Joe Bob from Problem 12 has a cousin Pete who consumes goods 1 and 2. Pete thinks that 3 units of good 1 is always a perfect substitute for 4 units of good 2. Which of the following utility functions is the only one that would not represent Pete's preferences? A) U(x₁, x₂) = 40x₁ + 30x₂ - 10,000. B) U(x₁, x₂) = 4x₁ + 3x₂ + 1000. C) U(x₁, x₂) = min{ 4x₁, 3x₂}. D) U(x₁, x₂) = 16x²₁ + 24x₁x₂ + 9x²₂. E) More than one of the above does not represent Pete's preferences.

Question

Quizplus · Accepted Answer

Pete's preferences are represented by a Cobb-Douglas utility function with perfect substitution between goods 1 and 2. This means that the marginal utility per dollar spent is always proportional for these two goods, regardless of their relative prices. In other words, the ratio of marginal utilities equals the ratio of prices, which is constant. 
If 3 units of good 1 is always a perfect substitute for 4 units of good 2, then the MRS (marginal rate of substitution) between the two goods is always 4/3. This can be seen from the fact that Pete is willing to exchange 3 units of good 1 for 4 units of good 2, or vice versa, without any change in satisfaction. 
The only utility function that does not have a constant MRS of 4/3 is C, which has a minimum function instead of a Cobb-Douglas function. Therefore, C does not represent Pete's preferences.

Joe Bob from Problem 12 Has a Cousin Pete Who