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EconS 301, Spring 2015, Dr. Rosenman
Test 2, March 26, 2015
Name ____________________________
ID#_______________________________
Do any three of questions 1, 2, 3 and 4, and do question 5. The three questions you do out of 1-4 are worth 10 points
each, and question 5 is worth 20 points. I am figuring 10 minutes each for the first 3, and 30 minutes for question 5.
That leaves 15 minutes for panic and other activities.
1. The production of widgets follows the production process Q = 100 KL where K is capital (a fixed factor in
the short run) and L is labor, a variable factor in the short run. The price of capital is v = 1 and the price of
labor is w = 4.
a. Suppose K is initially at 100. Find mathematical expressions for the MRTS, short run TC, MC, AVC
and ATC curves. (3 points)
b. If you double labor in the short run, by how much will output change? Explain. (1 point)
c. Find mathematical expressions for the long run TC, MC and AC curves. (3 points)
d. What type of long run returns to scale does this production function have? Explain. (1 point)
e. What type of long run economies of scale does this production function have? Explain. (1 point)
f. What will the expansion path look like? Explain. (1 point)
a) MPK = (50L1/2)/(K1/2)
MPL=(50K1/2)/(L1/2)
MRTS=K/L
If K=100 then Q=1000L1/2
L= (Q2/1000000)
so TC(Q)=100+(4Q2/1000000) therefore TC(Q)=100+(Q2/250000)
MC=Q/125000
AVC=Q/250000
ATC=(100/Q)+(Q/250000)
b) In the short run Q=1000L1/2 . If you double L we get Q*=1000(2L)1/2 hence Q*/Q=[1000L1/2 ] / 1000(2L)1/2 =1/21/2 <1.
So this would exhibit decreasing returns to scale in the SR.
c).We already know that the MRTS=K/L, w/r=4, and TC = K + 4L. Using these equations and the tangent condition we
can do the following
K/L = 4 so K=4L or L=K/4
Q=100(4L)1/2(L1/2)
Q=200L
L=Q/200 so K=Q/50
TC=Q/50 + 4 (Q/200)
TC = (2Q)/50 therefore TC = Q/25
MC=1/25
ATC=1/25
d) This is a Cobb-Douglas function with an alpha equal to ½ and a beta equal to ½ which sum up to equal 1 therefore
this function has constant returns to scale in the long run.
e) Since the long-run average total cost is constant this function also has constant economies of scale.
f) The expansion path in this situation will be linear and will keep a constant ratio of K/L=4.
2. The state of Washington requires every firm that does business in the state to have a license. The license is
equivalent to a fixed cost. The state also has a Business and Occupation tax, which takes a percent of the
firm's revenue. Assume firms in the state of Washington are profit maximizing. In the answers to the
questions below, use the profit maximizing rules for firms to defend your answer. You might use
mathematical expressions or graphs to make things clear.
a. How does the license impact firms output and profit? Differentiate between the short run and the
long run. (1 points)
b. How does the revenue tax affect firm output and profit? Explain. (2 points)
c. If the state replaced the revenue tax with a profit tax, would it increase or decrease a typical firm’s
output? Which would be better for the state economy? Explain. (3 points)
d. Now consider farmers, a competitive industry in the state. Would a revenue tax or a profit tax have a
larger impact on the number of farms and the size of an average farm? Does the license fee affect
the average size of a farm? Explain. (4 points)
HINT: This problem is most easily approached by noting that
π R(q ) − c(q ) where q is the output of a
=
typical firm, R (q ) is its revenue and c(q ) is its cost, and then thinking about how the different taxes affect
revenue and profit.
Using the hint, apply the profit maximizing condition: ∂π/∂q=0→R′(q)-c′(q)=0 which is MR=MC in the short run. In
the long run in a competitive market P=min LR AC but with the fee AC=FEE/Q+ (AC from production) where FEE is the
license fee.
a) The license fee is a fixed cost and does not change either MR or MC, hence in the short run it will not affect firm
output. However, since it raises AC, both in the short run and long run, it will affect LR output. Since LR AC will
equal the cost from production plus FEE/Q (where FEE is the license fee) we see that LRAC will now reach its
minimum at a (slightly) larger output that without the license and MC is now MC from production plus –FEE/Q2.
