 # Problem 1 Suppose a consumer has the following utility function: u...

## Question

Problem 1
Suppose a consumer has the following utility function: u = X1 1X2. Under this utility function the marginal rate of substitution with X1 on the vertical axis and X2 on the horizontal axis is the following: M RS = -x1/ x2
(a) write the consumer's problem if p1 = 2 and p2 = 1 and I = 200.
(b) what is the slope of the budget line assuming X1 is on the vertical axis?
(c) what must be true of the slope of the budget line and the MRS at the solution?
(d) what other condition must hold at the solution?
(e) solve for the demand for good 2 (x2*). This will be a number after you plug in all of the given parameter values.
(f) solve for xi by plugging (x2) into the budget constraint.
(g) what is the maximum utility the consumer can achieve (u*)?
(h) graph the solution and note the values of x1*, x2*, and u on the graph.
(i) If p2 increases to 2, solve for x1*, x2*, and u*.
(j) graph this solution on the same graph as in part g.

Problem 2 (a)
Assume cigarettes are a normal good.
Draw a graph showing an initial optimal bundle of other consumption on the vertical axis and cigarettes on the horizontal axis.
(b) Draw a new optimal bundle for the consumer after the government imposes a sales tax on cigarettes. Clearly label income and substitution effects.
(c) Does the tax reduce the amount the consumer smokes?
(d) Under what conditions would a tax on cigarettes actually increase smoking? (hint: there are 2 conditions)

Problem 3
(a) How can the cost of child day care services affect the labor supply of single mothers? Think about how child care costs affect the wage (take home pay for single mothers). Support your answer with a detailed graph showing how increased child care prices influence the optimal labor supply of women (graph these outcomes in consumption vs. leisure hours space and assume that total time is spent in either leisure or work). Clearly label income and substitution effects assuming preferences are quasilinear (leisure is a quasilinear good).

## Solution Preview

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Problem 1:
(a) the consumer problem can be written as the maximizing utility subject to a budget constraint.
Max U(X1X2) subject to
2X1 +X2 = 200
(b) The slope of the budget line is given as –P2/P1
(c) At the solution, the slope of the budget line must be equal to MRS
(d) The MRS (marginal rate of substitution) must be tangent to the budget line at the solution
(e) MRS(x1,x2) = -X1/X2 = -P1/P2 at the solution which gives the first equation as
X1/x2 = P1/p2
X1/X2 = 2/1
X1 = 2X2
X1-2X2 =0   -----------------------------(1)
The second equation is the budget equation given as
2X1 + X2 = 200 -----------------------------(2)
Solving the two equations we get
Multiplying equation (1) with -2 and adding it to (2) we get...

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