Assume That the Production Function for an Economy Is Given $r$

Essay

Assume that the production function for an economy is given by Y = AK^aH^bL ^{1- a - b}, where H is the stock of inventories. Then the marginal product of inventories (MPH) is given by MPH = bAK^aL^{1 - a - b}H^{b - 1}. If the stock of inventories does not depreciate, the price of inventories is the same as the price of output, and taxes are ignored, then the real "cost of capital" for inventories is just the interest rate r. a. Derive an expression for the "desired equilibrium stock of inventories" (H*) as a function of $r$ and output $Y$ by equating the cost of capital to $M P H$ . (Hint: First substitute the production function into the expression for $M P H$ to get $M P H = b Y / H$ .) If $r = 0.1 , b = 0.05$ , and $Y =$ 5,000 , what is the desired stock of inventories?
b. If $r$ rose to $0.12$ , how would the desired stock of inventories change?