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**Unformatted text preview: **Economics 21, Spring 2015 Prof. Luttmer Solutions to Quiz 3 1. Short Answer Questions (12 points) a. Deﬁne “the law of diminishing marginal returns.”
See book. b. Suppose there are 5 possible allocations, as indicated in the table below. Write down the letter(s) of the Pareto efﬁcient allocation(s): A 40
100
Utility of: N ,_.,_.,_. #93 O
MOON Co p—Ay—
\ll\)
4:
O
,_.,_. \l-l>b-)
N\l COO
v—Ay—Ab—i L11 0 4 O 30
20 100
70 100 110 120 03-5
DUI r—‘P-‘H
UIUIN Answer: Allocations B, C, and D are Pareto Efficient. An allocation is Pareto Eﬂicient (= Pareto Optimal) if there exists no other allocation that makes at least one person
better oﬁ’ and nobody worse of. To judge whether someone is better or worse of, you need to look at utility. Hours worked and after-tax earnings
deﬁne the allocation and determine utility, but they are not needed to answer this question once you know utility. Note thatA cannot be Pareto Efﬁcient because, allocation C makes Vered better oﬁ’ than allocation A without making
Rose or Zahra worse oﬂ. The same reasoning applies to allocation E. Allocation B is Pareto Eﬂicient because there is no other allocation that makes someone better oﬂ and nobody worse
of. To see this, note that Zahra would be worse 0]?" in all other allocations: A, C, D, or E. Similar reasoning applies to
allocation C (Vered would be worse of in any other allocation) and to allocation D (Rose would be worse oﬂ in any
other allocation). Page 1 of 7 c. In the country of Equilandia, the government decrees that no dairy farm may produce more than 1000 Gallons of
milk per day (because the government wants to “equitably” divide the production of milk across farms). Suppose
a dairy farm has the following marginal cost (MC), average variable cost (AVC), and average total cost curves
(ATC). The farm is a price taker. How much should the farm produce in the short run? Brieﬂy explain your Me. D 100 200 300 400 Note this is not the standard case because of the government decree. So you cannot blindly apply the ”rule ” MR=MC,
but you can successfully apply the reasoning that lead to the “rule ” MC =MR in the standard case. The correct
answer is that producing 1000 Gallons maximizes the farm’s proﬁt. There are several alternative ways of seeing this. 1. Compare MR to MC. MR=p because the farm is a price taker. If MR>MC, your proﬁts go up if you produce more.
MR>MC for any Q between 300 and I 000, so between 300 and 1000 you should try to produce more to increase your
proﬁt. Anywhere between 300 and 999.999 you can increase proﬁts by producing more, so those quantities cannot be
proﬁt maximizing. However, once you are at 1000 you cannot produce more (due to the decree/), so 1000 could
possibly be proﬁt maximizing. MR<MC for any Q between 0 and 300, so between 0 and 300 you should try to produce
less to increase your proﬁt. Anywhere between 0.00] and 300 you can increase proﬁts by producing less, so those
quantities cannot be proﬁt maximizing. However, once you are at 0 you cannot produce less (because Q<0 is not
possible), so 0 could possibly be proﬁt maximizing. So, we are left to decide between 0 and 1000 as possible quantities
that maximize proﬁts. AVC<P at Q=1000, so the revenue of producing 1000 covers the variable costs of producing
1000, so proﬁts are higher at Q=1000 than at Q=0. (In fact, proﬁts are positive at Q=1000 because P>AT C, and
proﬁts are negative at Q=0 because of the ﬁxed costs). Hence, Q=1000 is the proﬁt-maximizing quantity. BTW, at Q=300, your proﬁts are at a minimum. To see this: as soon as you produce more than 300, you get to the
region where MR>MC and your profits go up by producing even more; as soon as you produce less than 300, you get
