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Unformatted text preview: WELLESLEY COLLEGE
DEPARTMENT OF ECONOMICS ECONOMICS 20103 JOHNSON 1. P&R, page 307, #10: Suppose you are given the following information about a particular industry? Q0 = 6500 —100P Market demand
Q5 =1200P Market supply
2
C(q) = 722+ 2i?) ' ' Finn total cost function
2q . . .
MC(q = % F 11m marginal cost function.
ASsume that all ﬁrms are identical, and that the market is characterized by pure competition.
a. Find the equilibrium price, the equilibrium quantity, the output supplied by the ﬁrm, and
the proﬁt of the ﬁrm. Equilibrium price and quantity are found by setting market supply equal to market
demand, so that 6500'100P=1200P. Solve to ﬁnd P=5 and substitute into either equation to ﬁnd Q=6000. To ﬁnd the output for the ﬁrm set price equal to marginal 2
cost so that 5 = i and q=500. Proﬁt of the ﬁrm is total revenue minus total cost 200
5002 = — =5 —7 —
orll pq C(q) (500) 22 200 the market is 6000, and the ﬁrm output is 500, there must be 6000/500=12 ﬁrms in
the industry. = 528. Notice that since the total output in b. Would you expect to see entry into or exit from the industry in the longrun? Explain. What
effect will entry or exit have on market equilibrium? Entry because the ﬁrms in the industry are making positive proﬁt. As ﬁrms enter,
the supply curve for the industry will shift down and to the right and the equilibrium
price will fall, all else the same. This will reduce each ﬁrm’s proﬁt down to zero until
there is no incentive for further entry. c. What is the lowest price at which each ﬁrm would sell its output in the long run? Is proﬁt
positive, negative, or zero at this price? Explain. In the long run the ﬁrm will not sell for a price that is below minimum average cost.
At any price below minimum average cost, proﬁt is fiegative and the' ﬁrm is better off
selling its ﬁxed resources and producing zero output. To ﬁnd the minimum average
cost, set marginal cost equal to average cost and solve for q AZE+L
200 q 200
q 722 200 " q
q2 = 722(200)
q = 380 AC(q=380)=3.8. [1‘ Therefore, the ﬁrm will not sell for any price less than 3.8 in the long run. What is the lowest price at which each ﬁrm would sell its output in the short run? Is proﬁt
positive, negative, or zero at this price? Explain. The firm will sell for any positive price, because at any positive price marginal cost
will be above average variable cost (AVC=q/2000). Proﬁt is negative as long as price
is below minimum average cost, or as long as price is below 3.8. LR 80V lMKQ : 3.6 => WW ATC
Tm, LR <9}: 6500 ‘ WM} (@130 K . >§ _ :
‘£ Ovuke 3&0) \WQW “kg 2 V
a. MC 2 :1}:qu . To ﬁnd the proﬁt maximizing quantity, we set MC=P and get
0
PAZK . . . . .
% : 3 ° . The e1ast1c1ty of supply With respect to pnce is 0.5
W Plugging in the numbers into the supply function we get 3; = J1j . And so will p n" O
PS=P TVC=P 3‘: 2=341.36 , EH ‘
% % A g ‘ Phi (A?
