Question

Problem 2

A consumer has the following preferences regarding consumption and leisure time: ?(?, ?) = ? ∙ (24 − ?) Where ? is the quantity of an aggregated consumption good and ? are the supplied labour hours (working in a job) per day, and consequently, 24 − ? is the leisure time ?. The budget available for daily consumption is the sum of labour income and other fixed (daily) income with ? = price of the aggregated consumption good ? = wage rate per hour ? = other income

a) Formulate the utility maximization problem of the consumer taking into account the budget constraint with the Lagrange-approach.

b) Derive the demand function for the consumption good and the labor supply function of the consumer (i.e. the optimal quantities depending on ?, ? and ?ഥ).

c) Show that the demand- and labor supply functions are homogeneous of degree 0 in ?, ?, and ?ഥ.

d) How does labor supply react to a changing product price? What is your explanation for this?

Answer 1

a acc, f) = cf f leisure houss Bolo total income = total spending. M+ w (24 f) = p.c. fra=24, at Laborss Max u CCif) St B.C. 80 L= of + a[ M+ (24-f) w- pc] bl foc ah e f - Pa =0 & ah = c-frw=o. 80 feom foco f = ¢ ) for pc. from B.C. fw = Mt 24w-fw. 2fw= M+ 24 w. fx = (M+ 24 w) / 2w. ax = 84_fr = 24 - M - 12. Labor ss 12-M WX = 84 w-M)/2w. I Aine Ca= f'w ll TM+24w) Ang * = (M+24W) gp a CAM, AW, XP) = 24 aw-AM = x1 24w-M) aw) Хgю Homogeneous of degree o An

- NOW CCAM, AW, AP) = AM + 840w = ЗАР So Homogeneous of degree M+Qqw =XC CM, w, P) al zero. Boned dy Price changes, dan zo Laborss Labor doesnit Now as Boduct change, Buz labor ss dp responses to wage sate & not olf price