Answer to Question #220364 in Microeconomics for raju

Question

The technology for a firm is given as

Y=x^a  where 0 < a < 1

 a) Obtain the demand for x and supply of y for this firm.

b) Derive the profit function of this firm.

c) From determine its degree of homogeneity.

d) Obtain the Hessian using above and show that it is convex in p and w.

Answer 1

a.

$Y=x^a\\ a= \frac{\ln Y}{\ln X}\\ a=\ln (Y-X)\\ \frac{\partial \:}{\partial \:X}\left(\ln \left(Y-X\right)\right)=-\frac{1}{Y-X}\\ \frac{\partial \:}{\partial \:Y}\left(\ln \left(Y-X\right)\right)=\frac{1}{Y-X}$

b.

$TR(y) = py\\ (y) = TR(y) TC(y) = py TC(y$

c.

To find the polynomial degree, add up the exponents of each term and select the highest sum. The degree is, therefore a

d

If ηis a Borel subset of Ω and ρ>0, then the set P ρ(u,η) = {x+ρζ: x∈ η,ζ∈ ∂u(x)}