1. (8) Provide two examples of utility functions that imply the same ranking over bundles of goods (a monotonic transformation) as U(x,y) = xy.
Calculate the Marginal Rate of Substitution for the above utility function and both examples that you provide.
2. (8)
a.) On two separate sets of axes, draw two separate budget constraints for goods X and Y, one for which the price of X is increasing as consumption of X increases, and another for which the price of X is decreasing as consumption of X increases. b.) There is a potential issue with finding a unique utility maximizing tangency point for one of these cases. What is that issue? Hint: consider the tangency point for a typically shaped indifference curve.
3. Central High School has $60,000 to spend on computers (C) and other stuff (Y), so its budget equation is given by C + Y =60, 000, where C is expenditure on computers and Y is expenditures on other things. C.H.S. currently plans to spend $20,000 on computers.
The State Education Commission wants to encourage “computer literacy” in the high schools under its jurisdiction. The following plans have been proposed.
Plan A : This plan would give a grant of $10,000 to each high school in the state that the school could spend as it wished.
Plan B : This plan would give a $10,000 grant to any high school, so long as the school spent at least $10,000 more than it currently spends on computers. Any high school can choose not to participate, in which case it does not receive the grant, but it doesn’t have to increase its expenditure on computers.
Plan C
: Plan C is a “matching grant.” For every dollar’s worth of computers that a high school orders, the state will give the school 50 cents.
Plan D : This plan is like plan C, except that the maximum amount of matching funds that any high school could get from the state would be limited to $10,000. a) (12) Draw four graphs representing Central Secondary School’s budget set under each of the above four plans. Show the expenditure on computers in the horizontal axis, and the expenditure on other things, Y, in the vertical axis. In each graph , also draw the Central secondary School’s current budget constraint without any plan. For simplicity assume that the school will not spend more than $60,000 on computers under any plan. b.) (8) Suppose that the headmaster’s preferences for expenditures on computers and other things are given by the utility function U=CY 2 .
How much would the headmaster choose to spend on computers and other goods if it does not adopt any plans? Now calculate how much the headmaster will choose to spend on computers under plan A and plan C. c.) (3) What would the headmaster prefer: No plan, Plan A or Plan C? Why?

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Todd Odd enjoys goods X and Y according to the utility function:
U (x,y) = X 2 + Y 2 a. (5) Find the utility maximizing amount of X and Y in the standard way (by solving for the the tangency point between his indifference curve and his budget constraint) if Px=$3, Py = $4 and he has $50 to spend. What does his utility equal? b. (8) Graph Todd’s budget constraint and the basic shape of his indifference curve (don’t worry if it is exactly to scale). What would his utility equal if he spent all of his money on X? How does this compare to part a. Explain why this is the case and how it relates to your graph.
5) (28) Suppose Teri’s utility function is given by: U = X 1/6 Y 5/6 . a. If I=$1,000, Px = $25 and Py = $20, how much X and Y will she consume? b. What is her marginal utility of income equal at this point? c. Solve for her expenditure function. d. What is the minimum amount necessary for her to achieve utility equal to 50? e. If the price of Y increases to $30, how much is necessary to compensate for this change in price? f. Now suppose the price of Y is subsidized from $30 to $20. How much X and Y would Teri consume? What would this cost the government? What does her utility equal? g. Now suppose Teri were given this amount in part f. directly in cash, and there is no subsidy.
How much X and Y would Teri consume? What does her utility equal? What principle does this reflect?
6) (20) Suppose Dylan derives utility from the consumption of bottles of wine (W) and kilos of cheese C. He maximizes his utility subject to the budget constraint associated with his weekly income of $240. The price of wine is $12 and the price of cheese is $8. a) Determine the optimal consumption bundle if Dylan’s utility function is given by the following
Cobb-Douglas utility function:
2 1
U(W,C) = W 3 C 3 b) Determine the optimal consumption bundle if Dylan’s utility function is given by the following CES utility function:
U(W,C) =
W


C


 , where δ=0.5 c) Determine the optimal consumption bundle if Dylan is only concerned with calories consumed and has a utility function given by:
U(W,C) = 15W + 30C d) Determine the optimal consumption bundle if Dylan always drinks 4 parts wine with every 1 part of cheese, such that his utility function is given by
U(W,C) = min (W/4,C). e) Suppose the price of wine increases to $15. How much cheese and wine will he consume if his utility function is the same as in part d) (perfect complements). How much of this change in his consumption of wine is due to the income effect and how much of this change is due to the substitution effect?