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

question 69

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?

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Definitions:

Interneurons

Neurons that transmit signals between sensory and motor neurons, playing critical roles in reflexes and neuronal circuits.

Myelin Sheets

Protective coverings around nerve fibers in the brain, spinal cord, and peripheral nerves, made of myelin, which help in the rapid transmission of electrical signals.

Sensory Information

Data that is perceived by the senses (sight, hearing, smell, taste, touch) and transmitted to the brain for processing.

Axon Hillock

The cone-shaped area of a neuron where the axon originates and where action potentials are initiated.