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## Exercise 5.7

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**Exercise 5.7**MICROECONOMICS Principles and Analysis Frank Cowell March 2007**Ex 5.7: Question**• purpose: Illustration of labour supply model with very simple preferences • method: Consumer optimisation with endogenous budget constraint**Ex 5.7: Worker’s problem**• Basic 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**Ex 5.7: Worker’s optimum**• 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 + [a 1]`y = 0 • which implies ℓ = a + [a 1]`y / w • But this only makes sense if ℓ is non-negative • requires w≥ [1 a]`y / a • so optimal labour supply is**slope = w**Ex 5.7: Interior maximum • (leisure, consumption)-space x • Max value of leisure • Indifference curves • Non-labour income • Budget constraint • Optimum • w≥ [1 a]`y / a • 0 < ℓ*< 1 `y 1 ℓ 0 1 ℓ* 1**Ex 5.7: Corner solution**• Indifference curves as before x • Non-labour income • Budget constraint • Optimum • w< [1 a]`y / a • ℓ*= 0 • `y 1 ℓ 0 1**Ex 5.7: Points to note**• Treat leisure as a conventional good • Has a natural upper bound • where labour is zero • Have to allow for corner solution • where person chooses not to work • Type of equilibrium depends on • wage rate • non-labour income