ECON102: Principles of Macroeconomics (2021.A.01)

The Production Function

• GDP (Y) is produced with capital (K) and labor (N):

• Sometimes, I will modify the production function as follows:

• Realistic Example is a Cobb Douglas function for F(.):

Measurement

• Y is GDP (it is measured in dollars). As noted above, we want to measure Y in "real" terms.<<you should know what this means from lecture 2>>.
• For our Cobb Douglas production function, N is measured in number of workers and K in dollars:

• N can also be measured using
  total hours worked = number of workers × hours per worker
• Wage differentials can help to measure "effective labor supply", taking into account "skill" differentials.

N.B.: sometimes we will use N to denote total population (e.g. income per capitaY/N)

Graphical Representation 1

• Hold A and N constant (at levels A* and N*)
• Graph Y as a function of K

1. As K increases Y increases (the curve is upward-sloping)

2. As K increases the marginal increase in production decreases (the curve becomes flatter as K increases)

Graphical Representation 2

• Hold A and K constant (at levels A* and K*)
• Graph Y as a function of N

1. As N increases Y increases (the curve is upward-sloping)

2. As N increases the marginal increase in production decreases (the curve becomes flatter as N increases)

Aggregate Production Function: Fact 1

1. Constant Returns to Scale

Aggregate Production Function: Fact 2

2. Diminishing Returns to N and K

Cobb-Douglas:

• $\text{MPN = (1- α) A (K/N)}^{a}$

• $\text{MPK = α A (N/K)}^{1-a}$

Aggregate Production Function: Fact 3

3. Complementarities between A, K and N

Cobb-Douglas:

• $\text{MPN = (1- α) A (K/N)}^{a}$

• $\text{MPK = α A (N/K)}^{1-a}$

Aggregate Production Function: Fact 4

4. Elasticities and Income Shares

• Elasticity is the percentage increase in Y (dependent variable) resulting from a 

1% increase in X (independent variable), everything else constant

Cobb-Douglas:

• $\eta_{N}=(1-\alpha) \text { and } \eta_{K}=\alpha$

• Share of labor income out of total GDP is about 70%
  Share of capital income out of total GDP is about 30%

Two Notions of Productivity

• $\text{Labor Productivity = Y/N = A (K/N)}^{3}$

• $\text{Total Factor Productivity (TFP) = Y/F(K,N) = A}$ 

• Some of the above tend to make TFP procyclical (capital utilization) 

(Definition of Procyclical: Variable increases when Y is high, decreases when Y is low)

Simple examples (in words)

1. Technology: "It costs FedEx $2.40 to track a package for a customer who calls by phone, but only $0.04 for one who visits its website", says Rob Carter, the firm's technology boss.
2. Technology: "Airline kiosks reduce costs of boarding to less than a third". 
3. Management: Southwest’s oil hedging. Estimated oil price paid by SW: $31. United: $56. 
4. Infrastructure: Imagine what it takes to buy intermediate inputs from a different region with roads like in Nigeria.

Measure of Labor productivity

Productivity Levels

United States = 100