A short run equilibriam production function is given as Q= X½ were Q is output and X is input.
a)Let P represent output price and W represent the input price. Derive the prifit function
b)If a legitimate profit function is concave and positively linearlity homogeneous. is the profit functuon above legitimate? Show your working.
a) The profit function is given by Profit = PQ - WX
b) The profit function is not legitimate because it is not concave and not positively linearly homogeneous. To show this, we can take the derivative of the profit function with respect to X:
dProfit/dX = PQ' - W
Since Q' = 0.5X^-0.5, the derivative of the profit function is not linear and not homogeneous. Therefore, the profit function is not legitimate.