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Problem Set 3
Intermediate Microeconomics
Due Date: May 16, 2012 (Wednesday), before class
(Unless otherwise speci…ed, assume all quantities are continuous rather than
discrete. That means all goods are in…nitely divisible. No need to turn them to integers
when there are decimals.)
Question 1
Are the following statements true or false. Explain your answers brie‡y. The most
important thing is a reasonable explanation with the theory introduced in this class rather
than the true/false answer.
(1)
If one worker must work with one set of tools in a factory, and any more workers
or tools alone cannot increase output, then the production function can be represented
by y = g(L + K), where L is the number of workers, K is the number of sets of tools, and
g is a positive, increasing function.
0:6
(2)
Consider the production function y = x0:6
1 x2 : It is both decreasing marginal
products for every input, and increasing returns to scale.
(3)
f1 (K; L) = K 1=2 L1=2 : f2 (K; L) = 3KL: Both production function represents
the same technology, because they have same technical rate of substitution for all combinations of inputs.
(4)
For an increasing return to scale technology, when you cut all of the inputs by
half, you will get more than half of the original output.
(5)
The …rm owner has zero economic cost to use her own labor input in her …rm.
(6)
A real estate developer has bought a piece of land for apartment development
at a high price. However, after the developer bought the land, the economy turned bad
and the demand for apartment has decreased sharply. This developer will still sell the
apartment at a high price to avoid a loss.
(7)
An increase in …xed cost implies an increase in marginal cost.
(8)
If the marginal cost is falling with output, the average cost must be falling
with output too.
(9)
Four vending machines (like those selling canned and bottled drinks) can replace a worker selling in a small shop at all levels of each input. If the cost of using a
machine is half of the cost of employing a worker each month, the cost minimizing input
choice is to use only the vending machines.
(10)
When the wage of skilled labor is three times the wage of unskilled labor, if at
the current employment level, the marginal product of the skilled labor is only two times
of that of the unskilled labor, the …rm should use more unskilled workers to minimize
cost. (Just a assume a general convex technology.)
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(11)
For a cost minimizing …rm in the long run, if the manager expects the wage
will rise relative to the cost of machines, he will choose a machine of higher quality and
use less labor. (You may regard a higher-quality machine means more capital inputs. For
convenience, you may consider the target output does not change.)
(12)
The long run marginal cost must be lower than the short run marginal cost
curves of all levels of …xed inputs.
(13)
The long run average cost curve cannot cross any short run average cost
curves.
(14)
When pro…t is maximized, the cost of producing the required output level
must have been minimized.
(15)
In a competitive market, an individual …rm essentially faces a horizontal
demand curve even when the market demand is quite inelastic.
(16)
For a pro…t-maximizing …rm, if all input prices and output prices are increased
by 10% due to a general in‡ation, the optimal input and output levels are the same as
before.
(17)
If the manufacturing workers in Vietnam cost just half of those in China, but
they are perfect substitutes of Chinese workers. If other inputs can be obtained with equal
cost in Vietnam and in China, …rms would choose to locate their factory in Vietnam.
(18)
In advanced economies, the labor cost is relatively higher while the capital cost
is relatively lower than in developing countries. This means …rms in advanced countries
have a higher capital labor ratio than those in the developing countries.
(19)
If lemon now costs 2 times as before, and the prices of other inputs are
unchanged, the cost of a cup of lemon tea also becomes 2 times as before.
(20)
Isoquants of two di¤erent output levels cannot cross.
Question 2
Suppose we have two inputs x1 and x2 with prices w1 and w2 . Using an appropriate
diagram, derive how the conditional demand for each input changes for the following
changes given a certain output level y: Consider a general convex monotonic technology.
Determine also the change in total cost.
(a)
An increase in w1 holding w2 constant.
(b)
A decrease in w2 holding w1 constant.
(c)
An increase in target output level y. (Assuming normal inputs.)
Question 3
Consider the following production function with inputs L and K:
p
p
Q = ( L + K)2
The input prices of L and K, w and r repsectively, are both 1.
(a)
Is it constant, increasing or decreasing return to scale?
Consider our target production is Q = 64.
