midterm

Chapter 4: A Model of Production

Hikaru Saijo

University of California, Santa Cruz

4.2 A Model of Production

• Vast oversimplifications of the real world in a model can still allow it to provide important insights.

• Consider the following model

• Single, closed economy

• One consumption good

Setting Up the Model • Inputs

• Labor (L) • Capital (K )

• Production function • Shows how much output (Y ) can be produced given any

number of inputs

• Production function:

Y = F (K, L) = ĀK 1 3L

2 3

• Output growth corresponds to changes in Y .

• There are three ways that Y can change:

• Capital stock (K ) changes.

• Labor force (L) changes.

• Ability to produce goods with given resources (K , L) changes.

• Technological advances occur (changes in A). • TFP is assumed to be exogenous in the Solow model.

• The Cobb-Douglas production function is the particular production function that takes the form of

Y = KaL1−a

Assumed to be a = 1/3. Explained later.

• A production function exhibits constant returns to scale if doubling each input exactly doubles output.

• Standard replication argument

• A firm can build an identical factory, hire identical workers, double production stocks, and can exactly double production.

If the sum of exponents in the inputs. . .

• sum to more than 1 −→ returns to scale

• sum to 1 −→ returns to scale

• sum to less than 1 −→ returns to scale

Returns to Scale Example

Are these production functions IRS, CRS, or DRS?

1 Y = K 1 3 L

1 3

2 Y = L1.1

3 Y = K1.5L0.9

Allocating Resources

max K,L

Π = F (K, L) − rK −wL

• The rental rate and wage rate are taken as given under perfect competition.

• For simplicity, the price of the output is normalized to one.

• The marginal product of labor (MPL) • The additional output that is produced when one unit of

labor is added, holding all other inputs constant.

• The marginal product of capital (MPK) • The additional output that is produced when one unit of

capital is added, holding all other inputs constant.

MPL = ∂Y

∂L =

MPK = ∂Y

∂K =

If the production function has constant returns to scale in capital and labor, it will exhibit decreasing returns to scale in capital alone.

max K,L

Π = F (K, L) − rK −wL

• The solution is to use the following hiring rules:

• Hire capital until the MPK = r

• Hire labor until MPL = w

Solving the Model: General Equilibrium

• The model has five endogenous variables:

• Output (Y ) • the amount of capital (K ) • the amount of labor (L) • the wage (w) • the rental price of capital (r)

• The model has five equations:

• The production function • The rule for hiring capital • The rule for hiring labor • Supply equals the demand for capital • Supply equals the demand for labor

• The parameters in the model:

• The productivity parameter • The exogenous supplies of capital and labor

• Unknowns/endogenous variables:

1 Production function:

2 Rule for hiring capital:

3 Rule for hiring labor:

4 Demand = supply for capital:

5 Demand = supply for labor:

• Parameters/exogenous variables:

• A solution to the model

• A new set of equations that express the five unknowns in terms of the parameters and exogenous variables

• Called an equilibrium

• General equilibrium

• Solution to the model when more than a single market clears

Solving the Model

1 Find a market clearing (equilibrium) capital and labor.

2 Plug in the equilibrium capital and labor to the production function and get output.

3 Given equilibrium capital and labor, solve for prices.

Solving the Model

Solving the Model

Interpreting the Solution

• If an economy is endowed with more machines or people, it will produce more.

• The equilibrium wage is proportional to output per worker. • Output per worker = Y /L

• The equilibrium rental rate is proportional to output per capital. • Output per capital = Y /K

In the United States, empirical evidence shows:

• Two-thirds of production is paid to labor. • One-third of production is paid to capital. • The factor shares of the payments are equal to the

exponents on the inputs in the Cobb-Douglas function.

Y ∗ = F (K̄, L̄) = ĀK̄1/3L̄2/3

w∗L∗

Y ∗ =

r∗K∗

Y ∗ =

All income is paid to capital or labor.

• Results in zero profit in the economy. • This verifies the assumption of perfect competition. • Also verifies that production equals spending equals

income.

w∗L∗ + r∗K∗ =

4.3 Analyzing the Production Model

• Per capita = per person • Per worker = per member of the labor force

• In this model, the two are equal. • We can perform a change of variables to define output

per capita (y) and capital per person (k).

• Output per person equals the productivity parameter times capital per person raised to the one-third power.

y∗ ≡ Y ∗

L∗ =

• What makes a country rich or poor?

• Output per person is higher if the productivity parameter is higher or if the amount of capital per person is higher.

Draw graph for the per capita versions of the production function. Y = ĀK0.7L0.3 and Y = ĀK0.5L0.5

Draw graph for the per capita versions of the production function. Y = K − 3ĀL and Y = K + 3ĀL

Comparing Models with Data

• The model is a simplification of reality, so we must verify whether it models the data correctly.

• The best models:

• Are insightful about how the world works

• Predict accurately

The Empirical Fit of the Production Model

Development accounting:

• The use of a model to explain differences in incomes across countries

y∗ = Āk̄1/3

• Setting the productivity parameter = 1

y∗ = k̄1/3

• Diminishing returns to capital implies that:

• Countries with low K will have a high MPK

• Countries with a lot of K will have a low MPK , and cannot raise GDP per capita by much through more capital accumulation

• If the productivity parameter is 1, the model overpredicts GDP per capita.

Productivity Differences: Improving the Fit of the

Model

• The productivity parameter measures how efficiently countries are using their factor inputs.

• Often called total factor productivity (TFP)

• If TFP is no longer equal to 1, we can obtain a better fit of the model.

• However, data on TFP is not collected.

• It can be calculated because we have data on output and capital per person.

• TFP is referred to as the “residual.”

• A lower level of TFP

• Implies that workers produce less output for any given level of capital per person

4.4 Understanding TFP Differences

• Output differences between the richest and poorest countries?

• Differences in capital per person explain about one-third of the difference.

• TFP explains the remaining two-thirds.

• Thus, rich countries are rich because

• they have more capital per person.

• more importantly, they use labor and capital more efficiently.

Why are some countries more efficient at using capital and labor?

• Human capital

• Technology

• Institutions

• Misallocation