An important problem in consumer theory is the labour-leisure choice model. Assume that Fiona must decide how much of her time to spend working (i.e. generating income) and how much to spend enjoying leisure. Fiona prefers strictly more income to less, and also strictly more leisure to less. She is self-employed, and can dedicate as much or as little time to working as she desires. The number of hours dedicated to employment is E, and each unit of time spent working gives a wage rate of w, which is added to income. On top of that, Fiona has an initial endowment of income, y0, which is independent of the hours worked. Given that, Fiona must choose E and L to maximise her utility, which is u(y,L) = u(y0 +wE,L), and which is assumed to be strictly concave in the vector (y,L). Finally, the total number of hours available to distribute between work and leisure is T, that is, E + L T.

1. Formulate the Lagrangian for the problem, and find the first-order conditions and the complementary slackness condition.

2. Show mathematically that the constraint for the problem must bind in the optimal solution.

3. Analyse the comparative static effect of an increase in w upon L* and upon E*. What is the mimimal assumption you can make on utility so that this comparative static effect has an unambiguous sign?

4. Analyse the comparative static effect of an increase in y0 upon L* and upon E*. What is the mimimal assumption you can make on utility so that this comparative static effect has an unambiguous sign?

5. Assuming your conditions for non-ambiguous signs on the comparative statics hold, how do increases in w and in y0 affect the level of income that Fiona enjoys in the optimum?

6. How do increases in w and in y0 affect the level of utility that Fiona enjoys in the optimum? Do these results rely on your condition from above?