Question

7. A consumer has the following utility function for goods X and Y: U(X,Y) 5XY3 +10 The consumer faces prices of goods X and Y given by px and py and has an income given by I. (5 marks) Solve for the Demand Equations, X (px,py,I) and Y*(px,py,I) a. b. (5 marks) Calculate the income, own-price and cross-price elasticities of demand for X and Y

Answer 1

Q.7.

a)

Utility function for the consumer is given as:

U(X, Y5XY 10

Prices faced by the consumer and his income is given as:

px, Py, I Y

a)

Solving for optimal demands:

For this, we have to solve the consumer's following optimization problem:

U(X, Y) 3 5XҮЗ + 10 тахх,Y s.t. рхX + ру Ү —I

Setting up Lagrangian, we get:

L 5XY10(I - pxX - pyY)

Taking First order conditions we get:

5Y8 а -С - 5Y3 — Арх — 0 3 А 3 ох (1) рх

15XY2 15XY2 - py L (2) 0 pY

L 0 pxX pyY I (3

Equating (1) and (2), we get:

БУЗ 15ХY? Зрх X %3D руҮ рx рY

Substituting this in (3), we get:

ЗрхX %—D 13D X"(рх, ру, 1) 4px рхX + pуY — 13 рхX

Similarly, we get optimal demand for Y as:

31 PxXpyY I pyY = I Y*(pxPY, 4py

b)

$\text {Income elasticity of demand} = \frac{\text{\% change in quantity demanded }}{\text{\% change in income}} \\= \frac{\frac{\mathrm{d} Q}{Q}}{\frac{\mathrm{d} I}{I}} = \frac{I}{Q}\frac{\mathrm{d} Q}{\mathrm{d} I}$

For X:

XP I I 1 Х 4рх Income elasticity of demand X dI X

For Y:

I dY 1 Income elasticity of demand Y 4py Y dI I

Price elasticity of demand is defined as how responsive change in quantity demanded is with respect to change in its own price.

$\text {Price elasticity of demand} = \frac{\text{\% change in quantity demanded }}{\text{\% change in price}} \\= \frac{\frac{\mathrm{d} Q}{Q}}{\frac{\mathrm{d} P}{P}} = \frac{P}{Q}\frac{\mathrm{d} Q}{\mathrm{d} P}$

For X:

$\text {Price elasticity of demand} = \frac{p_X}{X}\frac{\mathrm{d} X}{\mathrm{d} p_X} = \frac{p_X}{X}\frac{-I}{4p_X^2} = -\frac{I}{4p_XX} = -1$

For Y:

PY dY Y dpy PY -31 Y 4py Y Price elasticity of demand - Y

Cross Price elasticity of demand is defined as how responsive change in quantity demanded is with respect to change in the price of other commodity.

$\text {Cross Price elasticity of demand} = \frac{\text{\% change in quantity demanded }}{\text{\% change in other's price}} \\= \frac{\frac{\mathrm{d} Q}{Q}}{\frac{\mathrm{d} P_0}{P_0}} = \frac{P_0}{Q}\frac{\mathrm{d} Q}{\mathrm{d} P_0}$

For X:

$\text {Cross Price elasticity of demand} = \frac{p_Y}{X}\frac{\mathrm{d} X}{\mathrm{d} p_Y} = 0$

For Y:

$\text {Cross Price elasticity of demand} = \frac{p_x}{Y}\frac{\mathrm{d} Y}{\mathrm{d} p_X} = 0$