Download 2016 FINAL EXAM Answer key

Simon Fraser University, Department of Economics, Econ 201, Prof. Karaivanov
2016 FINAL EXAM Answer key
Please let me know if you see any typos or mistakes.
I. TRUE or FALSE { no credit given without a short justi cation of your answer (6 pts
each)
1. FALSE. The person may have preferences for perfect substitutes and his IC coincide with his budget
line (for example, take u(x1 ; x2 ) = x1 + x2 and p1 = p2 ). Then any bundle on his budget line maximizes his
utility (he is indi erent among all the bundles on his budget line so might as well choose one at random).
2. FALSE. The rm maximizes its short-run pro ts py V C(y) F where V C(y) is variable and
F is xed costs. A (small) increase in the xed costs F should not change the rm's supply decision y
C(y)
which solves p = dV dy
= M C(y). The only possible exception would be if the rm's xed costs increase
so much that it wants to shut down so then its output would drop to zero. [give 2pts to students who
only discuss the shut down case; give full points if someone missed the shut down case but made otherwise
correct argument]
3. TRUE. We know that the SE always goes in the opposite direction of the price change, so the SE
will make the quantity demanded to decrease. A price increase also means less purchasing power (as if the
person had less income) which, since the good is inferior, implies an increase in quantity demanded.
4. FALSE. We know that LRAC(y)
SRAC(y) for all y. But SRAC(y) = AV C(y) + Fy , where
AV C(y) are the rm's SR average variable costs and F are its xed costs. Hence, we cannot say in general
(e.g., think of F=y large) that LRAC(y) < AV C(y).
5. FALSE. Plot a graph and note that the workers are suppliers in the labor market. The minimum
wage reduces the hours worked [think why]. Producers' surplus (here, the workers' surplus) can go up or
down depending on the elasticities of demand and supply so the policy e ect can go either way if we assume
that all workers keep their jobs but work less hours. However, due to the decrease in hours demanded in
the new equilibrium, it is quite likely that some formerly employed workers lose their job and they will be
worse o .
Problem 1 (30 pts)
(a) (4 pts) The budget constraint implies p1 s + p2 d = m therefore d = m
2.
(b) (6 pts) The budget line has equation 3s + d = 120. This is a straight line with intercepts (0; 120)
and (40; 0) and slope pp21 = 3. Plug the given numbers into the demand functions to nd the optimal
bundle: sA = 20 and dA = 60 (call it bundle A).
(c) (10 pts) Proceed as in class and the midterm. First, nd the optimal bundle (call it C) at the new
120
= 15 and dC = 60. We then need to nd Yoshi's `adjusted' income m0
price p1 = $4. We obtain mC = 2(4)
that keeps constant his original purchasing power at the new price (the income m0 with which he can just
a ord his original bundle). We have m0 = 4(sA ) + 1(dA ) = 140. Then, nd Yoshi's optimal bundle (call it
140
= 17:5 and dB = 140
B) at the adjusted income m0 and the new prices: sB = 2(4)
2 = 70. The income e ect
C
B
C
B
of the price change is s
s = 2:5 for soda and d
d = 10 for money spent on everything else.
B
A
B
The substitution e ect is s
s = 2:5 for soda and d
dA = +10 for everything else.
(d) (10 pts) Yoshi's new budget line after the tax and subsidy is 4s + d = 140 which is a straight line
with intercepts (0; 140) and (35; 0) and slope 4. Using the demand functions obtained earlier, his optimal
140
bundle (call it D) is sD = 2(4)
= 17:5 and dD = 140
2 = 70 { in fact the same as bundle B from part (c).
Yes, Yoshi is better o . Why? Note that at p1 = 4 and m = 140 Yoshi can a ord his original bundle A
from part b (since 4(20) + 1(60) = 140) but he prefers (optimally chooses) a di erent bundle { bundle
1
D. This implies that D A. At bundle D the government collects $1(sD ) = 17:5 in tax revenue while it
spends $20 on the subsidy { so the answer to the last question is No.
Problem 2 (30 pts)
p
(a) 5 pts. If the rm wants to produce y in the SR we must have: f (x1 ; x2 ) = y, or x1 (1) = y, so
the SR conditional input demand for workers is: x1 (y; x2 ) = y 2 . (Note that no cost minimization problem
needs to be solved here). The SR cost function is:
cSR (y; x2 ) = w1 x1 (y; x2 ) + w2 x2 = 8y 2 + 32(1) = 8y 2 + 32
(b) 6 pts. Given the SR cost function, we have: F C = 32; AC(y) =
32
y
+ 8y; AV C(y) = 8y; M C(y) =
dcSR (y)
= 16y. The MC and AVC curves are straight lines through the origin. The MC is twice as steep
and always above the AVC. The AC curve is like a parabola. Make sure that the MC crosses AC at its
minimum on the graph.
(c) 6 pts. Since MC is always above AVC and upward sloping, the SR supply curve is the whole MC
curve (p = 16y). Given that, the pro t maximizing output is where p equals M C(y), i.e. when 8 = 16y,
thus y = 1=2. Firm's pro ts at y are:
= py
cSR (y ; x2 ) = 8(1=2) 32 8(1=4) = 30 { the rm
makes a loss. Notice, however, that the loss is smaller than the xed costs which is why the rm does not
shut down.