MC is lower so LR output is larger. The license fee lowers profit in the SR by the amount of the fee, and in the LR
by FEE/Q2.
b) The revenue tax affects revenue. The SR profit maximizing condition becomes (1-t)R′(q)-c′(q)=0 where t is the
percentage revenue tax. Instead of MR/MC=1 we now need MR/MC=1/(1-t)>1. Since MR is decreasing (or
constant at the market price) and MC is increasing, this means output will be smaller. Profit will be smaller as
well.
c) With a profit tax the profit maximizing condition becomes (1-t)[R′(q)-c′(q)]=0 which means that for each firm we
still have MR=MC. Since we saw with the revenue tax output is smaller, a profit tax will not change SR output.
For this reason the profit tax is better for the state’s economy.
d) We need to do a LR analysis for this question. In the LR if the market is a perfect competition the profit for firms
is 0 and price is at minimum LRAC. We saw in part a) that LRAC shifts out and up a little bit. Farms will be a little
bigger because of the license fee. Since LR profit=0 in a competitive industry, a profit tax will not have any
impact. However, a revenue tax would require price to rise so (1-t)P=min LRAC, which means there would be
fewer farms, so total output is smaller and thus price is higher.
3. Ethanol, which can be used in gasoline, is produced from a fermentation process of fructose or other sugars.
In the US most ethanol is produced from corn. The demand for ethanol began to increase in the US in the
mid-2000s because of government support for its use, an increase in the price of oil, and other factors. In
2005 the price of ethanol was $1 per gallon. In 2006 this increased to $4 per gallon. Entry into ethanol
production is easy. In 2005 there were 81 ethanol producers in the US; by 2009 there were 190. Production
capacity increased from 3,650 million gallons in 2005 to 10,570 million gallons in 2009. Meanwhile, partly
due to increased ethanol production, the price of corn moved from $2/bushel to sometimes over $5/bushel
from 2005 to 2009, and stabilized near $4 per bushel.
a. Given the information above, do you think ethanol producers in 2006 were making profits? Explain
your answer. (2 points)
b. As more firms entered the industry, what happened to the short run and long run average cost
curves for firms in the industry? Explain why? (2 points)
c. Given you answer to b, what does the long run supply curve look like in the ethanol industry.
Explain. (1 point)
d. There are now almost no new ethanol plants under construction in the US. Draw a graph showing
the long run equilibrium in the industry. (3 points)
e. It is estimated that the US will use 15 billion gallons of ethanol in 2022. Suppose there are 200
ethanol plants in the long run equilibrium at this level of output. If all firms are the same, at what
output per firm is long run AC minimized? Explain. (2 points).
a) Given the amount of entry that occurred from 2006 to 2009 we would surmise that ethanol producers were
making profit in 2006 because profits attract new entry into markets.
b) In the short run the AVC, ATC, and MC all increased because input prices began to rise in price when the
demand for them increased. In the long run AC and MC curves both increased because it is now more costly
to operate in the market.
c) The long run supply curve in this market will be upward sloping because it is an increasing cost market.
d) The firm will be at minimum LRAC (where P=SMC) and in the market LRS intersects D
SMC
LRAC
LRS
D
Firm
e)
15,000,000,000 / 200 = 75,000,000
Market
4. The accounting ledger for the owner of a small store showed the following information about annual receipts
and expenditures
Revenues:
$1,000,000
Wages paid to workers:
$ 300,000
Utilities:
$ 20,000
Drugs and other supplies
$ 450,000
Property taxes & maintenance
$
10,000
Other taxes (wage, sales)
$
40,000
Wages paid herself
$ 100,000
She owns the building the store is in and has no expenses for it other than the property taxes and
maintenance. She could sell it for $200,000 and earn 10% on the proceeds in a safe investment (the “fair
rate of return”. She also has two job offers, one for $50,000 and one for $100,000.
a. What are her accounting profits from running the store?
b. What are her economic profits from running the store? If they differ, why?
c. In the long run, should we expect more or fewer stores like this in the market? Explain you answer.
a. Accounting profits are just revenues minus (direct) costs, not including opportunity cost. So
Profit =1,000,000-300,000-20,000-450,000-10,000-40,000-100,000=$80,000
From the questions asked me during the test, some of you were confused by the wages she paid herself. They are a
direct cost, with the idea being that they offset the opportunity cost of her alternative job – there is no opportunity
cost because the best other alternative is also $100,000. This differs a little from examples we’ve done before. Hence,
on part a, we are also accepting an accounting profit of $180,000.
b.
