Question

A consumer has preferences U=x₁¹/2X₂¹ and demand functions

X₁=Y/2p1; X₂=Y/2p2

The income is $100.

A) How many units of each good does the consumer buy if the price of each good is $1? What is the consumer’s utility?

B) How many units of each good does the consumer buy if the price of X1 is $4 and the price of X2 is still $1?

Answer 1

A consumer buys 50 units of each good at a price of $1, resulting in a utility of 50. When X1's price is $4 and X2's price is $1, the consumer buys $12.50 worth of X1 and $50 worth of X2, resulting in a utility of $25.

Step-by-step explanation:

To determine the number of units of each good the consumer buys, we need to set up an optimization problem where the consumer maximizes their utility subject to their budget constraint.

For part A, when the price of each good is $1, the consumer's demand functions become:

• X₁ = Y/2
• X₂ = Y/2

Substituting the consumer's budget constraint into the demand functions, we have:

• X₁ = $100/2 = 50 units
• X₂ = $100/2 = 50 units

So the consumer buys 50 units of each good, resulting in a total utility of:

• U = (√50)(√50) = 50

For part B, when the price of X1 is $4 and the price of X2 is still $1, the consumer's demand functions become:

• X₁ = Y/(2p₁) = $100/(2*$4) = $12.50
• X₂ = Y/(2p₂) = $100/(2*$1) = $50

So the consumer buys $12.50 worth of X₁ and $50 worth of X₂, resulting in a total utility of:

• U = (√$12.50)(√$50) = $25