Problem 3: Tax-Free Threshold [15 marks]
Consider the static model of the household discussed in Lecture 3. Suppose that instead of beingsubjecttoalump-sumtax, theconsumerfacesalabourincometaxwithatax-freethreshold. 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 τ 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)+π, if w(h −l) ≤ x,
C = w(h −l)−τ£w(h −l)−x¤+π, if w(h −l) > x,
where C is consumption, l is leisure, w is the wage, π 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.

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