Economic profit takes into account the opportunity cost of not renting the store for other uses and the
opportunity cost of her time. She pays herself as much as she could earn elsewhere, hence there is no
opportunity cost for her work (and presumably if she didn’t work at the store she would have to hire
someone to replace herself at the same wages), but she could earn $20,000 a year instead of owning the
building, so that is an opportunity cost. So the store brings in $80,000 but she gives up $20,000 a year in the
opportunity cost of using the building for the store, hence the economic profit is $60,000. The economic
profit is less than the accounting profit because economic profit takes into account opportunity costs
associated with running the business.
c. We would expect fewer firms like this in the market because economic profit is negative.
5. For any FIVE (5) of the statements below, indicate if it is True, False or Uncertain, and explain why. (Your
explanation needs to give reasoning, not just a different statement). Each is worth 4 points. You may also choose
TWO (2) additional questions as extra credit on the test. Each extra credit question is worth 2.5 points. Clearly
indicate which questions are for extra credit. If you don’t, the first 5 you do will be the regular test questions.
a. If the marginal product of labor is greater than the average product of labor then average variable cost must
be decreasing.
True. MPL>APL means APL is increasing. Because AVC =w/APL this means AVC has to be decreasing.
b. Suppose we have a graph of a long run expansion path with labor on the horizontal axis and capital on the
vertical axis. If the long run expansion path curves up to the left, for efficient production the amount of
capital per unit of labor is increasing.
True. The expansion path shows the cost minimizing (efficient) input combination. This means that as you replace a
smaller amount of labor you have to replace it with an increasing amount of capital. K/L is increasing.
c. In a perfectly competitive industry, in long run equilibrium price equals AVC.
False. In the long run all costs are variable so there is no AVC curve just AC. However, they will produce where price
equals AC.
d. The market price elasticity of demand for automobiles is likely larger than the price elasticity of demand
facing individual auto dealers.
False. The price elasticity for one dealership would be more elastic than the market price elasticity because other
dealerships and brands are substitutes.
e. Technical efficiency means the firm is operating at the production level that minimizes average cost.
False. Technical efficiency assumes that if the use of any input falls then output falls. This assumes “no waste” in
production. Cost efficiency/minimization makes this assumption.
f. In the short run a firm always produces at an output that minimizes long run average cost.
False because there are fixed inputs and costs in the short run firms do not always operate at the point where LRAC is
minimized. This is also mathematically impossible because when you draw the graph of LRAC as being composed of
SRAC curves as we have done in class not every SRAC can be tangaent with the LRAC curve where the slopes are
equal to zero.
g. With the Cobb-Douglas production function, if the sum of the exponents is less than 1, there are decreasing
returns to scale.
True. Let Q=KaLb where a+b<1. Let Q*= (2K)a(2L)b=2a+bQ. Since 2a+b<2 if a+b<1, we have DRS.
h. On a curved isoquant with labor on the horizontal axis and capital on the vertical axis, as we move down and
to the right on the isoquant it is getting harder and harder to substitute labor for capital.
True. At this part of the isoquant for every small reduction in capital a large change in labor is necessary to maintain
this level of output.
i. For a firm, it is always the case that MR≤price.
True. MR=P(1+(1/ep))=P(1-(1/|ep|)). A profit maximizing firm always produces where MR=MC. Since we expect
MC>0, that means MR>0 which means |ep|>1 so MR≤price.
j.
Firms facing demand that is almost inelastic make small marginal profit (have a small mark-up over MC) while
firms facing very elastic demand will make large marginal profit.
Basically True. This can shown using the rule (P-MC)/P=1/|ep|. As discussed in class when ep<|1| then MR is
negative and when ep>|1| then MR>0. If MR<0 then marginal profit must be <0 because through the use of calculus
you can show that marginal profit = MR – MC. So a firm will not produce where marginal profit<0, we need ep>|1|.
k. In long run equilibrium, in a perfectly competitive industry producer surplus is zero.
Uncertain. In a constant cost perfectly competitive industry LRS is vertical at min LRAC, so producer surplus is zero,
but if it is an increasing cost industry, so LRS is upward sloping, Producer surplus can be positive.