Question

2. Consider a pure exchange economy with two goods, wine (x) and cheese (y) and two con- sumers, A and B. Let cheese be the numeraire good with price of $1. Consumer A's utility function is UA(x, y) = 2.c + y and B's utility function is UB(x, y) = xy. A's initial allocation is 10 units of c and 0 units of y. B's initial allocation is 0 units of x and 30 units of y. (a) Put wine r on the horizontal axis and cheese y on the vertical axis. Measure goods for consumer A from the lower left and goods for consumer B from the upper right. Mark the initial allocation with the letter W. Draw the indifference curves for each person through this point. Calculate utility at this allocation for both consumers. Is the initial resource allocation consistent with Pareto efficiency? Explain. (b) Solve for the contract curve of Pareto efficient allocations in this economy and show this on your graph. (c) Find the competitive equilibrium prices and consumption for each type of consumer. Derive A's and B's demand functions (Marshallian). Calculate the equilibrium price of wine assuming price of cheese is $1. Using demand for wine, show that Walras' Law holds. Show the budget constraint and indifference curves at the equilibrium. Label the equilibrium E. Show that is on the contract curve. (d) Suppose instead, you wished to obtain the equilibrium El, where consumption of good x by A rises to x = 5, but is also on the contract curve. Find corresponding price of wine pi that must hold in this instance, and show the set of possible new endowments (there will be a line that shows the complete set) that would satisfy the Second Theorem of Welfare Economics, given Pi.

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Answer 1

Answer: Given that consider a pure exchange economy with two goods, wine (2) and choose Cy) and two concsumesls, A & B. (10,0 * wine (x), cheese (y) nome Baise - cheese MA (x, y) = 2x ty UB (2,y)=xy' (0,30) a) t has perfect substitok utility function & has a Doughe as " cobb " cheese A o- Initial allocation : wine Ic of A is in blue (straight lines) Icofa in green (Rectangular hyperbolas) utovery of A at (10,0) =) UA (10,0) = 20 +0= 20 ni dolay of B at Co, 20) – uB 0,30) 50 (30)

PE allocations : The Pe locus is given by culve oor. where both Ics are targed to each other suice w doesn't die on o, oz, it is not pe. This means we can find a paredo efficient superior allocation to w which will make atleast, strictly better off, no body borse off =7 to prove wis not p. &, a pareto su pesiog allocation can be given: w- & <10,0),(0,30)} a parto superior allocation can be - Ps= {(8,10), (2.2003 which gives u A' (8, 10) = 2(8) + 10 = 26 UB! (2,90). 2(20) = 40 Note un'ana - MB'>MB tence both individuals can be made better off by moving inside the box. & w is not pe b) 0,02 in the contract coove (usually itis known as - pardo efficiency locus) the condead cove is characterised by the following a each point on colve: scope of Icoff e scope of scope the MRSAMP3

MRSA =2 MRSR = dulax - 40 1 guldy xB 32 HB 22XB Ionel C a curve = 2xg) 5 equation of contact curve 0, oz is contract cosue ox inteams of anaya 30-40 = 2 (10-29) =>Ly9- 22 A=10) -) Demand function: het price vedea be (p.) * (tä,ya" ( B7) (23yapla alega, mes : slepe of Bodgetting whose BL = P(x) xy = 1/2 MR.say; n, slope of B2 =p I sey Tpty =2} P=2 y = px Polop this in B2

- Pxt Pe= M₂ .. (rds yd) { , To calculak equilibrium prices - The markets must clear. ;) het Pc2 Thon xn+ XB W, A+W, B (WA & we are initial endome M1 + 12 =10 ł 2p Now, M = 10pto, M2 = 0+30 P + 30=10 22 30 20 27 2P Nod possible, hence pis not less than a 1) hel P22 Then RA +RB²10 ( 04 ay) that a 10 lotio) + 15 = 10 25+7.5 = 10 hence maskel deges By waleas how, of market for a clears then assump marked for y also clears

waleas law. The value of excess demand in all market iso compelistue ogolibrism es price =) (P. 22) is price vector p (2, 11) . I es allocation (2.5,1s) (7.5,15) Now to show maitot of wine (y) satisfies walsesdan excess demand) Total demand for wino - Total endeomont of wine. 25 ie (15+15)-(0+30).0 value ofercess demanda) 1 P2 = 1 (0) = 0 hence walias low holds corollary of those are kmaelects with positive prices, - than k-1 markets clear, thank a market automatically clears. now, ag" => { (2.5, 15), (7.5, 1s)}, DE (211) l equilibrium - BL & ICA wincides, having same slope ICB is mas lood ingreon point of equilibriums is where shume in MeSAD MRS SAP- auto ma 67

Elies an the contract colve: contact curve equation => yo xxB Cşa) cais, is)) 15 =2(7.5) honce il lies on the contract colve equatio, seis Ps. d) det en in edease from 2,5 too, as Newce we know:RD 10-DA I lo-5-5 110-555 ad lies on the contract colve yo =268 9822(5)=20 ya = 30-43=20 .CE:{5,20).(5,10)} 48,5 {C5,20), (5,10)} to fuid peices, for markeds to clear, suice its ace, of satisfies the ling shilling demand function. Masshilli asedd for B => {

B's allocation is (s, 10) & Mr = 5, 42 = 10 - 42220 - 0 ob 142 no das com poli ve 2915X rum vengo :> 20710p - (P=2] price of wine 22 Second welfare thosem Evesy ps allocation can be obtained as a compolitie egoiliberoan given appopsika de distribution of ondowments. hel new endowments be Pop A: (0,"w.y) Bi(W2", way? also w, P+w, malo mw,9+w,9=30 since 42 = 20 we can say = 26,+ , Y = 20 Hance set of possible endowments = 7/2w, ?+w, 7-20) os in terms of it's endowment) 2 (10-w,?) +30-wy. - 1 20 20-20, P+30-6,920 =/2w, 4+ W, 9-30)