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 give me a solution and I need to see work too.

Explanation / Answer

(a)

As given in the question Y=4L
and we know that Profit = TR-TC

Total revenue is the product of price and quantity sold

TR = p•Y = p•4•L = 4pL

Total Costs are wage rate multiplied by Quantity of Labour

TC = w•L

Now Profit

Profit = 4pL - wL = L(4p-w)
Profit=0 because 4p=w

(b)

For maximization, the condition is MFC=MRP

MFC=w,

MRP=MPL•p

MPL=Y/L=4
MRP=4p
w=4p
w/p=4
Thus labor-demand is straight horizontal line at w/p=4

c)
Budget equation is given by income

which depends on wage level and quantity of labor supplied

=wL
0L1
C=y=I/p=wL/p

d)
Assuming that "C" denotes consumption:
U = C(1-L) = wL/p - wL²/p
w = 4p
U = 4L-4L²
0L1
U MAX
U/L=4-8L=0
4=8L
L=0.5
Thus individual labor-supply is vertical line at L=0.5

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