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Unformatted text preview: YORK UNIVERSITY — ECON2350 — V. BARDIS
ANSWERS TO PRACTICE SET 1 X0?) = 021(1)) + 1'20?) = (50 —p) + (25 —p/2) = 75 — 329/2 No; all three demands are equally elastic: 1’ 275—3p/2 50—p
p p
= _1 = _
6W ( )50—p 50—p P P 1
gm _ _§25—p/2 _ 50—p 2. (a) The inverse demands are p = 50 — x1 and p = 25 — x2.
(b) The aggregate demand function is 0, ifp250
X(p)= 50—p, if 25319350
(50—p)+(25—p)=75—2p if p325 3. (a)
M M 3M +M2
X(p) = _1 + _2 I 1—
p 3p 329
£92
_ —23M1 + M2 P _ 1
610 — p 3 3Ml+Mg —
320
(C)
1 M1 1 d 1 M2 1
61 Z —— : an 62 : —— : (d) Let M{ = M1/2 and M5 = M2 + M1/2. Substituting these in the demand function we get 3M{+M§ 1 X MIMI = —=—3M 2 M M 2
(Pa 17 2) 3p 3p( 1/ + 2+ 1/)
l l
= $(2M1 + M2) < $(3M1 + M2) = X(P7M17M2) As expected, aggregate demand decreases when money is taken from the person with the higher demand for the good and given to the person with the lower demand. (Is there a calculus way
of showing this?) If equilibrium is determined by equating supply and demand, the equilibrium price will be lower. m LODQ
[email protected]%=§:m=1mmm—pyzmbm—1mw
i:1 (b) From the inverse demand function of each person we can derive her demand function by solving for {L‘i in terms of p: 1
11125——p Therefore the aggregate demand is 1
XXP)=ih0mX5———§p)=:a000——axm 5. (a)
6p E g—Zi—j = —bAp_b_1 P Ap—b : —b (b) Revenue, R, is constant if b = 1: Izzzunszp—[email protected]*+1===Ap—H4==Am0=.4u)==A QAQ @ p
e:—1=: — —=—1
% q
a
:>— =0
ap+q
: MR=0 (b) On a graph, the linear demand q = a — bp is the line segment connecting the point on the
vertical axis (07 a / b) with the point on the horizontal axis (a, 0). The midpoint of this segment is a[2,a[2b. The elasticity of a linear demand curve is given by _3qp_ p
6——~———b
(919 q a—bp Setting the absolute value of the above equal to 1 gives: a
bp=a—bp=>p=2—b and for this value of p, q is
b a (1
CL — — : — 2b 2
Therefore the point of unitary elasticity is the midpoint of the linear demand. 7. (a) The ‘choke—off’ price (the lowest price that sets quantity demanded equal to zero) is 9. 13 = 10/2 2 5. C S is the area under the demand curve between the market price p = 2 and 15 = 5.
This is the area of the triangle with base equal to the quantity bought (1(2) 2 10 — 2(2) 2 6 and height equal to the price difference 13 — p = 3: GS = lea—M10 — 2p) = §<3><6> = 9 2
(Note: The above is equivalent to CS 2 ff q(z)dz, where z is the variable of integration.)
(b) M WTP and the inverse demand function are pretty much the same. The M TWP ‘comes
first’ but then optimal behavior by the consumer requires p = M WTP. The latter equation is
used to derive the demand function. So ﬁnding M WT P from the demand function means going backwards: solve for p from the demand function to get 1 25——
P 2‘1 and then simply replace p with M WT P 1
MWTP(q) = 5 — —q
—L CS 2 TWTP(q) —pq q 1 1
GS =/ MWTP(z)dz — pg 2 5g — Zq? —pq = 5(6) — 1(6)2 — 2(6) 2 9
0 n 1‘ X . The slope of the budget line is —pgc/py = —3/2 = —1.5 and the slope of the indifference curve is 10. —1. It follows that the consumer is willing to give up a maximum of 1 unit of good y to get one more unit of good 1;. But good x is more expensive than this in the market. Getting buying one unit of x means the consumer has to forego 1.5 units of y. Therefore, this consumer will buy all y and no x, that is7 :L'* = 0 and y* = I/py = 3. (Another way to answer this is to look at things
by focusing on good y: For one more unit of y the consumer is willing to ‘pay’ 1 unit of good 1;. It only costs her 2/3 = py/pw units of a", less than she is willing to pay. So she will spend all her income on good y.) An example of a quasi—linear utility functions is U(ac, q) = 10g — (12/2 + a". This utility function
is linear in ac and a polynomial in q. The marginal utility of good q and of good ac is given
respectively by M Uq = 10 — q and M Us: 2 1. (These are found by simply differentiating the
utility function with respect to good in question.) Thus the marginal rate of substitution is given by MRS = MUq/MUJ; = 10 — q. Suppose the price of good 1; is 1 and the price of good q is p.