to the region where MR<MC and your proﬁts go up by producing even less. 2. Think of the graph I gave you here as a ”zoomed—in ” part of the graph we saw in class (the rest of which I drew in
by hand above). Now you see that the unrestricted proﬁt maximizing quantity (so absent the decree) would be much
higher than 1000 Gallons (in fact 2000 Gallons, the way I happened to draw it). The closest that the ﬁrm can get to
the 2000 is by producing 1000, but we need to check producing I 000 is better than producing 0 or producing where
MC=MR. Note that P<ATC at Q=0 and at Q=300, so theﬁrm is making a loss at Q=0 or Q=300. But P>ATC at
Q=1000, so the ﬁrm makes a profit at Q=1000. Hence, Q=1000 is the proﬁt-maximizing quantity given the limits
imposed by the decree. 3. Because the following explanation is more mechanical, I like it less than the previous two explanations, but it is
correct nevertheless. Proﬁts are Q*(P—A TC). So, proﬁts are less than zero for any Q<800 because P<AT C for any
Q<800. Proﬁts are positive for any allowed Q>800 because P>AT C for any allowed Q>800. Between Q=800 and
Q=1000 proﬁts are increasing in Q (because both Q and P—ATC are increasing in that region). So the highest proﬁts
are at Q=1000 (given that Q>1000 is not allowed because of the decree). Page 2 of 7 2. Organic Lamb (10 points) There are many (potential) producers of organic lamb. Each (potential) producer of organic lamb has the
following total cost function: C(q) = 1000 + 5q + q2/40 where q is the quantity of lamb produced (in pounds
of lamb). The demand function for lamb is given by QD = 80,000 — 2,000p where p is the price per pound of
lamb. What is the long-run equilibrium in this perfectly competitive market for lamb: what is the price, what
is the total quantity produced in this market, and how many producers of lamb are there? MC = 5+ 2q/40
ATC = 1000/q +5+ q/40 In the long run (with identical producers), the price is determined by the zero—proﬁt condition. Proﬁts are zero when
AT C =p. Thus, producers will enter or exit this market until AT C=p holds. Each producer maximizes proﬁts, so sets
p=MC So, we have p=MC =AT C, which occurs at the minimum of the AT C. Find the quantity that each ﬁrm produces
by setting MC=ATC: MC =AC 5 + 2q/40 =1000/q +5 + q/40 q2/40= 1000
q2 = 40, 000
q: 200 So the minimum of the AT C is reached when a producer produces 200 pounds of lamb. Find the price by substituting q into the ATC or into the MC:
p =ATC = 1000/q + 5+ q/40 = 5 +5+ 5 = 15
(Double check: p = MC = 5 + 2q/40 =4+400/40= 15) Find the market quantity from the quantity demanded at p=15:
Q = 80,000 — 2,000p = 80,000-30,000=50,000 Find the number of ﬁrms by dividing the market quantity by the quantity produced by each ﬁrm :
N=Q/q = 50,000/200 = 250 Thus, to summarize:
p = 15 Q = 50,000 N = 250 Page 3 of 7 3. Pumpkin Queen (10 points) Consider the market for carving pumpkins in a faraway monarchy ruled by a queen with absolute power. In the
monarchy there are: 200 thousand households with kids, and each of them values one pumpkin at $20
200 thousand households without kids, and each of them values one pumpkin at $14
250 farmers on hilly land that can each produce a thousand pumpkins at a marginal cost of $17 per pumpkin
150 farmers on ﬂat land that can each produce a thousand pumpkins at a marginal cost of $ 9 per pumpkin The farmers have no ﬁxed costs relating to the production of pumpkins. Each household buys at most one pumpkin
and each farmer sells at most 1000 pumpkins. a. What are Social Surplus, the quantity of pumpkins sold, and DWL in the pumpkin market if this market is a
perfectly competitive market? The market is in equilibrium if when the quantity suppliedD' is the quantity demanded. This happens at p 1 7,
when only the families with kids demand a pumpkin (so QD is 200 thousand) and when all the ﬂat-land
