Exercise 13.7

1. Exercise 13.7 MICROECONOMICS Principles and Analysis Frank Cowell March 2007

2. Ex 13.7(1): Question • purpose: A simplified model of optimal income tax • method: Use tax parameters to determine individual budget constraint. Solve for individual optimisation. Put solution from this into government maximisation problem.

3. Ex 13.7(1): tax-transfer system • (pre-tax, disposable) income space • the no-tax line x • disposable-income schedule induced by the tax • break-even point • minimum-guaranteed income disposable income • marginal retention rate • No tax: x = y 1-t t y0 • x = [1  t][y y₀] + y₀ pre-tax income y0 y

4. Ex 13.7(2): Question method: • Standard optimisation • Take account of corner solution • Adapt from solution to Ex 5.7 (repeated here) Skip Ex 5.7 stuff

5. Ex 13.7(2): Worker’s problem (no tax) • In absence of tax constraints on worker are • x is consumption • `y is non-labour income • w is wage rate • ℓ is labour supply • Worker’s problem can therefore be written as • found by substituting from above into utility function

6. Ex 13.7(2): Worker’s optimum (no tax) • Take log of maximand to get • a log(wℓ +`y) + [1  a] log(1  ℓ) • Differentiate with respect to ℓ • This is zero if • wℓ + aw + [1 a]`y = 0 • which implies ℓ = a + [1 a]`y / w • But this only makes sense if ℓ is non-negative • requires w≥ [1 a]`y / a • so optimal labour supply is

7. Ex 13.7(2): Worker’s optimum (with tax) • Net wage is now • [1  t]w rather than w • Non-labour income is now • ty0 rather than`y • So we can modify previous result • to give optimal labour supply: • where

8. Ex 13.7(3): Question method: • Use labour-supply function from part 2 • Combine with a “break-even” condition for the government

9. Ex 13.7(3): Breakeven • To ensure that everyone works • must set tax parameters so that w0 > w1 • requires y0 > y1 • where y1 is given by • Net revenue raised in this system is given by • If the tax is purely redistributive, then this should be zero • If everyone works then this condition and ℓ* formula give:

10. Ex 13.7(3): Breakeven (more) • Take the breakeven condition: • Simplify to give • Use the definition of the mean of the distribution F: • Choosing t fixes guaranteed income ty0 that can be afforded

11. Ex 13.7(4): Question method: • Find poorest person’s disposable income using solution to part 3 • Find t that maximises this using standard FOC

12. Ex 13.7(4): income of poorest person • The after-tax income of the poorest person is given by • [1t]w0ℓ + ty0 • Using the expression for ℓ* this becomes • (if the person works) • α[1t] w0 +αty0 • In view of the net-revenue constraint this becomes • This is then the objective function • government with "Rawls" type objectives • max the min income

13. Ex 13.7(4): optimal t • Take the objective function • Differentiate with respect to t • and set equal to zero • We may eliminate w0 to get • This yields the quadratic

14. Ex 13.7(4): optimal t (more) • Take the quadratic for the optimal t • Use standard algorithm to get • Rearrange and ignore the irrelevant root: • Optimal tax rate increases with γ: • the larger is the mean wage (relative to the lowest wage)… • … the more well-off people there are to pay for transfers

15. Ex 13.7: Points to note • Optimal income tax problem based on standard labour-supply model • Ingredients • individual utility function • SWF • distributional assumptions • government budget constraint • Note simplification introduced by • linear tax function • max-min social welfare function