ISI problem

[nilanjan]

Can anyone tell me how to solve this?

Consider a closed economy in which household’s labor supply,L , to firms is determined by the amount which maximizes their utility U=C^α(1-L)^β

where α,β>0 and C denotes household real consumption expenditure which is taken to equal its wage income (the total labour time is normalized to unity).
• (a) Find the first order condition for utility maximization and also the household’s labour supply(L) for a given real wage rate(ω). Does L depend on ω? Explain your answer.
• (b) Assume that this economy is a Keynesian economy in which (real) investment expenditure (I) is autonomous and output (Y) is determined by aggregate demand, i.e. Y = C+ I. The aggregate production function is given by
Y=AL^θ
where A is a positive parameter, and θЄ [0,1]. Find the equilibrium values of Y for given value of I? What role does θ play here? Provide an intuitive explanation for this

[Dr. Strange]

U=(C^α)*((1-L)^ β )
PUT C=wL
DIFFERENTIATE W.R.T L
find optimum L
L(SUPPLY)=α/α+β....(1)


Now, in Production function =A*L^θ
For firm, profit= A*L^θ  -WL
Maximising gives,L(DEMAND)=(θ*A/W)^(1/1-θ)....(2)
Whichever L is small gives us the equilibrium L of economy and from that we can find out Y

As θ increases,L(demand) increases . For θ=1,Y is given by eqn 1(labour supply is limiting factor).For θ->0,Y is given by eqn 2.(labour demand is limiting factor)