Question

General Equilibrium: Consider an economy that can produce tacos (X) and hamburgers (Y). Let the production possibilities frontier (PPF) be

Y² = 100-4X² (Eq. 1)

or, equivalently

? = √100 − 4?² (Eq. 2)

(for positive values of tacos and hamburgers).

This means that the rate of product transformation (RPT), the number of hamburgers that must be given up to get one more taco along the PPF, is

− ??/?? = 4?/(√100−4?²)

a. Suppose initially that the price of X = 3 and the price of Y = 2. If the total value of hamburgers and tacos produced is maximized, given these prices, what combination of tacos (X) and hamburgers (Y) will be produced? What is the RPT at this combination of tacos and hamburgers?

b. Now consider consumer preferences for tacos and hamburgers. Suppose the aggregate utility function across all consumers is ? = √??. This means that the marginal rate of substitution, the slope of this indifference curve, is

???= (??/??)/(??/?Y) = (.5(??)^.5?) / (.5(??)^.5X) = Y/?. (Eq.4)

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.

c. To achieve general equilibrium, we need three ratios to be equal to each other: MRS = RPT = ?_X/P_Y

And we need this to occur somewhere along the PPF. We can find the equilibrium values of tacos (X) and hamburgers (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 Y 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 tacos and hamburgers in just the right combination to maximize utility given the constraint of the PPF?

Answer 1

A) @ At equilibrium : MRSxy= Px - may = Px ax Fy 1100-4x² or, 81 = 3(100-412) 42 Squarring both sides, 642²= a (100-42²) 7 (64+36) 22-ax100 2²=9 Setting x=3 in (2); we get y=8] Thus, (2-3, y =8) is the optimal combination. 4x3 = 12 = 12 = 2 2 2 J100-4X9 or, RPT, 82=3 = 1.5] No, this combination of (x, y) is not a consumer equilibrium, as the RPT obtained in part a) was because of opliurising the Community indifference curve (CIC); but here we are given individual consumer preferences, represented by the u() funtion. the vole of substituliou og

GE. MRS - RPT= Bc @ MRS - Probatusting From Or, VA- ОХ substituting in 2; we ger DX 1100-4x² From RPT, we have 4x² -xoay = DX 1100-4x² From @ & 6; 2100-422= 4x² 1100-42² 100-42² = 4x² 7 9,892- 100 2= 112.51 Betting a = √12,5 in ③ we have: Y = 1100 -50 = $50 = 2512.5 or y=-2/2,5 utility maximiping price ratio si Thus, the C) 3