Consider an economy with one consumer with a Cobb-Douglas utility function for consumption, x, and leisure, R: u(x, R) = X0.5R0.5. X is produced by one firm according to
X=, where L is amount of labour input. The consumer is endowed with 1 unit of labour/leisure. Let p and w denote the price of x and wage rate, respectively, and normalize p to be1
a. Find the real wage that clears the labour market.
b. Calculate the optimal consumptions of x and R and the profit of the firm