Question

Lisa is trying to decide whether she should buy stuff (c) with price p=1 and leisure (L) with wage w. her utility function = U(L,c) = L + 2c^1/2

Questions: 1a) write down the lagrangian function for lisa's utility maximization

1b) find the demand functions for leisure (L*) and stuff (c*) then find the labor supply funciton l* does matt work more than the wage rate rises?

Be sure to show your work, these answers need to be accurate if i'm going to learn from them please, thank you :)

Answer 1

Lisa is trying to decide whether she should buy stuff (c) with price p=1 and leisure (L) with wage w.

Her utility function is given as,

Now, Lisa has time endowment of 24 hours which she devides between Leisure (L) and Labor (l). Hence,

L+l=24........(1)

Lisa consumes stuff(c) with her labor income i.e. w.l. Hence, her budget constraint will be,

p.c=w.l

or, p.c=w.(24-L) {From equation 1}

or, p.c+w.L=24.w.......BL

This is Liza's budget constraint. She will maximize her utility U(c,L) subject to the budget constrat i.e.

Max subject to p.c+w.L=24.w

(1.a) The lagrangian function for Liza's utility maximization is,

where k>0.

This is the Lagrange's Equation for Lisa's utility maximization.

(1.b) Now we will solve the Lagrange's Equation for L* and c*.

Hencez the first order conditions or FOCs are

or, .......(2)

or, ........(3)

And,

or, p.c+w.L=24w.......(4)

Hence, deviding (2) by (3), we get

or,

as, p=1, then c*=

And, from equation (4) we get

p.c+w.L=24.w

or, 1. + w.L = 24w

or, w+L=24 {as

or, L*=24-w

Hence, demand function for leisure is L*=24-w and demand function for consumption is

c*=.

Hence, the labor supply function is

l*=24-L*=24-(24-w)

or, l* = w

Hence, the labor supply function is l*=w.

From the labor supply function we cam see that, when w increases, l* increases means Liza works more. Now we can see

dl*/dw = 1. Hence, the wage rate and labor supply increases equally. Hence, if wage rate rises, Liza increases labor supply by the same amount.

Hence, Liza works more when the wage rate rises.

Hope the solutions are clear to you my friend.