Lisa's preferences for good X and good Y are described by the following utility function: U(X,Y) = min(X,2Y), where X and Y are the respective amounts of the good X and good Y Lisa consumes.

Compute the numerical values for Lisa's optimal choices for good X and good Y. Show relevant steps and carefully explain your reasoning. Draw a graph that illustrates Lisa's optimal choice in this case. 4. Now consider how a change in income affects Lisa's choices. a. Suppose that Lisa's income changes from / = 20 to I = 40 while prices remain at Py = 1 and Py = 2. Calculate the numerical values for Lisa's optimal bundle in this case. b. Plot two points on Lisa's income expansion path when Py = 1 and Py = 2. c. Also plot two points on Lisa's Engel curve for good X when Px = 1 and Py = 2. Make sure to label your curves carefully, including what variable is shown on each axis of the graph. In each case, explain your answer. Include in your explanations a definition of the terms "Engel curve" and "income expansion path."