ECN 312 -Consider a firm which produces a single

Subject: Economics / General Economics
Intermediate Microeconomics ECN 312
Homework 2
Please hand in your solutions before the lecture on Monday, March 20. Please come to an
office hour or write me an email in case you have questions!
1. Consider a firm which produces a single output using two inputs according to the following production
function: y = min{2K, L}, where y is the firms output, K is machinery (measured in machine-hours)
and L is labor supply (measured in person-hours). Let r be the cost of one machine-hour and let w be
the wage rate (i.e. the cost of one person-hour).
(a) Does the firms technology exhibit increasing returns to scale? Explain!
(b) Draw the isoquant corresponding to output of 10 units. Be sure to label important features in the
(c) Suppose that r = 16 and w = 4. Determine the firms cost function c(y).
(d) On a diagram that has costs on the vertical axis and output and the horizontal axis, plot the firms
cost curve c(y), average cost curve AC(y), and marginal cost curve M C(y).
2. Short questions:
(a) If p · M P1 > w1 , (where p is the output price, M P1 the marginal product of factor 1, and w1 the
rental rate of factor 1), should the firm increase or decrease the amount of factor 1 in order to
increase profits? Explain.
(b) If a firm had everywhere increasing returns to scale, what would happen to its profits if prices
remained fixed and if it doubled its scale of operations? Show your work.
(c) A firm is currently earning negative profits on each good it produces. Claim: it is always optimal
for this firm to shut down production in the short run. True or false?
(d) As long as the marginal cost of production is greater than the average variable cost, the average
variable cost is increasing. True of false?
3. Consider the firm ACME which uses capital, K, and labour, L, to produce widgets according to the
following production function: f (K, L) = K 3 L 3 . Let r and w be the prices of capital and labour
respectively. P is the price of widgets. The markets for widgets, capital, and labour are all perfectly
(a) What is the technical rate of substitution between capital and labour for ACME?
(b) In the short run, the level of capital is fixed at K = K. Set up the short run profit maximization
problem. Calculate the short run optimal level of labour.
(c) In the long run, ACME can vary both K and L. Set up ACME’s long run cost minimization problem
by using the Lagrange method. What are the cost minimizing levels of K and L in the long run?
(d) Derive ACME’s long run cost function.
4. Ricardo produces widgets, using as inputs labor (L) and machines (K). His production function is given
by the following equation:
y = 10K 2/3 + L1/2 .
(a) What type of returns to scale (increasing/constant/decreasing) does Ricardos production function
exhibit? Explain.
At the end of last year, Ricardo bought his only machine for $1,000. He will use this machine for 5 years,
after which the machine will have no value. Ricardo will calculate depreciation linearly (depreciation
will be 20% of the initial value of the machine per year). This machine has no other use besides Ricardos
production of widgets, and, at this moment, Ricardo cannot buy any more machines.
(b) What is Ricardos annual fixed cost of production? Is the fixed cost sunk or not? Explain.
(c) What is Ricardos demand for labor as a function of the quantity he wants to produce annually?
(d) Assuming that wage equals 1, what is Ricardos annual total cost function? 5. Sallys firm produces granola bars with a fixed cost of 10 (this cost is already sunk). Her variable cost
function is V C = y 2 + 2y.
(a) Assuming the market for granola bars is competitive, derive Sallys supply function.
(b) What is Sallys revenue if the market price is 6? What is her profit? Does she want to stay in this
market? Explain.
6. In Freedonia, a single firm is in the domestic steel business. It sells steel at $680 per ton, well above the
wold price of $375 per ton. The firm is protected against foreign competition by incredibly high tariffs
that ban all imports.
The firm has never exported steel. The CEO argues: “The average cost of manufacturing steel, which
varies with the rate at which the firm produces steel, is never below $400 per ton. We simply cannot
make positive profits selling steel for $375 per ton.” Is this argument convincing? Provide a short answer
(at most 4 sentences).

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