08 Jun Question ECON 2020:B Intermediate Microeconomics I: Production
Question
ECON 2020:B
Intermediate Microeconomics I: Production
Problem Set # 1
Maya Papineau
Department of Economics, Carleton University
Fall 2015
Due Date: Tuesday, September 22, 2015
In this problem set, as in all following ones and the exams, the term marginal refers
to an infinitesimally small increment. In other words, use derivatives to determine
marginal products etc.
1
Short-run production (6 points)
Popsi Inc. produces soft drinks in quantity q in a manufacturing plant in Canada with labor input
L and machines K according to the production function
q = f (K, L) = 7L + (3/2)K 1/2 L2 − L3 /3.
In the short run, no additional machinery can be used, but workers can work overtime, so that the
labor input is variable.
(a) What is the short-run marginal and average product of labor for a given capital stock K? (2)
(b) For the following, assume that K = 4. For which value of L do the functions M PL and APL
intersect? What is the maximum of the APL function? What is the connection? (2)
(c) Where is the production function no longer defined? Why? (1)
(d) Does the law of diminishing marginal returns hold? Why? [One sentence is sufficient.] (1)
2
Long-run production (6 points)
Another firm in the same industry, Cake, considers setting up a plant in Canada and is thus
evaluating its long-run production possibilities there. Besides using machinery K, it can hire labor
L. The production function is
q = f (K, L) = K 0.5 L0.5 .
(a) What is the equation of the isoquant, i.e. K as a function of labor (and output)? (2)
1
(b) What is the marginal rate of technical substitution? (2)
(c) The manufacturer estimates that it will sell 12 units of its product. How many additional units
of capital do you need to substitute for one marginal unit of labor when L = 6? How many do
you need when you want to substitute for one marginal unit of labor when L = 12? In which
case do you need more additional capital?1 (2)
3
Production (6 points)
The firm Avanti produces custom-made sports cars that appeal to a certain clientele because of
their high price and thus exclusivity. Avanti has the following production function:
q = f (k, l) = k 2/3 l1/3 ,
where k and l are the capital and labor input respectively.
(a) What additional output does one additional marginal unit of capital produce when the capitallabor ratio is (k/l) = 8? (2)
(b) If you employ one less (marginal) worker, how many additional units of capital do you need
to produce the same output? Again, assume that the capital-labor ratio is (k/l) = 8. [Do not
assume this any further.] (2)
(c) How many units of labor does the firm need to produce q units of output in the short run when
the capital stock is fixed at a level of k = 5? (2)
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