Utility Function Implies Consumption of Not All Goods

Question

Suppose we have a utility function with three inputs, $j, k,$ and $s$ described by $$u(j,k,s) = A\ln(k^\alpha + \beta j^\alpha) + B\ln(s).$$ The price of $j, k,s$ are $p_j, p_k, p_s$, respectively, and we cannot spend more than amount $Y.$

We are asked to find conditions where the optimized result is to consume only two out of the three inputs. In other words, we wish to find conditions where one input is not consumed at all.

Is there a systematic way to approach this type of problem? I tried taking the marginal utilities with respect to each good, and finding a case where exactly one of the goods was negative for all $j, k, s \geq 0$, but I don't think that's the right way to go.

Answer 1

There might be a systematic way (e.g. some variation of the gradient method), but it is probably not trivial (see here), unlike this problem.

By checking $U(j,k,0)$, $U(j,0,s)$ and $U(0,k,s)$, you will find that one of them can actually never occur when utility is maximized. Then you can move on to examine the two goods where 0 consumption can occur, and compare their marginal rate of substitution to their price ratio. Whenever these are unequal in optimum, it signals a corner solution.