farmers and 50 hilly- land farmers supply pumpkins (s0 Q is 200 thousand). Note that at p —1 7, hilly-land
farmers are indifferent whether they supply or not (because the pricejust covers their cost), so they willingly
supply any quantity between 0 and 250 thousand Social surplus is the sum of the gains from trade (MB-MC for the pumpkins traded). For 150 thousand
pumpkins MB=20 and MC=9, so the gains from trade are $11 per pumpkin. For 50 thousand pumpkins,
MB=20 and MC =1 7, so the gains from trade are $3 per pumpkin. Social Surplus = 11 *150 thousand + 3*50 thousand= $ 1,800 thousand (i.e., $1.8 Million). DWL =0 because there are no unrealized gains from trade and there are no losses from trade. In other
words, there are no other allocations that would yield a higher Social Surplus. Note, BTW, that the values of MB of the households define a demand curve and the values of MC of the
farmers deﬁne a supply curve. So, this is really a supply and demand question (albeit slightly disguised). Of
course, this drawing is not needed for full credit because I did not ask for it. glam}? g5 ISO 200 #00 6? 4m FQOLCQ. Page4of7 The queen, concerned that scholastic achievement is suffering because kids spend too much time making elaborate
pumpkin carvings, issues a decree that bans households with kids from purchasing a pumpkin (offenses carry a 10-year
prison sentence!). To make sure farmers don’t suffer (too much), the order also includes a pumpkin subsidy of $7 per
pumpkin that is paid to farmers. b. What is Social Surplus, the quantity of pumpkins sold, and DWL in the pumpkin market when the queen’s decree
(so the ban and the subsidy) is in effect? With the pumpkin subsidy, flat~land farmers are willing to sell pumpkins for $2 (=9- 7) each and hilly-land
farmers for $10 (=1 7—10) each. Only households without kids are potentially interested in buying. There is
an equilibrium at p=$ 1 0 when QD=200 thousand and when all the ﬂat-land farmers supply pumpkins and 50
hilly-land farmers supply pumpkins, so QS=200. As before, hilly-land farmers are indifferent about supplying
when the price they receive ( =market price plus subsidy) is equal to their marginal cost. Social surplus is the sum of the gains from trade (MB-MC for the pumpkins traded). For I 5 0 thousand
pumpkins MB=14 and MC =9, so the gains from trade are $5 per pumpkin. For 50 thousand pumpkins,
MB=14 and MC=1 7, so the losses ﬂom trade are $3 per pumpkin. Social Surplus = 5 *150 thousand + (-3 ) *5 0 thousand: $ 600 thousand. DWL = highest possibly Social Surplus minus actual Social Surplus =
= $1,800 thousand — $600 thousand = $1,200 thousand (i.e., $1.2 Million) Another way to calculate the DWL is to notice that the costs of producing stayed the same (the same farmers
are producing) as in part (a), but that now households without kids rather than households with kids are
consuming, The MB for a household without kids is $6 (=20—14) less than for household with kid, so the
total benefits are 200 thousand times 6 = 1,200 thousand less than before. Hence, the DWL is $1,200 thousand (i. e., $1.2 Million). Page 5 of 7 4. Steel (13 points) The production of steel requires only labor (L) and capital (K). Assume that the price of a unit of labor is w and that
the price of a unit of capital is r. The steel factory takes these prices as given. The factory’s production function for
steel is given by: Q = F(K,L) = K2 L where Q denotes the number of tons of steel produced. a. [7 points] Use the method of Lagrange to ﬁnd this factory’s conditional (on Q) factor demand for L. Minimize cost subject to an output constraint. Set up the Lagrangian:
L =wL +tK + MQ—KZL) Take the ﬁrst—order conditions: d£/dK=r—Z,2KL=0 ¢:> t: AZKL (1)
dr/dL=w—AK2 =0 e W: 1K2 (2)
