# Problem set 4 of Economic Theory I

Nir Dagan, Esther Hauk, and Albrecht Ritschl

Due Monday, 11 May, 1998

1. Rafael has a utility function of u(x,y) =x2 + 2y2 (x square plus 2y square)
Note: In the printed Spanish version the goods are donated by x1 and x2.

1. Draw indifference curves of Rafael's utility function.
2. What bundle will Rafael choose to consume if px = 1, py = 2 and m = 6?
3. Are Rafael's preferences homothetic? are they convex?
4. Assume that py = 2 and m = 6. Is there a value of px in which Rafael has more than one optimal choice? If so, what is it?
5. Find the demand function for x, and draw the corresponding demand curve when py = 2 and m = 6.

2. Miguel consumes two goods (how not?) x1 and x2. He has an income of 50 Duros a week. If the price of good x1 increases, and the income and substitution effects go in opposite directions

1. may we conclude that x1 is normal or inferior, ordinary or Giffen? Explain your answer.
2. does his behavior violates the weak axiom of revealed preference?

3. Susana choses the bundle (6,6) when the prices are (6,5) and the bundle (10,0) when the prices are (5,5).

• Does Susana violate the weak axiom of revealed preference?
• Is one of the bundles revealed preferred to the other? And if so what is it?

4. Eva has the utility function: u(x1,x2) = ln(x1) + 0.5ln(x2).

1. Find the demand functions.
2. Draw an Engel curve of good x1.
3. Eva has an income of a \$100, and the prices are p1 = 2 and p2 = 1. Find the change in the demanded quantity of x1 if the price of x1 is now 1. Decompose the change in quantity demanded to income and substitution effects.