Representative consumers have the utility function U(C,Ls) = log(C) + log(1- Ls)

Question

Representative consumers have the utility function U(C,Ls) = log(C) + log(1- Ls)
Representative consumers have labor endowment of 1. So Ls must be less than 1
Representative firm’s production function is Y(Ld )=4 Ld

There is perfect competition in the markets (so price and wages are taken as given by firms and consumers)

a) Write down profit function of the firm (isn't it 4Ld-(w/p)Ld?)
b) Solve for the firms maximization problem and find the labor demand
c) Write down the budget constraint of the representative consumer
d) Write down the maximization problem of the representative consumer and find labor supply
e) Write down the market clearing conditions
f) Solve for equilibrium labor supply and labor demand
g) Solve for equilibrium goods demanded and goods supplied
h) Solve for equilibrium price and wage

please help this solution and I want to see work too.

Explanation / Answer

Answer)

a)            Profit function of the firm is given by:

Profit = Revenue - Cost

Profit = Ys – (w/p) x Ld

Where (w/p) = real wage and Ys = Y(Ld) = 4Ld

Thus, Profit = 4Ld – (w/p) x Ld

b)           

Max profit. = Ys – (w/p) * Ld

Profit, = 4Ld – (w/p)Ld

F.O.C, d /dLd = 4 – (w/p) = 0

(w/p) = 4

c)           

Budget Constraint is given by income which depends on quantity of labor supplied and wage level

I=wL , where 0L1

Thus, C=y=I/p=wL/p

d)           

Utility of consumers is U(C,Ls) = log(C) + log(1- Ls)

Using its monotonic transformation,

U = C(1-L)

U = wL/p - wL²/p

We have, w = 4p

U = 4L-4L² , where 0L1

Maximizing utility w.r.t L,

U/L=4-8L=0

4=8L

L=0.5

Thus labor-supply, L=0.5

e)

Goods market: Ys=Yd

Factor market: Ls=Ld

f)

Ls=Ld

Ls =0.5

L=0.5

Now, for the market, L= 0.5N

where N is the total labor population

g)

On individual level,

Income = I=wL=0.5w = 0.5*4p = 2p

Income = Y*p

2p=Yp

Y=2

and for production side:

Max Profit, = TR-TC

TR=TC

Y*p=4L*p=0.5*4p=2p

Thus, Y=2 is the equilibrium quantity.

h)

w/p=4

w=4p

p=0.25w

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