Ps  O 3/ 9“
43 =. \1 £5? 1 l 8 as
c. The industry is not in equill rium as the ﬁrm makes a positive proﬁt, equal to
PSTFC= 341.36 (48)3= 197.36 For long run equilibrium MC=ATC, 3wq2 2— +—géoy, which gives P= 36, AZKO r%A
=6. d. As technology improves, the costs will decrease and the ﬁrms will have
positive proﬁt. In the long run, we will see entry into the market. II II In the initial LR equilibrium, we know the typical ﬁrm earns zero economic
proﬁt. Thus, they must be producing where MC = ATC . Why? Well, use my
famous “box” from lecture. The combination of proﬁt maximization and perfect
competition mean that P = MR = MC . BUT, if ﬁrms earn zero economic proﬁt,
then P = ATC as well. Thus, by the transitive property, MC = ATC . Where
will this condition be met? If ATC is Ushaped (as it is here as you can check),
then MC = ATC at the minimum of ATC. SO, you can either ﬁnd the q* LR
initially by minimizing ATC with respect to q or by simply setting MC = ATC
and solving for q. I choose the latter: MC: P‘TC’ 7~ I
521, 1905+? ~> ”01> qD 7% Ll Now, use either the MC or ATC curve to ﬁnd P* LR: 1);: m”: slag“ 322: @ Now, with that P* LR , we know from the market demand curve that Q* = 256 — 2(64) = 128. With a q* LR of 2 , we must have n = 64 ﬁrms
initially. For all the diagrams on this problem, see the back pages of these
answers. Essentially, with a tax per unit of $4 , the TC function increases by an amount
equal to 4q . Thus, our new TC function becomes: TC = up 6% + £th Knowing therefore, that the SR supply curve is the market supply curve above
min AVC, we have the following SR supply curve for the individual ﬁrm: QP=MC : BloerLl NOW, however, we need to ﬁnd the MARKET SR supply curve to see what
happens to P* in the SR equilibrium initially after the tax is imposed. We know
that we still have 64 ﬁrms in the SR (entry or exit hasn’t yet occurred), so adding HORIZONTALLY becomes:
5  5R _ p, Ll— V
Q SR  H ‘ % ' 64' ”EE  (9.?“ 8 DON’T forget to solve for q ﬁrst before you “add,” because you’re adding q’s
and NOT adding P’s! Now, use this Qs SR along with the market demand
curve to ﬁnd the P** and Q**: (SLSSRZ Zl‘85 25" 2i2®~>
L‘ ’E
soc‘oi’e/W‘ Thus, the P* rises by LESS than the amount ofthe tax (sound familiar from
1017!?) Each individual ﬁrm produces LESS than 2 , and each incurs a LOSS of m . Thus, we know that from the SR to the NEW LR equilibrium, ﬁrms will
EXIT, P** will rise because the market supply curve shifts BACK and to the
LEFT, until each ﬁrm earns zero proﬁt again. Thus, our NEW LR equilibrium will once again be where MC = ATC . Let’s ﬁnd that again: M ”13% — Wig—M”, MC. WILD! It’s the same FIRM output level as in the LR equilibrium at the beginning
ofthe problem. What’s different?! Well, the P*** will be different, for one. Find it the same way as before:
12*“. MM: 5mg“; Ll .~ 32(2)+ ‘+ = So, now, with a P*** of $68 , we know that the new Q*** will be found by using
the same market demand curve as always (it’s not shifted at all ever in this
problem), so we have: Q*** = 256—2(68) = 256«136=. With q*** back to 2 , then , we have n = 60 ﬁrms in our new LR equilibrium, so
the tax caused 4 ﬁrms to exit the industry from SR to LR. Makes sense! [a (1. NOW, with the tax being lump sum, the new TC function becomes: TC = lto§+ was : may PM So, in the SR, because MR doesn’t change, NOTHING about P* and Q* changes!
HOWEVER, proﬁts now drop by the full $80, meaning that ﬁrms will exit once
more! How many? Well, ﬁnd where MC = ATC once again: MC = 32%): [60Vr [ﬂ : ATc: _ ‘b a.
“V Lelia 0H => B Thus, in that NEW LR equilibrium, each ﬁrm produces more, but because of exit,
the total Q* falls. WILD! See diagram at end of problem set answers for more! F”: 31%;”: 32.3 = <16
Q“. 256—1616): 296 \‘l2 = 6% “I”? @321 9:3: Jail/3 OqYH‘Sl &v(\’rz (NRA o5; Chg}, \Dq go WM (Kiev; o\ liaise QKNYSN“
I! ”ML .. ngﬂYkrs 9‘qu & la) (w W +o New Hg #3 My, comm E ~~'S'_ 4P
. Pl ' ' “
ugg1ngm% rl/Zwl/Z
l/Zrl/Z
c. From pan b P = %WT Plug P back into conditional demands in part a, and do r401 13 XIV ‘N ®;W\Q\Q,\. 2 U2 2 U2
u r t W .
get L : g—H—Z K = QT. Both functlons are homogenous of degree zero
16w 16r d] IF it»; afldim 0* 8&4“ am 9mm (03. TM MM {@‘G‘ CM‘M’QL‘9\ l/" 2”, l / 2r ;
TCWLd’TK: %ﬁ§’ 1/21/2 4PM . .
Proﬁt maximizing condition is P=MC, so P : %W—’ and %:r” which is
the supply ﬁinction from part b.
