Student Number:
QUIZ 4
Section A
Question 1. [5 marks] Consider the short-run production function
Q  25LK
a) [2.5 marks] Derive the cost-minimizing quantity of labor as a function of output and
capital. How does the cost-minimizing quantity of labor vary with Q ? And how does it
vary with K ?
Answer:
To find the cost-minimizing quantity of labor in the short-run, simply solve the
production function for L in terms of output and capital.
Q  25 LK
Q
L
25 K
Since output is in the numerator, the cost-minimizing quantity of labor will increase
as the quantity of output goes up. Conversely, since the fixed level of capital is in the
denominator, the cost-minimizing quantity of labor will decrease as the quantity of
capital goes up.
b) [2.5 marks] Assume the price of labor is w  50 and the rental rate of capital is r  10 .
Derive the short-run total cost function. How does the short-run total cost vary with Q ?
with K ?
Answer:
To find the short-run total cost function, plug the cost-minimizing quantity of labor and
the fixed level of capital into the cost function TC  wL  rK .
 Q 
TC  50 
  10 K
 25 K 
2Q
TC 
 10 K
K
1
Since output only shows up in the numerator of the left-hand term, total cost will
increase as the level of output goes up. The fixed level of capital shows up in both
terms. Since it is in the denominator of the left-hand term, as K increases, the lefthand term will fall, i.e., the labor costs will fall. At the same time, however, as K
increases the right-hand term will also increase, i.e., capital costs will increase.
Therefore, without knowing more about the current level of K , it is impossible to
determine whether total costs will increase or decrease.
Question 2. [3 marks] Consider the cost function Y=5*min{2*x1,3*x2}, x1 and x2 are
the two inputs and let w1 and w2 represent the input prices respectively. Y is the output.
Derive the conditional input demand functions, i.e. demand for x1 and x2. (Note that
generally conditional input demand functions are functions of Y and the prices of inputs,
but that does not hold true always).
Answer:
Note that this production function will have L shaped isoquants and the firms will always
produce efficiently only at the corner point of the L shaped isoquant. So the inputs used
will satisfy: x2 = (2/3) x1, which gives a straight line passing through the corner points of
the isoquants. Now note that if firm wants to produce Y units of output efficiently (i.e at
minimum cost), the production function gives: Y = 5*2*x1 = 5*3*x2, which gives the
conditional input demand functions as:
x1 = Y/10, and x2 = Y/15
Note that because of fixed proportions production function the conditional input demand
functions are independent of the prices of inputs. That is, for example, we must use
x1 = Y/10, if we want to produce Y units of output, no matter what the price of x1 and
x2.
Question 3. [2 marks] Derive the cost function (as a function of Y, w1 and w2) for the
production function given in Question 2.
Answer:
C(Y,w1,w2)=Y[(w1/10)+(w2/15)]
2