(d) (8 pts) Using the hint,
1=2 1=2
( 21 )x1 x2
x2
1
1
=
= or
1=2
1=2
4
x1
4
( 1 )x x
dy
2
1
2
so x1 = 4x2 at optimum. To nd the conditional input demands, use that output is y;that is, f (x1 ; x2 ) = y
and since x1 = 4x2 at optimum:
q
4x2 (x2 ) = y
y
LR
which implies xLR
2 (y) = 2 and so x1 (y) = 2y as the LR conditional input demands for producing y units
of output. The long run cost function is:
LR
cLR (y) = w1 xLR
1 (y) + w2 x2 (y) = 8(2y) + 32(y=2) = 32y
The LR supply curve is the portion of the LRMC curve above the LRAC curve. In this case however
LRAC = LRM C = 32 that is, the LR supply curve is a horizontal line at 32.
(e) (5 pts) The SR and LR supply curves cross when 32 = 16y which happens at y~ = 2. Then
cSR (~
y ) = 64 = cLR (~
y ). The reason why the SR and LR costs are equal is that at y~ when the rm optimally
supplies the same output level, the LR conditional demand for each input equals its SR conditional demand,
that is x2 = 1 is cost-minimizing for y~ = 2 and so in the long run the rm would not vary x2 away from 1
so its costs remain the same.
Problem 3 (30 pts)
low
(a) 6 pts. We have M C low (y) = dcdy = y 2 4y + 4 = (y 2)2 . Similarly, M C hi (y) = y 2 8y + 16 =
(y 4)2 . Use the p = M C(y) condition to derive the rms' supply functions: p = (y 2)2 implies
p
p
y low (p) = p + 2 and, similarly, y hi (p) = p + 4.
(b) 6 pts. Add up the rm supplies times the number of rms from each type to nd the industry
supply for any p:
p
p
p
S(p) = 10( p + 2) + 10( p + 4) = 20 p + 60
Use the given market demand to nd the equilibrium price:
p
p
150 10 p = 20 p + 60
2
and so p = 9. At that price low cost rms produce y low (9) = 5 and high-cost rms produce y hi (9) = 7
units each.
(c) 8 pts. In the LR industry equilibrium rms will produce at minLRAC and only those rms whose
2
low
minLRAC is smallest will remain operating. For the low-cost rms we have LRAC low = c y = y3 2y + 4.
To minimize it, take derivative and set to zero: 23 y 2 = 0 and so producing y~low = 3 achieves min LRAC =
2
1. Similarly, for the high-cost rms, LRAC hi = y3
4y + 16. Take derivative, set to zero: 23 y 4 = 0 and
hi
so y~ = 6 achieves minLRAC = 4. Clearly, the low cost rms have lower minLRAC and so, with free
entry and exit, only they will remain in the industry and the LRIE price will be pLR = 1. The high cost
rms exit since their minLRAC = 4 and they lose money at p = 1.
(d) 5 pts. At p = 1 market demand is 140 and remember each low cost rm produces y~low = 3 units
= 140
(the level that achieves minLRAC). Thus, D(1)
rms will operate [give full points if someone says
3
y~low
that it should be a round number, 46 rms].
(e) 5 pts. Now p = 4. Market demand is 130. At p = 4 each low-cost rm produces y low (4) = 4 units.
Thus, the number of rms operating is 130
4 [again, give full credit if someone writes 32 rms instead].
4 = 34 and so = y low (p LRAC(y low )) > 0
Firms make a pro t since 4 = p > LRAC(y low ) = 16
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Problem 4 (30 pts)
(a) 4 pts. The Edgeworth box is of size w1A + w1B = 12 by w2A + w2B = 8. The endowment point has
coordinates (4; 4). [Make sure the students label their graphs clearly.]
(b) 4 pts. We have mA = p1 w1A + p2 w2A = 4p1 + 4. Also, mB = p1 w1B + p2 w2B = 8p1 + 2.
(c) 7 pts. Consumer A considers the goods perfect complements at xed proportion 1:1, which means
A
that his preferences can be represented by the utility function minfxA
1 ; x2 g. This implies that at his
A
A
A
A
optimal bundle x1 = x2 and so, from the budget constraint, p1 x1 + x1 = mA which yields the demand
4p1 +4
mA
A
functions xA
1 = x2 = p1 +1 = p1 +1 = 4. For person B nding the demand functions is even easier
since we are given her Engel curves (the relationship between income and the optimal quantity chosen).
8p1 +2
mB
Using the Engel curve to express the quantity in terms of the income, we have: xB
1 = 3p1 = 3p1 and
B
16p1 +4
2m
.
xB
2 = 3 =
3
(d) 12 pts. A competitive equilibrium (CE) consists of allocations (list of quantities consumed) for each
A B
B
person, (xA
1 ; x2 ; x1 ; x2 ) and prices (p1 ; 1) at which:
(i) each person maximizes their utility at the CE allocation given the prices (p1 ; 1). That is,
j
xi (p1 ; mj ) = xji for i = 1; 2 and j = A; B (A and B demand the allocation at the CE prices)
A
B
B
A
B
(ii) the prices (p1 ; 1) clear both markets, that is xA
i (p1 ; m ) + xi (p1 ; m ) = wi + wi for i = 1; 2
[3 pts for a correct de nition of CE]
To solve for the CE price p1 , take market 2 for example, and set demand = supply
4+
16p1 + 4
= 6 or
3
16p1 + 4 = 6
8(1=8)+2
A
B
which implies p1 = 1=8. The CE allocation is: xA
= 8 and xB
1 = x2 = 4 and x1 =
2 = 2. Market 1
3=8
is cleared automatically by Walras' Law.
(e) 3 pts. No, it cannot. Since consumer A has preferences for perfect complements, in any CE it
A
must be xA
1 = x2 since the proposed allocation must be optimally chosen (demanded) by A given the
CE prices (see the argument about A's optimal choice made in part c). However, 6 6= 3.
3