2
P“ P‘” P2 P2 _ 2P
71': 4P(w3/gr1/8)(w1/8r3/8)_W(W)ﬁr(_wT2r—37)Tw1/2r1/2
'P 2
d7?_ L '_ _1 2.? _. 17 L
—_ﬁ _ / 5 ‘ : : ,—
dw 2— W iZY 2. Wk T’mmxﬁ
dlzK a 1, ( 2%
dr 2 W3]; : _—F7‘ __ ‘KX
figQ 4r? V WWW _ V “‘M
dP : TF/Vll
w 7? These conditions are called Hotelling s Lemma (in case you were wonderingl) Can
you provide the intuition here! 7 ' ' * No, this result does NOT violate the cost minimization assumptions. In the proﬁt
maximization problem, in response to an input price change, the optimal quantity
also changes, thus moving us to a different isoquant (the output effectl). Thus,
it’s possible for L to increase even in response to a drop in r because the output
effect (more q and thus more L) dominates the substitution effect (less L and more K because w/r has risen). See diagram below: Equilibrium occurs where quantity supplied equals quantity demanded, thus
where: P (3/ W)
120 — P = 2*P ‘10 P* = $40/heater and Q* = 80 heaters /week See graph below for CS and W CS: V2*80*80 = $3200 40 A
PS=‘/z*40*80 = $1600 / 20 \20 (\ﬁm) b. J erry’s price ceiling reduces quantity supplied to 40 heaters/week but increases , quantity demanded to 100 heaters per week. This price ceiling will incur DWL,
as well as reduce quantity from its original equilibrium level (thus more cold
people regardless of how consumes'the 40 heaters). Below, we have two
diagrams  one for the best case scenario of no black market (in itself probably a
bad assumption) and where those who value the good the most get the 40 heaters.
The other diagram below is the “worst case” scenario, in that there the ones who
value the good closest to $20/heater get the 40 heaters. In this second diagram,
start with the 100lh heater person and move backward along the quantity axis until
you’ve given the 40 heaters to the relevant consumers. Why is that the “worst
case" scenario? Well, in that case, the CS is its lowest because those who value
the good just at or above $20/heater wind up consuming it. Calculations for both
cases are next to the relevant diagrams: Eli/w.) BEST CASE SCENARIO: ‘1‘) go >®gKst> CS (Area A) = 1/2 * ( 100 + 60 )*40 = $3200. Note that the lost surplus from
consuming 40 rather than 80 heaters is exactly balanced by having to pay $20 for each of those 40 rather than $40 each. 80 too l'Uo yPS(AreaB)=‘/2*2O*4O = $400. DWL (Area C) = ‘/z * 60 * 40 = $1200. OR DWL is original total social surplus
[$4800 from (a)] minus the sum of CS and PS now ($3600). WORST CASE SCENARIO: R7 A
tk/ur)
\70 )
“'0 60 (0° \2.0 Q (\Q‘X‘S) CS=‘/2*40*40=$800
PS=‘/z*20*40 = $400 DWL = original total social surplus from (a) minus CS and PS here = $3600. With a $30/heater subsidy to consumers, find the new equilibrium price and
quantity using inverse demand and supply curves, noting that this subsidy shiﬁs
the demand curve up by $30/heater:
“New” Demand curve: P = 120  Q + 30
“Old” Supply curve: P = 1/2 * Q
Solving yields: 150 — Q = 1/2 * Q 150 = 3/2 * Q Q* = 2/3 * 150: 100 Plugging this equilibrium quantity back into the original demand and supply
curves yields the post subsidy prices to consumers and producers: P(consumers) = 120  100 = $20/heater
P (producers) = V2 * (100) = $50/heater Now, let’s calculate the CS, PS, Gov’t. Cost, and DWL in this subsidy case: CS(AreasABCE) = 1/z"‘ 100* 100 = $5000.
PS(AreasHCBG) = 1/2*100"‘50 = $2500
Gov’t. Cost = $30/heater * 100 heaters = $3000
DWL(AreaF)=‘/2*20*30 = $300. Verify that DWL can also be calculated by comparing the total social surplus in
(a) [$4800] with that in our subsidy case [$4500 = $5000 + $2500  $3000]. Sally’s plan thus does dominate even the best case scenario DWL while
eliminating the potential for a black market and warming up an extra 20 people
(assuming one heater per person)! Once again, perhaps the greatest negative
effect of price ceilings is the reduction in quantity exchanged. Subsidies, while potentially expensive to the government, increase equilibrium quantity and
eliminate price pressures in the marketplace (no black market pressure). 1 p G / Hr.)
