[color=rgba(0, 0, 0, 0.87)]Take the example with preferences over bananas (B) and apples (A) in [color=var(--jp-content-link-color)]Optimization

[color=rgba(0, 0, 0, 0.87)]The consumer solves the following problem:

[color=rgba(0, 0, 0, 0.87)]

max𝐴,𝐵s.t. 𝐵𝛼𝐴1−𝛼𝑝𝐴𝐴+𝐵≤𝑊maxA,BBαA1−αs.t. pAA+B≤W

[color=rgba(0, 0, 0, 0.87)]Fix <span class="MathJax" id="MathJax-Element-187-Frame" tabindex="0" data-mathml="pA=2" role="presentation" style="box-sizing: unset; display: inline-table; line-height: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-; min-height: 0px; border: 0px; position: relative;">&#119901;&#119860;=2pA=2 and <span class="MathJax" id="MathJax-Element-188-Frame" tabindex="0" data-mathml="α=0.33" role="presentation" style="box-sizing: unset; display: inline-table; line-height: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-; min-height: 0px; border: 0px; position: relative;">&#120572;=0.33 α=0.33. Make a grid of W between 1 and 3 and then plot the optimal ratio of B to A.