Question

Suppose that the total utility function of a consumer is given by TU(x,y) = 3x^2y, and the prices of X and Y are 1 Birr and 2 Birr per unit, respectively. If the income of the consumer is 600 Birr and if he spends all of his income on the consumption of commodities X and Y, find:

a) The optimum amount of X and Y that the consumer will consume at equilibrium.
b) The Marginal Rate of Technical Substitution (MRTSx,y).

Answer 1

To find the optimal consumption of goods X and Y, we use the total utility function and the budget constraint to perform a Lagrange optimization, resulting in the Marginal Rate of Technical Substitution which indicates the optimal combination of goods.

Step-by-step explanation:

To determine the optimum amount of X and Y that the consumer will consume at equilibrium given the total utility function TU(x,y) = 3x2y, and the prices and income constraints, we must set up a Lagrange optimization problem. This involves finding the marginal utilities of X and Y, setting the ratio of the marginal utility to the price of each equal to each other, and solving under the constraint of the budget.

The consumer's budget constraint can be expressed as 1x + 2y = 600. The first-order condition for maximizing utility under a budget constraint is that the marginal rate of substitution (MRS) is equal to the ratio of the goods' prices.

For the Marginal Rate of Technical Substitution (MRTSx,y), we differentiate the total utility function with respect to goods X and Y to find the marginal utilities (MUx and MUy), then find MRTSx,y as the ratio of MUx to MUy. This will show us how much of good Y the consumer is willing to give up to get one more unit of good X while keeping the utility level constant.