(b)
What are the marginal products of capital and labor respectively at K =
25; L = 9? Should the …rm use more labor or capital?
(c)
What are the marginal products of capital and labor respectively at K = 4; L =
36? Should the …rm use more labor or capital?
(d)
What is the optimal use of capital and labor given the production function
and price? What is the cost?
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(e)
Now if r = 2 but w = 1 as before, what is the optimal use of labor and capital?
How does it di¤erent from before? (Leave your answers in decimal places if it is the case.)
(f)
Now, consider a technological advancement such that
p
p
Q = 2( L + K)2
and w = r = 1, then what is the optimal level of K and L? Any change in K=L? What
is the new cost?
(g)
Now, consider a technological advancement that only promotes the productivity of capital
p
p
Q = ( L + 2K)2
while w = r = 1: What is the optimal level of K and L? How does K=L change? What
is the new cost?
Question 4
Consider the production process with 2 inputs and 1 output. The production function
0:4
is given by y = x0:4
1 x2 . The input prices are w1 and w2 respectively. Consider the case
of long run where both factors are variable. The output price is denoted as p.
(Please leave the numbers in decimals or fractions.)
(a)
First, consider the pro…t maximization problem directly. Derive the optimal output function and input demand functions in terms of input prices w1 , w2 and
output price p. How do they change with the input and output prices? Do they make
sense?
(b)
Now, consider the cost minimization problem. Derive the conditional input
demand functions and the cost function in terms of w1 , w2 and y. How do they change
with input prices and output level? Does it make sense?
Next, we would like to separate the e¤ect of an input price change to input demand
into substitution e¤ect and output e¤ect. To demonstrate, let us have the initial prices
w1 = 1, w2 = 1 and output p = 1:
(c)
Calculate the pro…t maximizing output level and the corresponding optimal
input levels.
Now, consider a rise in price of input 1 so now w1 = 2. Other prices are unchanged.
(d)
What are the corresponding pro…t maximizing output level and input levels?
(e)
Under the new prices, if we hold the output level at the previous level (as in
(c)), what are the input levels for both inputs? How does it change compared to part(c)?
This is the substitution e¤ect.
(f)
The di¤erence between (d) and (e) is due to the change in output, so we call
it the output e¤ect. Calculate the output e¤ect to factor demands.
Question 5
Consider a product can be produced with the following production function with
capital K and labor L
y = K 0:3 L0:5
with the price of capital r = 1 and price of labor w = 1:
(a)
First consider the long run where both capital and labor can be adjusted to
minimize cost. Derive the total cost function in terms of output y.
(b)
What are the long run average cost curve and the long run marginal cost curve?
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(c)
If you decide to produce y = 16, what are the optimal levels of capital and
labor?
After the decision is made, the capital level is …xed at that in part (c), and the …rm
cannot change it for some time. Thus now the …rm can only change labor if it wants to
change the output. The …rm is now in the short run.
(d)
Now …nd out the short run total cost in terms of output y, with the level of
capital …xed at the level at (c).
(e)
What is the short run average total cost curve? What is the short run marginal
cost curve? (Note: here curve means they are functions of y.)
(f)
At y = 16, what is the short run average total cost? What is the short run
marginal cost? Is the average cost the same as in the long run? Is the marginal cost the
same for short run and for long run? (Compare using your results in (b).)
(g)
At y = 16; what is the slope of short run average total cost? What is the slope
of long run average cost? Are they the same? Does the long run average cost curve meet
the short run average cost curve at the latter’s minimum?
(h)
At y = 10; what is the short run average cost and short run marginal cost?
Are they the same as the corresponding long run cost at this quantity?
Now, consider a change in price of labor w = 2 instead of 1.
(i)
Obtain the long run average cost curve and marginal cost curve. How does it
di¤erent from above?
(j)
Calculate the optimal level of capital and labor input if we target y = 16: How
does it di¤erent from above?
(k)
Calcuate the short run average and marginal cost curves given the level of
capital …xed at that in (j). How does it di¤erent from above?
[Beware of rounding errors. Rounding errors may make them look slightly di¤erent
while they are the same.]
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