A simple theorem on the utility function Claim: For a two-good system, if the utility function is of the simple form: U=U1+U2 then neither 1 nor 2 can be inferior. Proof: (a) lemma: in a two-good system, there are only three possible combination (N,N), (N,I), (I,N) where "N" stands for normal and "I" for inferior. So two goods cannot be both inferior. I won't be bothered to prove this lemma and take it for granted. (Beat me?) We also need the following two principles. (b) First, we know "the principle of maximizing utility" is equivalent to "equalizing marginal utility per dollar", and then is equivalent to "MRS=relative price" where MRS stands for marginal rate of substitute. (c) Second, "diminishing marginal utility" implies for Q (dU/dq)(Q)>(dU/dq)(Q'), namely the marginal utility is a monotonic decreasing function in quantity q. Alright, reductio ad absurdum: Assume good 1 is inferior. We test this claim by increasing the real income. By (a), good 2 has to be normal. Let the original quantities at which utilities are maximized be Q1, Q2. Income effect tells us at the new tangent point with maximized utility, if the quantities are Q1', Q2' then Q1>Q1' and Q2'>Q2.
Then based on (c), we conclude (dU1/dq1)(Q1')>(dU1/dq1)(Q1) and (dU2/dq2)(Q2')<(dU2/dq2)(Q2). Therefore, MRS for these two cases cannot be equal. However, the relative price is the same for these two cases. So based on (b), we arrive at a contradiction. The claim follows. QED Remark: If one good is inferior in a two-good system then the utility function must assume an involved correlated form.
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