d£/dk=Q-K2L=0 ¢:> Q=K2L (3) These are 3 equations in 3 unknowns (x1, K, and L). Solve them for K and L:
Divide (I) by (2): r/w = 2L/K CID r/2w = L/K 6? K = (2w/r) L (4) Substitute (4) into (3): Q = K2L ¢> Q = [(2w/r) L J2 L = (2w/r)2 L3 e L = (r/2w) ”3 Q” So: L(Q, w, r) = (r /2w) ”3 Q” (5) BTW: K(Q, w, r) = (2w/r) L = (2w/r) (r/2w) ”3 Q” = (2w/r)”3 Q” Check that the conditional factor demands make economic sense. You don ’t need to do this to get full credit, but if you
get a wrong answer of a form that does not make economic sense, I take off more point than if it is a wrong answer
that does make economic sense. The former is an economics mistake; the latter merely a math issue. S0, to check that
the answer makes economic sense: (1) The conditional factor demands are decreasing in their relative price, so L is
decreasing in w if we hold r constant. (ii) If both r and w double, the conditional factor demands stay the same. (iii) At
least one of the two conditional factor demands is increasing in Q. b. [1 point] For which ratio of input prices does the factory use the same quantity of labor as capital (i.e., K=L)? See equation (4) above. If L/K=] , then r/2w=1, s0 r/w=2. Page 6 of 7 c. [5 points] Now suppose the production function is given by: Q = F(K,L) = Min(K2 L, K L2). Find this factory’s
conditional (on Q) factor demand for L. Note: you can answer this question without doing new calculations by
relying on your answers in parts (a) and (b). This is a very hard question, so don ’t worry if you missed it. However, you can exploit that KZL and K L2 are
symmetric. Note that for K>L, we have K2 L > K L2 , Q = Min(K2 L, K L2) = K L2. So for K>L, the production fn. is Q = K L2.
Note that for K<L, we have K2 L < K L2, Q = Min(K2 L, K L2) = K2 L. So for K<L, the production fn. is Q = K2 L.
(0fcourse, for K=L, Q = K212 = KL2 ) From part b, we know that for Q= K2 L, the ﬁrm chooses K=L when r/w=2. Because conditional factor demands move
in the opposite direction of relative factor prices, the ﬁrm chooses K<L' if r/w>2. So, for r/w>2, the relevant factor
demand for labor comes from the production function Q= K2 L, and we found this in part a: L(Q, w, r) = (r/2w) ”3 Q” for r/w >2 By symmetry, we know that for Q= K L2, the ﬁrm chooses K=L when w/r=2 (note NOT r/w=2). Because conditional
factor demands move in the opposite direction of relative factor prices, the ﬁrm chooses L<K if w/r>2 ¢> r/w < 1/2.
S0, for r/w < 1/2, the relevant factor demand for labor comes from the production function K L2 . By symmetry, the conditional factor demand is:
L(Q, w, r) = (2r/ w)” Q16 for r/w < 1/2 (This is the expression for K from part a but with r and w exchanged) For 1/2 S r/w S 2, the firm neither wants to choose K<L nor K>L. Hence, K=L. IfK=L, then Q = K3 = L 3 . This means that L: Q”. So to summarize: L(Q, w, r) = (Zr/w)” Q15 for r/w < 1/2 (i. e., for w/2r > 1)
L(Q, w, r) = Q” for 1/2 5 r/w s 2
L(Q, w, r) = (r/2w) ”3 Q” for r/w >2 (i. e., for r/2w > 1) A graph helps illustrate what is going on (not needed, of course, for full credit but these solutions are also a teaching
tool). The min(a, 1)) function creates an isoquant that has a kink at K=L. Thus, there is a range of relative prices (1/2 S
r/w S 2) for which the iso-cost line is tangent at K=L (so where the condition demand for L = Q”, and does not vary
with r or w). T 0 see that we have a kink, notice that part (b) tells us that at K=L, the MTRSK/brL = —_2 for the
production function f(K,L)= K2 L. By symmetry, the MT ESL/m- I; = -2 for the production function f(K,L)= K L2; so the
MT RSKﬁ” L = -1/2 for this production function. Because the slopes ( =MRT S) of the two isoquants are different at K=L, we have a kink 1’ V
\\
/\ Page 7 of 7 ...

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