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Unformatted text preview: Department of Economics
Columbia University W3211
Fall 2011 P r ob l e m Se t 6
I n t e r m e d i a t e M i c r o e co n o m i cs
P r of . Se y h a n E A r k on a c
1. Sam and Erica are starting a new restaurant in Portland, Oregon. While Sam plans to do the
cooking himself, he will need to employ workers and machinery to produce food. He
estimates his production function as:
q = 15L K
Sam is able to accumulate $10,000 to finance the business. Workers cost $10 and capital
(a) If Sam wishes to produce the most output with the finances available, how much labor and
capital should Sam employ. Use a Lagrangian to solve this problem.
(b) Does this bundle of capital and labor also minimize the costs? Explain using a graph.
2. What is the last dollar rule for cost-minimization? Provide a brief explanation (in words) as
well as the corresponding mathematical equality. If the firm is producing at a point where
the isocost line is st e epe r than the isoquant, what does the last dollar rule imply (i.e., where is
the last dollar most productive, L or K) and how should the firm alter its capital and labor in
the long run?
3. Suppose Ralph hires workers at his supermarket at a wage of $12/hour. Ralph currently has
10 checkstands (i.e., capital) with a rental rate of $10/hour. Production of customers served
(i.e., output) is determined by the hourly production function f(L , K ) = 0.5 L 3/4 K 2
For the questions that follow, the number of checkstands is fixed. Show your work clearly.
(a) If Ralph wants to serve 400 customers per hour, how many workers must he employ?
How much will it cost to serve 400 customers per hour?
(b) Derive Ralph’s short-run cost function with the 10 checkstands.
(c) Derive the equations for the MC, AC, AVC, and AFC.
4. A paper company dumps non-degradable waste into a river that flows by the firm’s plant. The firm estimates its production function to be:
Q = 6 K P,
where Q = annual paper production measured in pounds, K = machine hours of capital, and P
= gallons of polluted water dumped into the river per year. The firm currently faces no
environmental regulation in dumping waste into the river. Without regulation, it costs the
firm $7.50 per gallon dumped. The firm estimates a $30 per hour rental rate on capital. The
1 firm produces 600 million pounds of paper per year. For this problem, consider the lon g- r u n
production of output.
(a) Determine the firm’s optimal ratio of wastewater to capital.
(b) Given the firm’s output of 600 million lbs, how much capital and wastewater should the
(c) How much will it cost the firm to produce the 600 million lbs of paper?
(d) The state environmental protection agency plans to impose a $7.50 fee for each gallon
that is dumped (this is in addition to the current cost of $7.50). Assuming that the firm
intends to maintain its same output level, how much capital and wastewater should the
(e) How much will the firm pay in fees? What happens to the firm’s cost as a result of the fee?
5. Consider a firm with two technologies to choose between when producing output. The cost
function when using technology 1 is given by:
2 c1(q) = 3600 + 65q + 36q
The cost function when using technology 2 is given by:
c2(q) = 900 + 900q +q
Assume that the firm can only implement one of the two technologies at a time.
(a) If the firm wishes to produce output at the lowest per-unit cost, which technology should
it choose and how much output should it produce?
(b) Which technology should the firm choose if it wishes to produce 15 units of output?
What about 25 units of output?
6. A firm produces output according to the following function:
q = f (L, K) = L K
The cost of labor is $9 per hour and the rental cost of capital is $4 per hour.
(a) With the given prices, use the Lagrangian method to compute the optimal (costminimizing) capital to labor ratio (K/L) for the firm.
(b) Suppose the firm wishes to produce 72 units of output. How much capital and how much
labor does the firm employ?
(c) What is the total cost of producing 72 units of output?
(d) Suppose that the firm suddenly decides to double the quantity of output but only has a day
to complete the order. Therefore, in that time, the amount of capital is fixed but labor hours
are not. How much will it cost to produce 144 units of output? How much would it cost if
the firm could also vary capital? Compute as well as providing a graph (isocost/isoquant)
illustrating the optimal bundles.
7. Suppose that all firms in a constant-cost industry have the following long-run cost curve:
c(q) = 4q2 + 100q + 100
The demand in this market is given by QD = 1280 - 2p. Suppose the number of firms in the
market is restricted to 80
(a) Derive the supply curve with this restriction. Find the market equilibrium price and
2 quantity with the restriction.
(b) If firms are allowed to buy and sell these permits in an open market, what will be the
rental price of permits? Will firm’s that own permits make profit? Briefly explain.
(c) How much deadweight loss is generated by the permit system? Provide a graph showing
the region of this deadweight loss.
(d) Suppose the government abandons the permit system and simply imposes a fixed fee on
firms in the market. If the fee is set equal to the permit price you found in c., what will
be the equilibrium price, quantity, number of firms and deadweight loss? F ol l ow i ng q u est i ons w i l l not b e g r a d e d , t h e y a r e f o r you to p r a c t i c e a n d w i l l b e d i sc usse d a t t h e
r e c i t a t ion :
6. Ch. 7 question 22
Ch. 7 question 7
Ch. 8 question 6
Ch. 7 question 32
Ch. 7question 33
Ch. 7question 34 3 ...
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