L ,K ) is
diminishing, constant, increasing, or none of these.

a.   Q = LK , for Q=4

b.   Q = L K , for Q=2

c.   Q = L2 3 K 1 3 , for Q=8
Econ 3070
Prof. Barham
d.   Q = 3L + K , for Q=3

e.   Q = min{3L , K } , you chose Q

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