Final answer:
The Marginal Rate of Substitution (MRS) at the bundle (X = 4, Y = 1) for the utility function U(X,Y) = MIN(2X, 5Y) is 2.5. This is calculated based on the ratio of the coefficients of Y to X within the constraint set by the MIN function.
Step-by-step explanation:
The question asks to find the Marginal Rate of Substitution (MRS) at a given bundle for the consumer's utility function U(X,Y) = MIN(2X, 5Y), where X = 4 and Y = 1. The MRS is the rate at which a consumer can give up some amount of one good in exchange for another good while maintaining the same level of utility.
Given the utility function U(X,Y) = MIN(2X, 5Y), we know that utility is maximized when 2X = 5Y. At the bundle (X = 4, Y = 1), utility is determined by the good for which the amount times its respective coefficient is smaller. In this case, it's 2X since 2(4) = 8 and 5(1) = 5. Thus, the utility is constrained by the X term.
To keep the same level of utility if Y is increased, X must be decreased in such a way that the product 2X remains equal to 5Y. Therefore, the MRS is the ratio of the marginal utility of X to the marginal utility of Y, which, due to the MIN function, equals the ratio of the coefficients of Y to X. Consequently, the MRS at the bundle (X = 4, Y = 1) is 5/2 or 2.5.