<p>There might be a systematic way (e.g. some variation of the gradient method), but it is probably not trivial (see <a href="https://economics.stackexchange.com/q/24702/1601">here</a>), unlike this problem.</p>

<p>By checking <span class="math-container">$U(j,k,0)$</span>, <span class="math-container">$U(j,0,s)$</span> and <span class="math-container">$U(0,k,s)$</span>, 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. </p>

CC BY-SA 4.0
2018-10-15T20:55:07.907Z
2018-10-15T20:55:07.907Z
Giskard
<p>Suppose we have a utility function with three inputs, <span class="math-container">$j, k,$</span> and <span class="math-container">$s$</span> described by <span class="math-container">$$u(j,k,s) = A\ln(k^\alpha + \beta j^\alpha) + B\ln(s).$$</span> The price of <span class="math-container">$j, k,s$</span> are <span class="math-container">$p_j, p_k, p_s$</span>, respectively, and we cannot spend more than amount <span class="math-container">$Y.$</span></p>

<p><strong>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.</strong>  </p>

<p>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 <span class="math-container">$j, k, s \geq 0$</span>, but I don't think that's the right way to go. </p>

<microeconomics><optimization>
Utility Function Implies Consumption of Not All Goods
lithium123
# Utility Function Implies Consumption of Not All Goods

(Asked by: username_0 on 2018-10-15)

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.

**Tags:** microeconomics, optimization

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## 1 answers

### Answer by username_1 on 2018-10-15. Score: 3 | Accepted answer

There might be a systematic way (e.g. some variation of the gradient method), but it is probably not trivial (see [here](https://economics.stackexchange.com/q/24702/1601)), 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.


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