# Given that both tax regimes raise the same amount of revenue explain why utility di ers

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1) Suppose a consumer has the budget constraint y = p1x1+p2x2, where y is income, x1 and x2 are the demands for goods 1 and 2, respectively, and p1 and p2 are their respective prices. Below are the Marginal Rate of Substitution (MRS) between goods 1 and 2 under di erent preferences.
U(x1; x2) = p
x1x2; MRS = x1 x2 (1)
U(x1; x2) = r x1 x2
; MRS =x1x2(2)
b)Using the budget constraint and the conditions on the MRS derive the demand functions for each case (Remember to use the appropriate sign con- vention for each case).
c) Given your answers to b), for each case plot the income o er curve, Engel curve, price o er curve and demand curve (for both p1 and p2) for x2 making sure to label the graph in each case correctly.
d)Provide some intuition about the di erence in the graphs under di erent preferences.
2) Suppose a consumer has Cobb-Douglas preferences given by U(x1; x2) = x 1x(1 )
2 with MRS = (1 )x1x2
a) Are these preferences Homothetic? What does that mean and what are its implications?
b) Assume the usual budget constraint with the price of both 1 and good 2 normalized to 1. Draw your indi erence curves for = 1=2.
c)On the same graph plot the indi erence curves for = 1=3.
d) Draw the income o er curves for each.
e) Do the same as in c and d but increasing p1 = 2.
d) Discuss the e ect of the changes
3) Consider the utility function U(x1; x2) = p
x1x2 with a MRS = x1 x2
.
Assume the price of good 2 is normalized to one and the price of good 1 is p and income is y so the budget constraint is given by px1 + x2 = y.
a). Solve for x1(p; y); x2(p; y) and U(x1(p; y); x2(p; y)).
b) Now suppose y = 10 and p = 3. Draw your optimal point by showing the tangency of your indi erence curve and budget constraint.
c) Consider an increase in p from 3 to 4. Using the Slutsky Equation, solve for the substitution e ect, the income e ect and the total e ect of the change in p.
d) Draw the results you get in c) on the graph you drew in b) making sure to label all of the changes correctly, outlining each step (substitution e ect, income e ect, total e ect).
e) Solve for the change in your utility level , U(x1(p; y); x2(p; y)) and the demand for x1(p; y) in both nominal and percentage terms? Are they the same or di erent? Can you provide some intuition for the result?
4) Repeat question 1) but now assume that income is not exogenous but the consumer has the endowment (!1; !2) where !1 and !2 represent how much of good 1 and good 2 the consumer has. Suppose initially !1 = 5 and !2 = 5. Thus you want to take into account the additional endowment e ect that occurs with a change in price.
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5) Consider the following set of bundles and prices

p1 p2 x1 x2
1 1 5 4
2 1 4 5
1 2 5 4

a) Put together a chart as in Table 7.2 in the book.
b). Identify which bundles are preferred. For those bundles that are not chosen, what can you say about them?
c) Are there any violations of WARP?
6) Suppose that an individual makes an income of y = \$50; 000. There is a probability, p that an accident will occur which will cost a = \$20; 000, and with probability (1 p) the accident doesn’t occur at there is no cost to the individual. The individual can purchase insurance I at a cost of \$10,000 and if they do the insurance company will reimburse them for 75% of the cost
of the accident. The individual is deciding whether or not to purchase the insurance.
a) Consider the case where the individuals utility is given by
U(net income) = p (net incomewith accident)+(1p)(net incomewithout accident)
Solve for the value of p that the consumer is indi erent between buying and not buying insurance. In words describe what the implications of your value of p are.
b) Now suppose the individuals utility is given by
U(net income) = p ln [u(net incomewith accident)]+(1p)ln [(net incomewithout accident)] Again solve for the value of p. How does it di er from the case in a). What is the intuition behind the result?
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7) Suppose there are 2 people in an economy and that a government has to raised \$5 in revenue. The government has two choices they can charge a at tax, T of \$2.50 per person or can charge a sales tax, t, on good 1 so that the unit price of a good is p1 + t. Each household has the same Cobb-Douglas utility function u(x1; x2) = p
x1x2 with the MRS = x1 x2
and the same budget constraint y = p1x1+p2x2. Suppose p1 = p2 = 1 and y = 10.
a) Solve for the demand and utility for each consumer under both the at tax situation and the sales tax situation. What is the value of t that is needed to raise \$5 of revenue?
b) Under which tax situation is utility higher? Given that both tax regimes raise the same amount of revenue explain why utility di ers
c) Show your results in b) graphically.

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