Consider the static model of the household discussed in Lecture 3. Suppose that instead of being subject to a lump-sum tax, the consumer faces a labour income tax with a tax-free threshold. More precisely, the representative consumer pays no tax on wage income for the first x units of real wage income, and then pays a proportional tax on each unit of real wage income greater than x (in Australia, for example, this threshold is $18,200 per annum).

The consumer's budget constraint is then given by:

C = w(h l)+, if w(h l) x,

and C = w(h l) w(h l) x +, if w(h l) > x,

where C is consumption, l is leisure, w is the wage, denotes dividends, and h is the total amount of time available.

1. Draw the consumer's budget constraint on the (l,C) plane.

2. Suppose that the government reduces the tax-free threshold x. Using diagrams, determine the effects of this tax change on the optimal choices of the consumer under two cases: (a) The consumer pays a positive tax before x is reduced. (b) The consumer does not pay any tax before x is reduced, but pays tax after x is reduced. Explain your results in each case by alluding to income and substitution effects.