ECON 2000 Chapter Notes - Chapter 17: Real Interest Rate, Fixed Investment, Marginal Product
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There are three types of investment spending:
Business fixed investment includes the machinery, equipment, and structures that
businesses buy to use in production.
Residential investment includes the new housing that people buy to live in and that
landlords buy to rent out.
Inventory investment includes those goods that business put aside in storage, including
materials and supplies, work in process, and finished goods.
Business fixed investment
The largest piece of investment spending. Includes everything from fax machines to
factories, computers to company cars. The standard model of business fixed investment is
called the neoclassical model of investment. This model examines the benefits and
costs to firms of owning capital goods. To develop the model, imagine that there are two
kinds of firms in the economy. Production firms produce goods and services using
capital that they rent. Rental firms make all the investments in the economy; they buy
capital and rent it out to the production firms. This is not usually the case in the real
world, however it simplifies our analysis.
The firm rents capital at a rental rate R and sells its output at a price P; the real cost of a
unit of capital to the production firm is R/P. The real benefit of a unit of capital is the
marginal product of capital MPK, the slope of the production function. The firm rents
capital until the marginal product of capital MPK falls to equal the real rental price of
capital R/P in order to maximize profit. The downward sloping demand for capital
(MPK) curve reaches equilibrium with the vertical capital supply curve, and the real
rental price of capital (Y-axis) adjust to equilibrate supply and demand.
The marginal product of capital for the Cobb-Douglas production function is MPK =
Since MPK = R/P in equilibrium, we can write R/P = αA(L/K)1-α. This expression shows
the following: the lower the stock of capital, the higher the real rental price of capital; the
greater the amount of labour employer, the higher the real rental price of capital; the
better the technology, the higher the real rental price of capital.
Firms that rent out capital have three costs for each period of time that it rents out capital:
1. When a rental firm borrows to buy a unit of capital it intends to rent out, it must pay
interest on the loan. If PK is the purchase price of a unit of capital and i is the nominal
interest rate, then iPK is the interest cost. iPK is the interest cost whether the rental firm
buys capital with loans or with its own cash. If it uses its own cash, it loses out on the
interest it could have earned by depositing this cash in the bank. In either case, the
interest cost equals iPK.
2. While the rental firm is renting out the capital, the price of capital can change. The
cost of this loss or gain is –ΔPK (the minus sign is here because we are measuring costs,
3. While the capital is rented out, it suffers depreciation. If δ is the rate of depreciation,
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then the dollar cost of depreciation is δPK.
The total cost of renting out a unit of capital for one period is therefore:
Cost of Capital = iPK – ΔPK + δPK.
= PK(i - ΔPK/ PK + δ)
The cost of capital depends on the price of capital, the interest rate, the rate at which
capital prices are changing, and the depreciation rate.
In this case, ΔPK/ PK equals the overall rate of inflation π. Because i – π equals the real
interest rate r, we can write the cost of capital as:
Cost of Capital = PK(r + δ)
We want to express the cost relative to other goods in the economy. The real cost of
capital, the cost of buying and renting out a unit of capital measured in units of the
economy’s output, is:
Real Cost of Capital = (PK/P)(r + δ)
This equation states that the real cost of capital depends on the relative price of a capital
good PK/P, the real interest rate r, and the depreciation rate δ.
Determinants of investment
Consider a firm’s decision about whether to increase or decrease its capital stock. For
each unit of capital, the firm earns real revenue R/P and bears the real cost (PK/P)(r + δ).
The real profit per unit of capital is:
Profit Rate = Revenue – Cost = R/P – (PK/P)(r + δ) = MPK – (PK/P)(r + δ)
The rental firm makes a profit if the marginal product of capital is greater than the cost of
capital. It incurs a loss if the marginal product is less than the cost of capital.
The change in the capital stock, called net investment, depends on the difference
between the marginal product of capital and the cost of capital. If the marginal product
of capital exceeds the cost of capital, firms find it profitable to add to their capital stock.
If the marginal product of capital falls short of the cost of capital, they let their capital
ΔK = In [MPK – (PK/P)(r + δ)], where In is the function showing how much net
investment responds to the incentive to invest.
We can now derive the investment function. Total spending on business fixed investment
is the sum of net investment and the replacement of depreciated capital. The investment
I = In [MPK – (PK/P)(r + δ)] + δK
Business fixed investment depends on the marginal product of capital, the cost of capital,
and the amount of depreciation. The investment curve slopes downward because a
decrease in the real interest rate lowers the cost of capital, increasing the incentive to
invest. Any event that raises what business managers expect the marginal product of
capital to be increases the profitability of investment and causes the investment schedule
to shift outward. When the capital stock reaches a steady state, we can write: MPK =
(PK/P)(r + δ). In the long run, the marginal product of capital equals the real cost of
Taxes and investment
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