1. A consumer's preference ordering over baskets of goods (x1, x2) includes the following: i) she is indifferent between (5, 90) and (35, 10); ii) she strictly prefers (20, 50) to (5, 90).
Check whether convexity of preferences is satisfied over these three alternatives.
2. Draw the map of indifference curves corresponding to the following utility functions:
a) U(x1, x2) = (x1+ x2)2 ; b) U(x1, x2) = x1+ x2 ; c) U(x1, x2) = x1+ 2x2 ;
d) U(x1, x2) = x1 (x2+1) ; e) U(x1, x2) = min {x1, 2x2} ; f) U(x1, x2) = x1
3. Determine which of the following utility functions represent the same preferences:
a) U(x1, x2) = x1 x2 ; b) U(x1, x2) = ln x1+ ln x2 ; c) U(x1, x2) = x1+ (x2)2 ;
d) U(x1, x2) = 3 ln x1+ 2 ln x2 ; e) U(x1, x2) = (a x1 x2)1/2 with a>0 ;
f) U(x1, x2) = (x1)3 (x2)2 ; g) U(x1, x2) = (x1 x2)1/4 .
4. Suppose a consumer obtains utility only from quantities of x1 and x2 that exceed the minimum subsistence levels given by (x1, x2), according to the utility function U(x1, x2) = (x1 - x1)a (x2 - x2)b , where the exponents a and b are positive parameters.
Draw the indifference curves, find the expression for the marginal rate of substitution and show whether the function is homothetic or not.
5. Consider the utility function U(x1, x2) = ln x1 + x2. Find the equation of indifference curves and represent them graphically. Are they convex? Calculate the marginal utilities and the marginal rate of substitution. What is special about them? Is this a homothetic function? Show a trajectory along which the MRS would be constant.
6. If we observe once a consumer choosing (x1,x2) when (y1,y2) is available, are we justified in concluding that (x1, x2 ) ? (y1, y2 ) ? Explain briefly.