Then utility maximization requires that MRS = p/ 1 or simply MRS = p which in turn gives
10 — q = p. We can solve for quantity as a function of price directly (without usingn the budget constraint) to get q = 10 — p. This is the demand function for good q and it is linear in price. 11. See notes for part (a) and (c). Using the familiar steps the answer to part (b) is C(w,7“,q) = 2x/qu, MC 2 AC 2 2x/w7“. 12. Here we assume that the total cost function C (q) is twice differentiable. We know that AC (q) =
C(q)/q and MC(q) = dC(q)/dq. To minimize AC(q), we begin by setting its ﬁrst derivative wrt
to (1 equal to zero: dAC _ 0
L
Using the deﬁnition of AC (q) = C (q) / q, the above condition becomes 1
(MC(<I) — A0(q)) = 0
9—
Suppose the above has a solution denoted by qr > 0 and recall that the sufﬁcient second order condition for a minimum is dZAC
> 0
(if
Thus we have d2AC dMC dAC
l l
= —— M C — AC — — — —
dq, q2< (q) <q>> + q ( dq dq ) Evaluating the above derivative at qr makes the ﬁrst term on the right hand side disappear and
the second term reduces to dg2 _ E, dg > 0 it is suﬁicient to have % > 0. d2AC _ 1 (dMC) d2 AC
dq2 Therefore, for 13. (i) Each ﬁrm maximizes proﬁt given by 7r, 2 pg, — C ((1,) where i denotes ﬁrm 2'. Thus by setting
the derivative of proﬁt wrt to output equal to zero we get the familiar condition of p 2 MC. In
thismraseﬂenhane P = 10 + 2% Solving for (1,, we get the supply of each ﬁrm q, = —5 + .51). This equation is valid if the ﬁrm’s output response to price is such that pq 2 VC (q), where VC (q) 2 10¢] + (12 is the individual
7 ‘ A III 1 1" U, I: A“I. I‘ q,‘ I‘I I‘III A.,,. i A A and thus produce zero output. This requires that p 2 AVC (q) Now we know that p = M C ((1)
must hold for any q > 0 that the ﬁrm supplies. Thus the ‘shut—down price’ of the ﬁrm 1; is given
by p = min(AVC). Since AVC = VC/q = 10 + (1 here, then min(AVC) = 10 (where q = 0)
and so 12 = 10. Now we can properly specify the ﬁrm’s supply as q, = —5 + .51) if p Z 10 and g, = 0, if Q < 10.
(ii) We can add to the above the possibility where the cost function given represents long—run costs and the ﬁxed—cost of 100 is interpreted as an entry—fee so it can be avoided by leaving the
industry. In that case a ﬁrm will operate in the market, that is it will choose to enter and to
remain in the market, only if price exceeds min(AC). Only in that case the ﬁrm can quarantee no losses in the ’long run’. Thus the long—run supply function of the ﬁrm entails a shut—down
price of p = min(AC). By setting AC 2 MC and solving for q, we ﬁnd that AC is minimized
by setting q = 10. Thus min(AC) = 100/10 + 10 + 10 = 30 and so 12 = 30 in the long run. (iii) The aggregate supply with 200 ﬁrms is Q = 200g, => Q = 200(—5+.5p) => Q = —1000+100p
forp210andQ=0forp<1O (iv) In the long run, new ﬁrms will enter if p > 30 since the proﬁt of each existing ﬁrm will be
positive. Likewise, ﬁrms will exit the industry if p < 30 since each ﬁrm would receive negative
proﬁt. Thus in the long run the number of ﬁrms will adjust so that the price will be equal to 30 and the demand for the good at that price, for any amount demanded, is satisﬁed. This implies
that the long—run supply is an horizontal line at p = 30, that is, it is perfectly elastic. (v) For each ﬁrm PS, 2 7T, + FCi, that is, producer surplus is equal to proﬁt plus ﬁxed cost
which in turn implies PS, 2 pg, — VC(q,). Thus if p = 40, we have (1, = —5 + .5(40) 2 15 and
therefore PS, 2 (40)(15) — 10(15) — (15)2 = 225. Since the ﬁrms are identical, the total producer
surplus is PS 2 200PS¢ = 45, 000. ...
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 Summer '08
 Bardis

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