1. Mike is using quasilinear utility function from sugar x and other stuff N: U(x, N) = 20√x + N. The price per unit of sugar is $p, and the price of other stuff is $1 per unit. Mike has an income of Y = 100. Assume that sugar is a continuous good.
(a) Assuming an interior solution, show that Mike's demand function Dx(p) for sugar is given by Dx(p) = 100/p2(100 divide P square) . Explain why this demand function is valid only as long as p > 1.
(b) If instead p ≤ 1, then Mike has a corner solution. At this corner solution, what is his consumption of N and x, as a function of p?
(c) Sketch Mike's demand curve for candy. Given a price of p, how is Mike's consumer surplus defined? Feel free to use your diagram as an explanation, but give a precise verbal definition as well.
The price of sugar used to be p0 = 2. A tax on sugar has led to the price of sugar increasing to p1 = 2.5. It can be shown by integration of Mike's demand curve, that his consumer surplus as a function of price is given by CS(p) = 100/p (100 divide P) as long as p ≥ 1.
(d) Using consumer surplus as a welfare measure, what is the Mike's welfare loss due to the tax?
(e) At price p1, how much sugar does Mike demand? Given this quantity, show that the amount of tax revenue being raised from Mike is equal to 8.
(f) Compare the value of Mike's welfare loss due the tax to the amount raised in tax revenue. Which is larger in magnitude? Can you explain why they are not the same? (Hint: Consider the effect of price changes.)
(g) Can you suggest an alternative policy that be better for both Mike and for the taxation authority? Assume the taxation authority care only about tax revenue, and Mike about his welfare.
2. Rackspace (a cloud-computing company) is relocating an engineer from San Antonio to San Francisco. Because of the difference in prices between San Antonio and San Francisco, Rackspace is offering a relocation premium to the salary of the engineer. To determine the size of the premium, the company uses the engineer's consumption bundle in San Antonio (given prices here), calculates the cost of buying this bundle in San Francisco, and then offers a premium exactly equal to the difference.
Is this is the smallest premium that Rackspace could offer that would not make the engineer worse off from relocating? Why or why not?
How would the amount by which one could reduce their calculated premium depend upon the nature of the engineer's preferences?