Question

in class next week. I strongly recommend making a good effort We will discuss these questions to work through the problems on your ow answers in cl should help you review some key on consumer theory (our main topic for today and next week). The second question is on monopoly pricing (which we will cover later in the semester, but is good for math n and/or with fellow students before we discuss the specifically, get out some paper and write down your answers. Both questions mathematical ideas. The first question will also get you started ass - revi ew). 1) Suppose you have the following Cobb-Douglas utility function for mugs of root beer and slices of pizza: You have S100, all of which you will spend on pizza and root beer. Pizza costs $1 per slice, and root beer costs $1 per mug. a) What is your budget constraint? (Write down the equation.) b) Using your budget constraint and the utility function given above, derive an expression to indicate your utility as a function of root beer only (with income fixed at $100). The expression you derive should indicate the level of utility you reach with $100 spent and Xtb of root beer consumed c) Starting with your answer to part (b), use calculus to determine your utility-maximizing quantity of root beer. d) What is your utility-maximizing quantity of pizza? e) Check the second order condition to make sure your answer to part (c) is a maximum.

Answer 1

U(x_rb, x_p) = x^.5_rb x^.5_p Budget constraint = > p_rb middot x_rb + p_p middot x_p = I 1.x_rb + 1.x_p = 100 x_p = 100 - x_rb Put this x_p into utility function u = x^.5_rb (100 - x_rb)^.5 utility is maximized at du/dx_rb = 0 du/dx_rb = 0.5x_rb^-0.5 (100 - x_rb)^.5 + 0.5 (100 - x_rb)^-.5 (-1) x_rb^.5 = 0 1/x_rb^0.5 (100 - x_rb)^.5 - x_rb^.5/(100 - x_rb)^.5 = 0 (100 - x_rb) - x_rb = 0 100 - 2x_rb = 0 x_rb = 50 utility maximizing quality of root beer = 50 x_p = 100 - 50 = 50 utility maximizing quantity of Pizza = 50