[Fieldsend]

Economists, please help

Finn has the following utility function:
U=√C*L
where C represents real consumption spending and L hours of leisure time consumed daily (i.e. T = 24). As a result, his first-order condition for utility maximization can be written as a simple ratio of the arguments in his utility function:
MRS=C/L
a) Show that Finn’s indifference curves will be convex to the origin (e.g. choose U = 10 and show that the resulting indifference curve is convex). Under what conditions will this result in a unique equilibrium? (HINT: a graph will definitely help here.)

b) Suppose Finn has $96 of nonlabour income each day (he comes from a wealthy family) and faces the minimum wage of $12 per hour. Set the price of consumption (P) equal to one, and do the following:
(I) Write down Finn’s budget constraint. Show his first-order condition for utility maximization.

(ii) Compute Finn’s optimal daily consumption of leisure (L*), hours of work (H*) and consumption of goods and services (C*).

(iii) Derive Finn’s labour supply function (i.e. show how his optimal hours of work would vary with the wage rate he faces in the labour market). Compute Finn’s reservation wage. (Hint: Repeat the steps from part (ii), but keep the wage rate as a variable).

(iv) Sketch out Finn’s labour supply curve. What can you conclude about the relationship between his substitution and income effects as the wage increases?