1) For the following equations, find the first derivative and local maxima and minima where x > 0.
a. Y(x) = x2- 3x.
b. Y(x) = (√x -1)2 - √x.
2) Show that given a fixed perimeter C, the rectangle with the largest area is a square. (Note: The formula for the area is the objective function. The formula for the perimeter gives the constraint.)
3) Find δF/δx, δF/δy and δF/δz as appropriate:
a. F(x,y) = x2 + y.
b. F(x,y) = x2 + y2 +xy.
c. F(x,y) = x2y2
d. F(x,y,z) = (x+z)/(y2+z).
4) Find the market equilibrium (p*, Q*) for the following demand and supply curves:
a. Qd = 100 - p, Qs = p - 50.
b. Qd = 600 - 10p, Qs = 2p.
c. Qd = 100 -3p2, Qs = 10p - 20.
d. Qd = 16/p, Qs = p
5) Sketch the demand curves in 4a and 4b at the bottom of the page. Be sure to clearly label the choke price, the p and Q axes and the Q intercept.
6) If the demand curves in 4a and 4b represent two segments of the same market, write down the market demand curve. Be unusually careful in specifying the demand curve - watch your choke prices. Sketch the market demand curve.
7) Give an example of a good which violates the more-is-better principle. Explain your answer. (Just saying, "Pizza, because a person gets full," is not a full credit answer. After all, many of us have been known to put the leftovers in the refrigerator for later. Think of a good where there is a right amount that makes one better off, but where too much of a good thing would NOT be wonderful.)
8) Perloff Ch. 3, 1.4. Show one plausible shape for Don's indifference curves. Yes, there are many possible correct answers. Show one. You are being asked to express an opinion.
9) Suppose Shoney can buy French Fries for $2 and Burgers for $5 and that Shoney must stick to a budget of $20 for both goods. Write down an equation for Shoney's budget constraint and sketch this line showing the quantity of Burgers on the horizontal axis and, similarly, French Fries on the vertical axis. Yes, it is a straight line. Yes, it has horizontal axis and vertical axis intercepts.
10) Find the MRS for each of the following Utility Functions. Sketch the indifference curve U=12 for all 3 and verify that they are the same. One sketch will do.
a. U(X,Y) = √X*Y.
b. U(X,Y)= XY2.
c. U(X,Y)=X2Y4.
Since the slopes of all the indifference curves are the same at an optimal bundle, and all are cobb-douglas preferences, the set of preferences curves are the same for all cases.