Chapter 3: National Income (Long-Run Neoclassical Theory)

Home , Production function, Profit (economics)

The economy uses inputs (or factors) of labor (L) and capital (K) to produce the output or GDP (Y). The wage and rent are denoted by W and R, respectively.

We want to answer two questions

Q1: How to use K and L to produce Y? So where does national income come from? Q2: How are W and R determined? So how is the national income distributed among owners of labor and capital?

Mathematically, the production process is captured by the production function of Y = F(K, L)

We assume (1) ̅ ̅, so the quantity of both inputs are fixed in long term. (2) Both K and L are fully utilized. So there is no unemployment. Assumptions (1) and (2) imply that______(3) Let

( ) ( )

denote the marginal product of labor,

( ) ( )

the marginal product of capital. As K or L rises, Y rises. Mathematically ______

(4) As K or L rises, Y rises but at decreasing rate. This assumption is called decreasing marginal product. Mathematically, ______. We can draw the graph (Figure 3-3) that represents a typical production function as

(5) The production function has the property of constant returns to scale. That is,

( ) (1)

So if both inputs are doubled ( =2), the output is doubled as well. A production function with constant returns to scale has the property of

(2)

This result is called Euler’s theorem. Proof (Optional)

(6) The firm is competitive in both output market and input market. In other words, the firm takes the output price (P) and the input prices (W, R) as given.

Two steps to determine the real wage, W/P

Step 1: The revenue of a typical firm is______The input cost is______The profit is______

Step 2: The firm maximizes profit by letting the first (partial) derivative of profit with respect to labor equal zero. So the first- order (necessary) condition is

Or equivalently,

(3)

Equation (3) states that when the labor market is in equilibrium, the real wage, W/P, adjusts so that it is equal to the marginal product of labor. Equation (3) corresponds to Figure 3-2:

In a similar fashion we can show that when capital market is in equilibrium,

(4)

Equation (2) (3) (4) jointly imply that

(5)

So that the economic profit =0. See the equation on page 51. Remember result (5) is derived from the assumptions we made (such as constant returns to scale). If capital is owned, instead of being rented, then the accounting profit, which is the sum of economic profit and RK, is nonzero. See the equation on page 55.

Critical Thinking: Case Study on page 55 Consider the effect of black death (a massive deduction in labor). What happens to Y? What happens to MPL? What happens to MPK? (treat land as capital) What happens to W/P? What happens to R/P? Can you show your answer using math and graph?

A commonly used production function is called Cobb-Douglas production function which is given by

( ) (0< ) (6) where parameter A measures the state of technology. Shows that

What happens to MPK if K rises? What if L rises?

Capital income = MPK*K = ______

Labor income = MPL*L = ______

So determines the share of income that goes to capital, (1- ) the share for labor.

( )

So the Cobb-Douglas production function has the property of (1) Constant return to scale

(2) Constant factor share, i.e.,

(3) ( ) . So the marginal

productivity of a factor is proportional to its average productivity.

Figure 3-5 shows that most recent US data are consistent with the Cobb-Douglas production function with .

Critical thinking: Case Study on page 58

Neoclassical theory implies that real wage = ______Cobb-Douglas production implies that MPL = ______Jointly, we expect that real wage rises at the same rate as ______

(Hint: FYI on page 26 indicates that if Z = XY, then percentage change of Z = percentage change of X + percentage change of Y, so percentage of product equals the sum of percentage changes)

Next we want to answer the questions that how the output from production is used, or in other words, what determines the demand for goods and services.

We assume a closed economy (so NX = 0). The national income accounts identity states that

Y = C + I + G

We just show that ̅ ( ̅ ̅)

Let T denotes the tax. We assume consumption (C) depends positively on disposable income (Y-T): C = ( ) The consumption function is depicted in Figure 3-6. The slope of the consumption line is marginal propensity to consume, which takes a value between 0 and 1. So as disposable income rises by one dollar, consumption rises by an amount less than one dollar.

The investment (I) depends negatively on real interest rate (r), see figure 3-7.

The government expenditure (G) and Tax (T) are both exogenous. So ̅ ̅

On the goods and service market,

The supply = ______

The demand = ______

The goods market is in equilibrium when ______. Mathematically we have

̅ ( ̅ ̅) ( ) ̅ (7)

This equilibrium is achieved via the adjustment of real interest rate, r.

Alternatively we can consider the market for loanable funds:

The demand for loanable funds = ______

The supply for loanable funds = private saving + public saving

Private saving = ______

Public saving = ______

The market for loanable funds is in equilibrium when

Figure 3-8 depicts the equilibrium in loanable funds market.

Critical Thinking: What happens to real interest rate if G rises? See Figure 3-9