08 Jun Question Exercises | Practice on the producer
Question
Exercises | Practice on the producer
1. Exercise 1 – A rm manufactures a product in quantity Y , using quantity L (hours)
of labor and quantity K of capital. The technology is represented by the production
function:
f (L; K) = L + minfL; Kg
The price of capital is xed and equal to 1. The hourly wage is !.
(a) Draw the isoquant associated to the production of Y = 1.
(b) To produce a quantity that is not equal to zero, is it still indispensable to use
the capital? Is it still indispensable to use the labor?
(c) What is the optimal combination of inputs:
(a) when ! > 1;
(b) when 0 < ! < 1;
(c) and when ! = 1?
(d) Give the expression C(Y ; !) of the long run cost function of the rm in each
of the 3 previous cases.
Exercise 2 – Consider a rm producing some output in quantity Y using 3 factors:
labor L, energy E and capital K. The labor and energy are variable costs and the
capital is a xed cost in the short term.
The production curve is:
(
Y = (K
Y =0
1
1
1
1) 4 L 4 E 4 si k 1
otherwise
The unit price of each factor is equal to 1.
(a) Give the expression of the marginal and average cost functions in the short
run.
(b) Infer from the preceding results, the marginal and the average cost functions
in the long urn.
(c) In this question, we are considering the long run. The company maximizes its
pro t. It forecasts that the unit sales price of its product is going to be p = 6.
What capital stock does it select?
(d) Suppose that the company has acquired the capital stock determined in question c, and consider now the short run, the capital stock being xed at the
latter amount. The price nally turns out to be lower than expecxted: p = 5.
What quantity does the company decide to produce? What quantity would it
have produced if it had forecasted a price of 5 to begin with?
(e) Compute the pro t loss due to the forecast error on the sales price.
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Exercise 3 – We consider a market where identical rms operate in a context of
perfect competition. We consider the long run context. All rms have the same
long run cost function, composed with a quadratic part and a lump-sum :
C(Y ) = 5Y 2 + 320
The cost of entry (for new entrants) on this market is 3600. This is a one-time only
expenditure and cannot be recovered. This cost corresponds to data base software
purchases allowing to track orders and payments. Such an asset cannot be resold.
As the companies have an interest rate of 5%, such a cost is equivalent to an annual
charge of 180.
(a) What is the entry price of a generic new entrant rm? (Hint: the minimum
point of a function with expression f (Y ) = aY + b=Y is reached when Y =
q
b=a:)
(b) What is the exit price of a generic rm if we suppose that the entry cost is
sunk?
(c) What is the exit price of a generic rm if we suppose that the set-up investment
can be sold at price 1700 (we’ll suppose the same discount rate of 5%)?
(d) Suppose the demand for the product is expected to expand, and be given by
the expression:
Qd = 900 p
What will be the long run equilibrium price? How many rms operate on this
market in the long run?
(e) There is a brutal fall of demand; the new demand is given by the expression:
Qd = 240
p
What will be the new long run equilibrium price? How many rms exit?
Exercise 4 – On a local market, Firm A is producing glass out of capital, labor and silicium (we neglect other intermediate variable inputs). Capital is xed
for the next two years (the short run in that industry). Labor and silicium are
perfect complements. Firm A needs 1 ton of silicium per produced ton of glass, and
a worker is producing 100 tons of glass per year. Hence the Firm A’s short run
production function (for the xed capital) writes:
Y = f (L; S) = min f100L; Sg
L, the quantity of labor used, is measured in number of workers; S, the quantity
of silicium used, and Y , the quantity of glass produced, are measured in tons. The
price of silicium is 100$ a ton, the price of labor is 15,000$ per year per worker.
(a) Give the expression of the average variable cost (AVC) curve of Firm A.
2
Next year, a potential rival, Firm B, whose AVC is constant equal to 220$ a ton
of glass, is contemplating entering on this local market with a penetration price of
230$ a ton of glass.
(b) What is the rationale of this penetration price?
Forecasting this potantial entry, Firm A turns toward its supplier of silicium, Firm
C, to warn that next year, it might not order the usual 10,000 tons of silicium, but
maybe less, and maybe none (in case of shut-down). The supplier then wants to
impose a `Take-or-Pay’ contract to Firm A:
either Firm A takes the usual 10,000 tons of silicium at the standard 100$ a
ton,
or in case it has to reduce production and take only, say, N tons of silicium
(N smaller than 10,000), Firm A will be charged the usual 100$ a ton for the
N tons that it actually takes, and have to pay 80$ for the (10,000-N ) tons of
silicium that it does not want (need) any more.
Firm A accepts and warns Firm B that it signed such a Take-or-Pay contract. Then
Firm B renounces entering the considered market.
(c) Explain why Firm B renounces. (Hint: it may be usefull to give indications
on the new penetration price that Firm B should post to enter the considered
market.)
(d) Would it be possible to reach the same aim (pushing Firm B to renounce) by
guaranteeing Firm A’s worker a compensation in case of unemployment? What
level of unemployment compensation would make it?
Exercise 5 – A shoe manufacturer F holds a monopoly on the market. It faces the
following demand:
Q = 10 p
where Q corresponds to millions of pairs of shoes purchased and p the price per
pair. The production cost is 2$ a pair.
(a) What is the pro t maximizing price ? How many pairs are sold at that price?
The Trade Commission does not approve of this monopolistic situation. It thus
proposes to break the company into two distinct entities. Firm F1 sells the right
shoes at price p1 whereas rm F2 sells the left shoes at price p2 . Demand for the
pairs remains the same:
Q = 10 p1 p2
The marginal production cost for each shoe is 1$.
3
(b) Each rm maximizes its own pro t in taking for granted the price posted by
the other rm. Give the expression of p1 as a function of p2 . Give also the
expression of p2 as a function of p1 .
(c) Compute the new equilibrium (price and total quantity of pairs of shoes sold).
(d) Has the decision to break the rm in two increased consumer surplus?
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