2. The consumer maximizes utility U = X15 yo.5 . They have $M to spend on good X, which costs $Px per unit and good Y, which sells for $10 each. A. Write the LaGrangian equation for the consumer's problem maximize utility subject to the budget constraint. B. Find the First Order Conditions (FOC's). Solve the first two FOC's for 1. Then set those two equal and solve for X or Y (always your "choice"). C. Use your solution from part C and the 3'd FOC to find the optimal amount of the other good (Y or X) to max utility. (Note - Both X* and Y* will be functions of M.) Side note - If you plug in a price and look at the relationship between the optional X and Income, it is called the "Engle Curve". It quantifies the relationship between the Quantity of X demanded & Income Changes. D. Plug in M= $100. Py = 10 and Px = (2.5, 5, 10). Plug this into your solution for X*. E. Then draw the other 3 budget constraints and use the graph to approximate the optimal amount of X at each price. Do you come up with an answer that matches your math here? F. Use the next graph to plot the 3 points (X* , Px) on the 2"d graph. What is this called?3. A firm produces and sells two differentiated products X & Y. (Recall differentiated products mean Monopolistic Competition, so firm demand curves slope downward.) The demand for good X is Px = 72 - 4x and Py = 60- 2Y. The cost function is C(X, Y) = (2X2 + 3XY + Yz). A. Set up and solve the profit max problem to find solutions for the optimal amount X and Y. B. ON YOUR OWN: Add a quota of 7 for the sum of X and Y. Set up the Lagrangian equation and compute the new optimal amounts of X and Y