A consumer spends all her income of R120 on the two goods A and B. Good A costs R10 a unit and good B costs R15. What combination of A and B will she purchase if her utility function is U = 4A0.5 B0.5?
Question
A consumer spends all her income of R120 on the two goods A and B. Good A costs R10 a unit and good B costs R15. What combination of A and B will she purchase if her utility function is U = 4A0.5 B0.5?
Solution
The consumer's utility function is U = 4A^0.5 B^0.5. This means that the consumer derives utility from consuming goods A and B in a certain proportion. The exponent on A and B (0.5) suggests that the consumer likes to consume these goods in equal proportions.
Given that the consumer has R120 to spend and the prices of goods A and B are R10 and R15 respectively, the consumer's budget constraint is 10A + 15B = 120.
To maximize utility subject to this budget constraint, the consumer should set the ratio of the marginal utilities of A and B equal to the ratio of their prices. The marginal utility of A is 2B^0.5 / A^0.5 and the marginal utility of B is 2A^0.5 / B^0.5.
Setting the ratio of the marginal utilities equal to the ratio of the prices gives:
(2B^0.5 / A^0.5) / (2A^0.5 / B^0.5) = 10 / 15
Solving this equation gives A = B.
Substituting A = B into the budget constraint gives 10A + 15A = 120, or 25A = 120. Solving for A gives A = 4.8.
Since A = B, B = 4.8 as well.
So, the consumer will purchase 4.8 units of good A and 4.8 units of good B to maximize her utility given her income and the prices of the goods.
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