I need help with Utility and marginal utility HELLLPPP?

Question

Claire has the utility function, U(x,y) = min {5x, 2y}. What is the marginal utility of x and y? Are those two goods complements or substitutes for her

Answer 1

This is discrete function, so increasing quantity one of goods will give us following results:

If 5X < 2Y then marginal utility of X is MU≤5

If 5X ≥ 2Y then marginal utility of X is MU=0

If 5X ≤ 2Y then marginal utility of Y is MU=0

If 5X > 2Y then marginal utility of Y is MU≤2

These two goods are complements, just draw few iso-utility graphs and add budget lines, you will see that optimums (tangent of budget lines to iso-utility) are located at points there 5x=2y.

Answer 2

First of all, I would argue that they are substitutes: the way these goods enter Claire's utility function (which is called Leontief or fixed proportions for your reference) tells us that if we increase the consumption of, say, good x, then we have to increase the consumption of y as well ( by the amount of their ratio) to stay on the optimal path: this is exactly the opposite of substitution; if you do not increase the consumption of y as well, then the increase in x is worthless, since you would get the same utility with unnecessary (excessive expenditure)  These goods are definitely complements. Thus, as I noted above, the optimal solution requires that 5x=2y, and nothing else.