See diagram below: \
\10 Quickly, We can look at the facts that the price to consumers falls by $20/heater
while the price to producers rises by $10/heater. Therefore, consumers receive
the lion’s share (2/3) of the subsidy beneﬁt. CS in (a) is $3200, while in (c) it’s
$5000. PS in (a) is $1600 and in (c) it’s $2500. The own price elasticities of
supply and demand at the subsidy equilibrium are: [aQ/aP]*[P/Q] = —1*(20/r00) = .2 [aQ/aP]*[P/Q] = 2*(50/100) =1 If, indeed, the more inelastic curve bears the relatively greater burden of any tax,
it should follow that the more inelastic curve receives the relatively greater beneﬁt
of any subsidy. Our calculations bear this out  demand is much more inelastic
that supply here. Because demand is more inelastic than supply, any subsidy
equilibrium will produce a greater drop in price to consumers than it will an
increase in price to suppliers. Thus, consumers here enjoy the greater subsidy
beneﬁt. Synthesize this result with the tax case. m ?. a. With the TC function given, ifthe ﬁrm is an input price taker, we can solve for the output elasticity oftotal cost by inspection. TC is log linear in q , making
that output elasticity of total cost equal to the value of the exponent on q (isn’t
that synthesis a WONDERFUL thingl). Thus, that output elasticity of total cost here is 2 . 31c QNZH.
o; 5%,‘ngjf: /a, ( >qD—[email protected] TC ’bSZMxQ TC/«Q (WNW) ,QD
Furthermore, from lecture, we also decided that the output elasticity of total cost
could be simpliﬁed to: as WC/a
fL: {3% ﬂ»: MC/ATC
\ C/% Because, then MC > ATC ifindeed the value ofthe ratio is 2 , then we know
that MC exceeds ATC for this ﬁrm, or equivalently that ATC is RISING! A
rising ATC curve is consistent with DISECONOMIES of scale, because TC must
be rising faster than q for additional output. So, doubling q MUST mean ATC
rises by MORE than double, reﬂecting diseconomies of scale. b. TC 1 ( WT) q; ijrh SRQDMQWXLQ LQmmm, W+f
x 357C  Z (WiﬂY ’ W'YUU
L W W + «W M
2, W?+wv»wv
 (kg (WHY >
 L 1
a @44va CR
C WAY X L z X— : \N‘“ 2.
\R L () WNW? Cb “8% K () waycb
We had 3m e/ng‘wﬁm W’s & V5 M
PER 51:: 1 Y1 & K» _ W; qO (WWW 0]; 6 WHY /
E W5 Wm Wm M & 3“ WWI; L: L & K); W
CE) WHF /\ 3 H“ [75mm Aﬂqaga UdeQSQU‘GS RN 30}
1/ \ <74)? (2L) + WK 2 Sit/1+ sz CWCWM Mk \o/w was &
Aisecmm‘uu g m ”I ﬁrm 19% ND 8 . a. TC(q, w) = 0.5q2 —10q + w Equilibrium in the entrepreneur market requires
Q = 0.25w= Q01: n or w = 4n. Hence C(q, w) = 0.5q2 —10q+ 4n.
MC: q — 10 AC=.5q—10+:¢£
q
In longrunequilibrium: AC = MC so q—10=.5q—10+:4£
‘1
.quﬂ gm:
‘1 Q3 : nq : n(P + 10). Have 3 equations in Q, n, P.3‘Since an 8n and Q = n(P+ 10), we have
n 8n:n(P+10) P=J87—10.Q021500~50P=150050J£4_n+500 :2000—50J872Q52n 871 or (n+50)\/8—n:2000 \) Um n : 50 (= # of entrepreneurs) anJ8—n:1000 W
q=Q/n:20 EKKVX
P=q—10:10 8““ 0R
w_=4n=200. b. Algebra as before, (n + 50)\/8—n = 2928, therefore n = 72 Q=nJ87=1728 —_
q=Q/n=24 \Mr 35W ’17er
P=q~10:14 z
w:4n:288 W” CPHO This curve is upward sloping because as new ﬁrms enter the industry the cost curves
shift up: AC : 0.5q — 10 + (4n/q) as n increases, AC increases. >(SEE LPiST wagiiog ng oimgprmtw
To Vme JW QWDJWJ/ (H? We Imrmse
wags in We 6Q w LR 3W5 \{xr ...
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