Question

Show that if a consumer's utility function is given by U = 2X + Z, and prices are pX = 2 and pZ =1, there is no unique utility maximizing solution regardless of income level. What does this tell you about X and Y as commodities?

Answer 1

The utility function given

U= 2X +Z tells us that the goods are perfect substitutes because it is of the form U(X,Z) = aX+bY which is the utility function of perfect substitutes.

Also from the given utility function, we can easily find the MRS which is (-)a/b = 2.

Now we are given that regardless of income, there won't be a unique maximising solution. This implies that more of one solution exists. Also we know that price ratio is equal to MRS, so this means that budget line completely coincides with indifference curve.

The indifference curve in case of perfect substitutes is a straight line rather than a curve convex to origin.

So having the slopes same of both implies that both would be coincident. And hence instead of a unique solution, the consumer would be willing to consume any bundle lying on both indifference curve and budget line.

(You can comment for doubts)