1.  Consider two utility functions u and v over consumption bundles (x,y) ∈ R, which are defined as

u(x,y) = x0 . 1 y0 .9 ,  and

v(x,y) = ln(x) + 9ln(y).

Recall that ln is the natural logarithm function that is defined on R++ , such that ln(1) = 0,  =  , and ln(xa yb ) = aln(x) + bln(y).

(i) Derive the marginal utilities of u and v with respect to x and y .

(ii) Denote the marginal rates of substitution of u and v by mrsu  and mrsU , respectively, and derive

these functions. What does your result tell you about the shape of the indifference maps for u and for v?

(iii) Find a function f that maps R++  to R so that v(x,y) = f(u(x,y)) = [f ◦ u](x,y), i.e., so that

the composition of f and u is equal to v . Find the derivative of f .

(iv) Explain the implications of your results from parts (ii) and (iii) regarding (constrained) utility-

maximising bundles for u and for v given a fixed constraint set.

2. Consider again Keiko’s optimal study time problem from the lecture, but now assume that she aims to derive her optimal bundle (h,y) consisting of Studying h and Learning y, so that the trade-offs she must balance are between a “bad” h and a “good” y .  Assume that she still has 16 hours per day at her disposal, so that h ∈ [0, 16], and that her utility over bundles (h,y) is given by

u(h,y) = (16 − h)0 .4 y0 .6 .

The production function that describes how Studying h is converted into Learning y is given by the relation

y = f(h) = aln(h),

where a ∈ R++  is a strictly positive (fixed) parameter.

(i) Since Keiko’s utility function u only values strictly positive levels of Learning y, what does the functional form of the production function f imply about minimal required study times for Keiko.

(ii) Derive the marginal utilities of u with respect to h and y, and the corresponding marginal rate

of substitution. Describe how the mrs and the associated indifference curves differ from the case of bundles of“goods” (x,y) considered in lecture, and interpret your answers.

(iii) Derive the marginal productivity for Keiko’s production function f, and determine whether it

exhibits diminishing or increasing marginal productivity. Identify the feasible frontier for Keiko’s constraint set expressed in terms of bundles (h,y), and find the corresponding marginal rate of transformation. Interpret your answer.

(iv) Sketch Keiko’s indifference map and her feasible frontier (remember part (i)!), and for the case where a = 5, use the mrs = mrt rule and Wolfram Alpha (or any alternative computer algebra system) to derive a numerical approximation of her optimal bundle (h* ,y* ).