# macroeconomics chapter three

## Terms

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Factors of production
The factors of production are the inputs used to produce goods
and services. The two most important factors of production are
capital and labor. In this module, we will take these factors as
given (hence the overbar depicting that these values are fixed).
K (capital) = K
L (labor) = L
Production function
The available production technology determines how much output is produced from given amounts of capital (K) and labor (L). The production function represents the transformation of inputs into outputs. A key assumption is that the production function has constant returns to scale, meaning that if we increase inputs by z, output will also increase by z.
We write the production function as: Y = F ( K , L )
Constant returns to scale
A key assumption is that the production function has constant returns to scale, meaning that if we increase inputs by z, output will also increase by z.
If the production function has constant returns to scale, then
doubling the amount of equipment and the number of workers doubles
Factor prices
The distribution of national income is determined by factor prices.
Factor prices are the amounts paid to the factors of productionâ€”the
wages workers earn and the rent the owners of capital collect.
Because we have assumed a fixed amount of
capital and labor, the factor supply curve
is a vertical line.
Competition
The goal of the firm is to maximize profit. Profit is revenue minus
cost. Revenue equals P Ã— Y. Costs include both labor and capital
costs. Labor costs equal W Ã— L, the wage multiplied by the amount
of labor L. Capital costs equal R Ã— K, the rental price of capital R times
the amount of capital K.
Profit = Revenue - Labor Costs - Capital Costs
= PY - WL - RK
Then, to see how profit depends on the factors of production, we use
production function Y = F (K, L) to substitute for Y to obtain:
Profit = P Ã— F (K, L) - WL - RK
This equation shows that profit depends on P, W, R, L, and K. The
competitive firm takes the product price and factor prices as given
and chooses the amounts of labor and capital that maximize profit.
Marginal product of labor (MPL)
The marginal product of labor (MPL) is the extra amount of output the
firm gets from one extra unit of labor, holding the amount of
capital fixed and is expressed using the production function:
MPL = F(K, L + 1) - F(K, L).
Diminishing marginal product
Most production functions have the property of
diminishing marginal product: holding the amount of capital
fixed, the marginal product of labor decreases as the amount of labor
increases. Like labor, capital is subject to diminishing marginal product.
The increase in profit from renting an additional machine is the extra
revenue from selling the output of that machine minus the machineâ€™s
rental price: D Profit = D Revenue - D Cost = (P Ã— MPK) â€“ R.
Real wage
The firmâ€™s demand for labor is determined by P Ã— MPL = W,
or another way to express this is MPL = W/P, where W/P is the
real wageâ€“ the payment to labor measured in units of output rather
than in dollars. To maximize profit, the firm hires up to the point
where the extra revenue equals the real wage.
Marginal product of capital (MPK)
firm decides how much capital to rent in the same way it decides
how much labor to hire. The marginal product of capital, or MPK,
is the amount of extra output the firm gets from an extra unit of
capital, holding the amount of labor constant:
MPK = F (K + 1, L) â€“ F (K, L).
Thus, the MPK is the difference between the amount of output produced
with K+1 units of capital and that produced with K units of capital
Real rental price of capital
To maximize profit, the firm continues to rent more capital until the MPK
falls to equal the real rental price, MPK = R/P.
The real rental price of capital is the rental price measured in units of
goods rather than in dollars. The firm demands each factor of production
until that factorâ€™s marginal product falls to equal its real factor price.
Economic profit vs. accounting profit
The income that remains after firms have paid the factors of
production is the economic profit of the firmsâ€™ owners.
Real economic profit is: Economic Profit = Y - (MPL Ã— L) - (MPK Ã— K)
or to rearrange: Y = (MPL Ã— L) - (MPK Ã— K) + Economic Profit.
Total income is divided among the returns to labor, the returns to capital,
and economic profit. If the production function has the property
of constant returns to scale, then economic profit is zero. This conclusion
follows from Eulerâ€™s theorem, which states that if the production function
has constant returns to scale, then
F(K,L) = (MPK Ã— K) - (MPL Ã— L)
If each factor of production is paid its marginal product, then the sum
of these factor payments equals total output. In other words, constant
returns to scale, profit maximization, and competition together imply that
economic profit is zero.
Cobbâ€“Douglas production function
The Cobbâ€“Douglas production function has constant returns to
scale (remember Mankiwâ€™s Bakery). That is, if capital and
labor are increased by the same proportion, then output
increases by the same proportion as well.
Next, consider the marginal products for the Cobbâ€“Douglas
production function. The MPL :
MPL = (1- Î±)Y/L
MPK= Î± A/ K
The MPL is proportional to output per worker, and the MPK is
proportional to output per unit of capital. Y/L is called average
labor productivity, and Y/K is called average capital
productivity. If the production function is Cobbâ€“Douglas, then
the marginal productivity of a factor is proportional to its average
productivity.
An increase in the amount of capital raises the MPL and
reduces the MPK. Similarly, an increase in the parameter

