Given:

Utility Function:

U(x₁, x₂) = - 1/x₁- 1/x₂

Budget Constraint:

P₁x₁ + P₂x₂= y

Where: x₁, x₂= Quantities of Goods Consumed

P₁ = Price of Good x₁

P₂ = Price of Good x₂

y = Consumer Income

Course Hero Expert Answer:

For Utility Maximization, we have the following LaGrangian Function:

Z = U(x₁, x₂) + (y - P₁x₁ - P₂x₂)

Z = - 1/x₁- 1/x₂+ (y - P₁x₁- P₂x₂) (1)

For the First Order Condition, we partially differentiate equation w.r.t. x₁, x₂& , then make them equal to zero.

Z/x₁²- P₁= 0

or

1/x₁² - P₁= 0

P₁= 1/x₁²

= 1/P₁x₁²

= 1/ P₁x₁² (2)

Z/x₂ = 1/x₂²- P₂= 0

P₂ = 1/x₂²

P₂= 1/x₂²

= 1/P₂x₂² (3)

Z/ = y - P₁x₁ - P₂x₂ = 0

y = P₁x₁ + P₂x₂ (4)

This is the First Order Condition for Utility Maximization.

1. Suppose a consumer seeks to maximize the utilityr function . . l l L [13.32: = . II I? subject to the budget constraint 33:11"? P2352 =11 where .r, and I, represent the quantities of goods consumed, p. and p; are the prices of the two goods and 1' represents the consumer's income. (d) (10) Show that the second-order condition for maixmization is satisfied