A single father inspired by tennis greats Venus and Serena Williams has decided that his daughters Xelna and Yuna should grow up to be professional tennis players. He seeks to maximize their wealth and fame such that his utility function is represented by U(X,Y) where the success of Xelna and Yuna are quantified as X and Y respectively. He knows that future success is an increasing function of both talent and training (assume function is same for both children) while talent t is determine exogenously. The father can choose the amount of training to provide to each child by purchasing lessons l at a price p. The childrens innate talent is denoted as tx and ty while lessons are lx and ly. In addition to our normal assumptions about rational utility functions assume that Ux = Uy whenever X = Y and U(0,0) = 0. Assume diminishing marginal returns from success from lessons such that second derivatives have properties Xtl<0, and Ytl<0. Do not consider utility from other goods or services besides his daughters success and assume that he has exogenous wealth M to allocate between lessons for Xelna and Yuna. a) What is the parents utility maximization problem in terms of objective function, decision variables and constraints b) Show mathematically that the father will choose lessons so as to equalize the marginal contribution of lessons to the success of Xelna with the marginal contribution of lessons to the success of Yuna. Explain why the condition makes sense c) Suppose the following assumptions are true 1. tx < ty 2. Xtl> 0 and Ytl> 0 3. X(l=0,t=tx) = Y(l=0,t=ty) = 0

In other words 1.) Xelna has less innate talent that Yuna. 2.)Talent has a positive effect on the marginal impact on lessons on success. 3.) Without lessons neither child can achieve any success no matter their innate talent. Given these assumptions to which child will the father provide more lessons?. Demonstrate your answer mathematically or graphically and explain the intuition behind your result