Final Answer:
Malachi's optimal bundle, satisfying the Slusky equation for rentals, involves renting 3 DVDs and purchasing 5 cups of coffee.
Step-by-step explanation:
Malachi's utility function is U(R,C) = R^0.75 * C^0.25. Given his budget constraint of $16, where the price of a DVD rental (R) is $4, and the price of coffee (C) is $2 per cup, we can formulate the budget constraint equation as 4R + 2C = 16.
To find the optimal bundle, we need to maximize Malachi's utility subject to this budget constraint. Since we only need to show the Slusky equation is satisfied for rentals, we can focus on the DVD rentals (R). The first-order condition for utility maximization is ∂U/∂R = λ * ∂B/∂R, where λ is the Lagrange multiplier, and B is the budget constraint.
Taking the partial derivatives, we get 0.75 * R^(-0.25) * C^0.25 = λ * 4. Solving for λ, we find λ = 3 * R^(-0.25) * C^0.25.
Substituting λ back into the budget constraint equation, we have 4R + 2C = 16 * 3 * R^(-0.25) * C^0.25. Simplifying, we get R^0.75 * C^0.25 = 3. This implies R^0.75 = 3/C^0.25.
To satisfy the Slusky equation, the marginal utility of DVD rentals should be proportional to the compensated marginal utility, which holds true in this case. Solving for R and C, we find R = 3 and C = 5, resulting in an optimal bundle of renting 3 DVDs and buying 5 cups of coffee.