Question

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5. General Equilibrium: Consider an economy that can produce bowls of chili (X) and hot dogs (Y). Let the production possibilities frontier (PPF) be Y = 100-4X (Eq. 1) or equivalently Y = V100 - 4X2 (Eq. 2) (for positive values of chili and hot dogs). This means that the rate of product transformation (RPT), the number of hot dogs that must be given up to get one more bowl of chili along the PPF, is OY = 4X 700472 (Eq. 3) a. Suppose initially that the price of X = 3 and the price of Y = 2. If the total value of hot dogs and chili produced is maximized, given these prices, what combination of chill (X) and hot dogs (Y) will be produced? What is the RPT at this combination of chili and hot dogs? (5) b. Now consider consumer preferences for chili and hot dogs. Suppose the aggregate utility function across all consumers is U = XY. This means that the marginal rate of substitution, the slope of this indifference curve, is MRS = x - YE! (Eg. 4) EXY What is the marginal rate of substitution at the levels of production of X and Y that you found in (a)? Will this combination of (X,Y) and prices be an equilibrium-specifically, will it maximize utility given this utility function and these prices? Explain. (5) c. To achieve general equilibrium, we need three ratios to be equal to each other: MRS = RPTP And we need this to occur somewhere along the PPF. We can find the equilibrium values of chili (X) and hot dogs (Y) by writing the MRS from Eq. 4, substituting in for Y from Eq. 2 (to get an expression in X alone), setting this ratio equal to the RPT (Eq. 3) (which is an expression in X alone), solving for X, and deriving from that solution and Equation 2. Use this approach to find the values of X and Y that equalize the RPT and the MRS along the PPF. What are these values? What is the price ratio at these values of X and Y-the price ratio that draws out production of chili and hot dogs in just the right combination to maximize utility given the constraint of the PPF? (10)

Answer 1

Ans.) PAGE: DATE Q Rout Pouce of x = 3 Pouce of y = 2 I the producen must try to maximize the value of so volue doutput - Bx_X + RyY output V= 3x+24 _ Differentiale went to X əv_= 3+Q_ay - 0 Ox - ƏY=__44_ Cowen) ax_V100-4x2 __ Putting this eg" in = 0 3 + 2 1 - 4x I V100-4x2] V = X = 3+3 - 4 x 2125-8² av ax 3 - 4x__- 125-82 Puting av za Jax 3-4X0 25 -X2

3= - 4x V25 -82 3 va5-x² = 4x Squarung both the sides 9625-x2)= 16x2 225 - 9x2 = 16x2 225 = 25.x2 25x2 = 225 2-225 25 [x 91 and X=13_ X is output produced it can helbe negative So we rejectx = -3 1 x=3 Puthing x= 3 in y= V100-4x2 y = V 100-C4) (3) Vido-36 = √64 | Y= 8 )

PAGE: DATE: 1 RPIL JX=3 403) 4x V 100-4x2 V100-403)2 Y=8 2_AC3) %2 01 b.peut - 8 = 2.667 MRS 1 Jal x= 3 This point of production does nou maximize the utility because MRS (slope old is mot same as RRT CSlope oppe). This implies at this point Icis mol tangomi to PPF/PPC y isovolueline 1 -> siope of ic is higher than slope_of PRESO. LIC Wilbe more sicepeathan LPĖ. IC and PPF both lies on some so value line because value of consumphion must be equal to value of producties

PAGE: DATE: cpaut) MRS = RPI = Y X ux Vico-4X2 Y= - 4x² = 2x2 VIOO-4x2_ V05-x2 Squasung both sides vast y2 = 4x4 (25-32) Puthingvalece of y2_in_Y2=100-4x2 y2 = 100-4x2_ 4x4 = 100 - 4x2 25-X2 x4 = toe (@5-x2) 25-X2 x9 = (25-x2) C25-*). x 4 = 625 - 25x2-25x2 + x x 4-X4=_625 -5 0x2 50X? = 625 x2 = 12.5 x = 3.536] appeloximately

Putting back. X= 12.5 18 y? = 100 - 4x? V2 = 100-4612-52 y2 = 100 - 50 Y = 50 Y = √50 1,2 1 Y=7072 | approximately MRS = RPT= px RRT = y = Px. x Py 7.072 = Px 3.536 py approximately Darice vratio = Px = 2 Ру As._Px - MRS= RPI are same --this pauce 9ato draws OLLL porodu.ction of chill and holdoqın just the ought combinatics fo_maxlampeutlity quen the coastrant oppf

isovalueline/ BL with PAGE: Price Rolo = 1.5 DATE: 1 isovoue line IBL with pouc@ ratio Ру = + point E both PPF and la - Que tangent