Given production function ,Y=X^a in maximum profit .
a) calculate explicitly of the profit function and verify that its homogeneous and convex (P,W)
The profit function is given by P = Y - WX, where Y is the output, W is the wage rate, and X is the amount of labor used.
The profit function is homogeneous of degree 1 in (P,W) since P = Y - WX can be written as P = Y/W - X, which is a linear function of (P,W).
The profit function is convex in (P,W) since the Hessian matrix of the profit function is positive semi-definite. The Hessian matrix is given by:
H = [2aX^(2a-2) -W, -X; -X, 0]
Since the determinant of the Hessian matrix is 0, the Hessian matrix is positive semi-definite and the profit function is convex in (P,W).