Question

In: Economics

Jacinto consumer two goods: A and B, his preferences can be represented by the Cobb-Douglas function:...

Jacinto consumer two goods: A and B, his preferences can be represented by the Cobb-Douglas function: U (X_A, X_B) = 2X_A * X_B², where X_A represents the units consumed of good A and X_B the units consumed of good B. Consider generic prices for the goods P_A, P_B and an income of m

A) Find Jacinto's demand functions for good A and B

B) if P_A = 60, P_B = 90 and m = 540:

i. What are the optimal quantities of goods A and B that Jacinto should consume?

ii. What is the maximum utility that Jacinto can obtain?

iii. Represent graphically the optimal choice of Jacinto

Expert Solution

U = 2XA.XB²

Budget line m = PA.XA + PB.XB

(A) Utility is maximized when MRS = MUA / MUB = PA / PB

MUA = U / XA = 2XB²

MUB = U / XB = 4XA.XB

MUA / MUB = XB / 2XA = PA / PB

2PA.XA = PB.XB

Substituting in budget line,

m = PA.XA + 2PA.XA = 3PA.XA

XA = m / 3PA

Again,

m = (PB.XB / 2) + PB.XB = (3/2).PB.XB

XB = 2m / 3PB

(B)

(i) XA = 540 / (3 x 60) = 3 and XB = (2 x 540) / (3 x 90) = 4

(ii) U = 2 x 3 x (4)² = 6 x 16 = 96

(iii) Budget line: 540 = 60XA + 90XB, or 18 = 2XA + 3XB [Dividing by 30]

When XA = 0, XB = 18/3 = 6 (Vertical intercept) and when XB = 0, XA = 18/2 = 9 (Horizontal intercept)

In following graph, AB is the budget line and IC0 is the indifference curve tangent to AB at point E with optimal quantity of XA being XA0 (= 3) and quantity of XB being XB0 (= 4).

Rahul Sunny answered 2 years ago

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