Intermediate Microeconomics
Due:
1.
June 2, 2004
June 9, 2004
Prof .Li-Chen Hsu
Problem Set #8
(20 points) Is the following statement true or false? Briefly explain your answer.
(a) If the production function is f ( x1 , x2 )  min{x1 , x2 } , then the cost function
is c( w1 , w2 , y)  min{x1 , x2 }y .
(b) A firm use a single variable input x to produce outputs according to the production
function f ( x)  300 x  6 x 2 . This firm has fixed costs of $400. This firm’s short- run
marginal cost curve lies below its short-run average variable cost curve for all
positive values of x.
(c) The cost function C ( y )  10  3 y has marginal cost less than average cost for all
levels of output.
(d) The cost function C ( y )  100  3 y 2 has marginal cost less than average cost for all
levels of output.
(e) If marginal costs increase as output increases, then the average fixed cost curve will
be U-shaped.
2.
(5
points)
If
the
production
function
is
given
by
and
the
prices
of
f ( x1 , x2 , x3 , x4 )  min{x1 , x2 }  2 min{x3 , x4 }
inputs ( x1 , x2 , x3 , x4 )  (2,1,5,3) . What is the minimum cost of producing 1 unit of
output?
3.
(8 points) An orange grower has discovered a process for producing oranges that
requires two inputs. The prices of these two inputs are w1  $5 and w2  $2 , respectively.
What are the total costs of producing 40 units if
(a) The production function is Q  min 2 x1 , x2 
(b) The production function is Q  min  x11/ 2 , x21/ 2 
4.
(16 points) A competitive firm has short-run cost function c( y)  y 3  2 y 2  5 y  6 . Write
down equations for:
(a) The firm’s average variable cost function.
(b) The firm’s marginal cost function.
(c) At what level of output is average variable cost minimized?
(d) Graph the MC curve and AVC curve for this firm, being careful to label the key
points on the graph with the numbers specifying the exact prices and quantities at
these points.
20038-1
5.
(20 points) The production function is Q  L2/ 3 K 1/ 3 . The price of labor and capital are w
and r. In the short run, the amount of capital is fixed at K  K 0 . Please answer the
following questions.
(a) Derive the short run conditional factor demand of L.
(b) Derive the short run total cost function, marginal cost function and average cost
function.
(c) If w  1, r  2, K 0  4 , answers (a) and (b) again.
(d) Derive the long run conditional factor demand function of L and K.
(e) Derive the long run total cost function, marginal cost function and average cost
function.
(f) If w  2, r  3 , answers (d) and (e) again.
6.
(5 points) A firm has two factories. One factory has the cost function c1 ( y1 )  2 y12  90
and the other has the cost function c2 ( y2 )  6 y22  40 . If the firm wishes to produce a
total of 32 units as cheaply as possible, how many units will be produced in the second
factory?
7.
(5 points) A firm produces Ping-Pong balls using two inputs. When input prices are ($15,
$8) the firm uses the input bundle (17, 70). When the input prices are ($12, $24) the firm
uses the input bundle (77, 5). The amount of output is the same in both cases. Is this
behavior consistent with WACM? Why?
8.
(6 points) A firm has the long-run cost function C (Q)  4Q 2  64 .
(a) What is the long –run supply curve of the firm?
(b) When will the firm supply a positive amount of output?
9.
(8 points) A firm has the short-run cost function c( y)  2 y 3  16 y 2  64 y  50 .
(a) What is the short-run inverse supply function of the firm?
(b) When will the firm produce a positive amount of output?
10. (7 points) A competitive firm’s production function is f ( x1 , x2 )  12 x11/ 2  4 x1/2 2 . The price
of factor 1 is $1 and the price of factor 2 is $2. The price of output is $4. What is the
profit-maximizing quantity of output?
20038-2