l. bonsmer a representative consumer who has preferences over an aggregate consumption good c and leisure l. Her preferences are described by the utility function: U((:, l) = ln(c) + ln(l) The consumer has a time endowment of h hours which can be used to work at the market or enjoyed as leisure. The real wage rate is 111 per hour. The worker pays a proportional wage tax of rate i, so the worker's aftertax wage is (1 t)w. The consumer also has dividend income of 71'. (Note that there is no lumpsum tax here.) (a) Set up the consumer's utility maximization problem. (b) Analyze what would happen to consumption and leisure if the tax rate if increased: (a) show the effect of an increase in t on a graph. (b) Explain the economic intuitions for your results 2. Suppose that a consumer can earn a higher wage rate for working\"overtime.\" That is, for the rst (1 hours the consumer works, he or she receives a real wage rate of MI, and for hours worked more than (1 he or she receives 1102, where 102 > 1111. Suppose that the consumer pays no taxes and receives no nonwage income, and he or she is free to choose hours of work. (a) Draw the consumer's budget constraint, and show his or her optimal choice of consump tion and leisure. (b) Show that the consumer would never work (1 hours, or anything very close to q hours. Explain the intuition behind this. (c) Determine what happens if the overtime wage rate, 1172, increases. Explain your results in terms of income and substitution effects. You will need to consider the cases of a worker who initially works overtime and a worker who initially does not work overtime. 3. Suppose that the representative consumer's preferences change, in that his or her marginal rate of substitution of leisure for consumption increases for any quantities of consumption and leisure. (a) What effects does this have on the equilibrium real wage, hours worked, output, and consumption? (b) Do you think that preference shifts like this might explain why economies experience recessions @er