MPL = (1- Î±) A KÎ± Lâ€“Î± or, MPL = (1- Î±) Y / L

and the marginal product of capital is:

MPL = Î± A KÎ±-1L1â€“Î± or, MPK = Î± Y/K
We can now confirm that if the factors (K and L) earn their
marginal products, then the parameter Î± indeed tells us how much
income goes to labor and capital. The total amount paid to labor
is MPL Ã— L = (1- Î±). Therefore (1- Î±) is laborâ€™s share of output Y.
Similarly, the total amount paid to capital, MPK Ã— K is Î±Y and Î± is
capitalâ€™s share of output. The ratio of labor income to capital
income is a constant (1- Î±)/ Î±, just as Douglas observed. The
factor shares depend only on the parameter Î±, not on the amounts
of capital or labor or on the state of technology as measured by the
parameter A.
Disposable income
(Y - T)
Consumption function
C = C(Y- T)
Marginal propensity to consume
The marginal propensity to consume (MPC) is the amount by which consumption changes when disposable income (Y - T)
increases by one dollar. To understand the MPC, consider a shopping scenario. A person who loves to shop probably has a large MPC, letâ€™s say (.99). This means that for every extra dollar he or she earns after tax deductions, he or she spends \$.99 of it.
The MPC measures the sensitivity of the change in one variable (C) with respect to a change in the other variable (Y - T).
Nominal interest rate
The quantity of investment depends on the real interest rate, which measures the cost of the funds used to finance investment. When studying the role of interest rates in the economy, economists distinguish between the nominal interest rate and the real interest rate, which is especially relevant when the overall level of prices is changing. The nominal interest rate is the interest rate as usually reported; it is the rate of interest that investors pay to borrow money.
Real interest rate
The real interest rate is the nominal interest rate corrected for the effects of inflation.
The investment function relates the quantity of investment I to the real
interest rate r. Investment depends on the real interest rate because the
interest rate is the cost of borrowing. The investment function slopes
downward; when the interest rate rises, fewer investment projects are
profitable.
NATIONAL DEMAND:
The demand for the economyâ€™s output comes from consumption,
investment, and government purchases. Consumption depends on
disposable income; investment depends on the real interest rate;
government purchases and taxes are the exogenous variables set by
fiscal policy makers
Now, letâ€™s combine these equations describing supply and demand
for output Y. Substituting all of our equations into the national
income accounts identity, we obtain:
Y = C(Y - T) + I(r) + G
and then, setting supply equal to demand, we obtain an equilibrium
condition:
Y = C(Y - T) + I(r) + G
This equation states that the supply of output equals its demand,
which is the sum of consumption, investment,
and government purchases.
National saving
First, rewrite the national income accounts identity as Y - C - G = I.
The term Y - C - G is the output that remains after the demands of
consumers and the government have been satisfied; it is called national
saving or simply, saving (S). In this form, the national income accounts
identity shows that saving equals investment.
To understand this better, letâ€™s split national saving into two parts-- one
examining the saving of the private sector and the other representing
the saving of the government.
(Y - T - C) + (T - G) = I
The term (Y - T - C) is disposable income minus consumption, which is
private saving. The term (T - G) is government revenue minus government spending, which is public saving. National saving is the sum of private and public saving.

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