HW6 - Washington State University

Homework 6
EconS 301, Spring 2015 Dr. Rosenman
The value of each question is given below
1. Suppose production is guided by the fixed proportion production function Q=min{2K,L}. Let K be
the fixed factor.
a. Graph the long run expansion path, and the short run expansion path if K=100. Why can
you do this without knowing input prices? (2 points)
b. Let PK=2 dollars per unit, and PL=4 dollars per unit. Find the long run total, average and
marginal cost curves. Can you find short run cost curves as well? Why or why not? (3
points)
c. What are the economies of scale? (1 point)
2. Inventory problems offer a simple example relating production to cost. Let Q be the annual flow
of goods through storage or service. This could be, for example, light bulbs for a university. We
will assume that the flow is met by inventory (I) and reorder (R). Thus, the production function
is simply Q=RI. Inventory is the amount kept on hand, and R is the number of times the storage
must be replenished. Inventory capacity is the fixed factor, reordering is the variable factor. Let
the cost of maintaining a unit of storage for a year be $1.00 per unit, and let the cost of each
reorder equal $100.00.
a. Suppose I is fixed in the short run at 1000. Derive the short run cost function, and the
average total cost, average fixed cost, average variable cost and marginal cost of output.
Find the short run total cost for Q=10000 and for Q=12000. (4 points)
b. Derive the (long run) optimal input combination, and use this ratio with the production
function and the total cost function to find the long run total cost function. Find the
long run total cost for Q=10000 and Q=12000. Why are these different from the short
run costs? (4 points)
c. Using the production function find the returns to scale, and the long run cost function
find the economies of scale (look at the function for long run ATC). Are they consistent?
How will the short run and long run cost functions change if the price of a reorder
increases to 121 dollars? (2 points)
3. Auto malls are "super markets" for cars, where large multi-franchise dealers offer a variety of
competing brands. Thus, a single dealer might offer Porche, Saab, BMW, Mercedes Benz,
Cadillac and a myriad of other luxury cars under one roof - an unusual arrangement for brands
that usually compete with one another. The owner of one auto mall, Motor Werks in Illinois, is
quoted in the Wall Street Journal as saying "I have one chief financial officer, one general sales
manager, and one owner - me. That's where the economies start paying off." Auto malls also
allow firms to remove redundant advertising and administrators. Explain what the expansion
path for auto malls might look like, and why we are indeed talking about economies of scale and
not increasing returns to scale in this example. (2 points)
4. Venture capitalists weary of risk often turn to well-established low-tech firms as places to invest
their money. One strategy is to buy up a number of smaller firms in the same field and combine
them into one larger firm. In doing so, the venture capitalists can follow two general strategies;
allow the firms to operate in a manner similar to the way they did as independent firms, or
consolidating the operations of the different firms into a single entity under a single
management, often combining physical plants as well. Explain why the former strategy is more
like a scale change which could be analyzed with returns to scale, but the second strategy
presents a problem more amenable to analysis by economies of scale. Also, discuss why you
may see different effects on average cost between the two strategies, and why it is not clear
which strategy is superior. (3 points)
5. A state interested in raising revenue is considering three taxes. One is a fixed license fee which
all firms in the state would be required to pay. The second is a tax on output, where each unit
produced would be taxed a set amount. The third tax would be a labor tax. Firms would pay a
tax equal to a portion of the wages it paid. Using graphs, contrast how each of these taxes
would affect short run and long run cost curves, including total cost, marginal cost, and where
relevant, variable and fixed cost. (4 points)
Extra Credit (5 points)
Q= KL where K is capital (the fixed factor) and L is labor (the variable factor). Let PK=PL=1.
a. If K=16, find the short run total, average, average variable and marginal cost functions.
b. Find the long run total and average cost functions.
c. Determine the type of returns to scale and economies of scale this production function shown
by this production function.