Optimal consumption bundle?

Question

I need to find the optimal consumption bundle of this utility function:

U(C,T)= 3C+2T

s.t. 2C+1.25T=50, where the price of C is 2$ and the price of T is 1.25 and income = 50

this is a perfect substitute equation and the MRS would be 3/2 right? (MUC/MUT). So when it comes to the condition that MRS= PC/PT, i hit a problem, MRS is 3/2, PC/PT is 2/1.25, 1.5 does not equal 1.6. Am i on the right track here or have I completely stuffed up somewhere?

Answer 1

Since your U is linear in both C & T and they are perfect substitutes, you will spend all of you money on EITHER C or T.

You can either buy 25 Cs or 40Ts. So your optimal bundle solves max(3*25,2*40)=80.

So, you would buy 40 Ts & 0 Cs.

Normally, the approach that you took (MRS=Price Ratio) is the correct approach. This however is one of the few special situations where the optimization rule is different. Other examples of special situations include perfect compliments and functions with atypical properties such as structural breaks.