Final answer:
Leonard's MRS, income elasticity, and demand functions for two goods are derived from his utility function, which involves marginal utilities and budget constraints. MRS is the negative ratio of the marginal utilities. Demand functions and income elasticity depend on the ratio of marginal utility to price and income changes.
Step-by-step explanation:
In this context, the student's question pertains to the determination of the marginal rate of substitution (MRS), the income elasticity of demand, and the demand functions for two goods based on Leonard's utility function U(x₁,x₂) = ln(x₁) - x₂. To find the MRS, we need to calculate the ratio of the marginal utilities of the two goods. The interior demand functions can be derived by maximizing the utility function subject to the budget constraint p₁x₁ + p₂x₂ = m, which can then be used to find the income elasticity for both goods.
The marginal rate of substitution (MRS₁₂) is found by taking the derivative of the utility function with respect to each good, U₁ = 1/x₁ and U₂ = -1, and then taking the negative ratio of these two marginal utilities, which yields MRS₁₂ = x₁. For the interior demand functions, we apply the utility maximization rule that at the optimal choice, the ratio of the marginal utility to price of good 1 should be equal to the marginal utility to price of good 2.
To find the income elasticity of demand, we take the percentage change in quantity demanded of a good in response to a percentage change in income. This is calculated as (dx₁/dm)(m/x₁) for good 1 and (dx₂/dm)(m/x₂) for good 2, where dx₁/dm and dx₂/dm are the derivatives of the demand functions with respect to income m.