The Aggregate Production Function

1
The Aggregate Production Function

• Requires Assumptions about
• What input aggregates to use
• How inputs affect output

2
The inputs

• GDP f(L, K, MFP)
• Contribution of L (Labor)
• Contribution of K (capital)
• Multi-Factor Productivity (MFP)
• Know-how not specific to K or L contributions
• Important but unobservable input into production

3
How inputs affect output

• Diminishing returns to K or L
• Marginal productivity of K falls as K increases
  (holding L constant)
• Marginal productivity of L falls as L increases
  (holding K constant)

4
How inputs affect output

• MFP exogenous and has constant returns
• Constant returns means infinite MFP means
  infinite output, so there is infinite demand
• MFP exogenous means supply is not affected by
  economic activity or by prices (demand)

5
How inputs affect output

• K affects GDP with a delay (lag)
• Implication Firms adjust L if want to change
  output in the near-term

6
Deriving Labor Demand from the production
function for GDP

• Assume perfect competition
• Wage Pmpl
• w/p mpl
• w/p real wage
• Assume diminishing returns to L
• As L up, mpl falls
• Therefore firms willing to pay lower w/p as add L

7
Things that shift Ld

• Change in technology
• Change in amount of capital in use
• Change in quality of capital in use
• Aka capital deepening

w/p
Ld
L
8
Labor Supply

• Assume people can either enjoy leisure, do
  market work, or do work outside the marketplace
  that earns no pay (home production)
• Opportunity cost of working is leisure/home-produc
  tion foregone
• As labor time increases, leisure/home-production
  time falls
• Assume diminishing returns to leisure and
  home-production in utility
• As leisure/home-production time falls, it gets
  more valuable
• Need to receive a higher w/p if are going to give
  up more leisure or non-market work

9
Things that shift Ls
w/p

• Demographic changes
• Changes in unemployment, welfare, or social
  security benefits
• Cultural changes (entry/exit of women/men from
  workforce)

Ls
L
10
Labor market equilibrium
w/p

• At (w/p), everyone who wants a job has a job
  (LsL)
• At (w/p), the marginal worker is getting paid
  exactly what s/he is worth (purchasing power
  equal to output produced at L)

Ls
(w/p)
Ld
L
L
11
Labor market dynamics moving to equilibrium

• We assume perfect competition always holds true
• We are always on Ld
• Example technology improves

w/p
Ls
(w/p)
Ld1
L
L
12
Labor market dynamics moving to equilibrium

• We assume perfect competition always holds true
• We are always on Ld
• Example population decreases

w/p
Ls
(w/p)
Ld1
L
L
13
Problem with the model no equilibrium
unemployment

• Obvious fact in the data that there is
  unemployment in equilibrium
• Perfectly competitive model implies no
  unemployment in equilibrium

14
Why labor markets dont satisfy perfect
competition

• Search frictions imply no pressure on wages
  even though workers and firms are refusing to
  match with each other
• Not all jobs or employees are alike
• It takes time to find a good match
• Its worth it to take time to find a good match!

15
Why labor markets dont satisfy perfect
competition

• Hiring and firing costs put a wedge between the
  wage paid and marginal productivity of labor
• It may be cheaper to hold onto workers even if
  w/p gt mpl
• It may too expensive to hire workers even if
  their marginal productivity would equal the
  equilibrium real wage

16
Models of labor market frictions can explain

• unemployment in equilibrium
• why identical workers may earn different wages at
  different firms even in equilibrium
• why employment rises (falls) about 2 quarters
  after GDP rises (falls)

17
The market for capital

• Unlike work, capital goods are durable
• New capital goods are called investments
• The Fixed I in the AE approach to GDP
• Not all capital is used each period (capacity
  utilization varies over time), and some old
  capital is useless (depreciates)
• Were ignoring these two things in Econ 2!
• Well study capital markets more carefully in
  Chapter 7 but introduce them now

18
Investment Supply Savings

• Investment Supply is just the amount of money
  supplied to buy investment goods
• Firms either use their own Savings or they borrow
  someone elses Savings in order to pay for new K
• Investment Supply Savings

19
The Investment Supply Curve

• The Opportunity Cost of Saving is Consumption
  forgone
• The real return from Saving is the real interest
  rate r
• Assume diminishing returns to C
• As S up, C gets more valuable
• Require higher r to give up more S

20
Things that shift Savings

• Change in income
• Change in expected future income
• Demographic changes
• Change in spending

r
IsS
I
21
Why r is the cost of new capital

• Case 1 Firm buys capital good with own S
• Opportunity cost is r could have earned elsewhere
  
• Case 2 Firm borrows someone elses S
• Actual real cost is the interest rate must pay on
  the loan (r)
• Case 3 Firm rents the capital good
• Under perfect competition, rental rate cost to
  the capital owner r

22
Deriving Investment Demand from the production
function for GDP

• Assume perfect competition
• r mpk
• r is the real interest rate
• Assume diminishing returns to K (I)
• As I up, mpk falls
• Therefore firms willing to pay lower r as add I

23
Things that shift Id

• Change in technology
• Change in the quality of capital
• Long-run changes in employment
• Changes in the quality of labor

r
Id
I
24
Capital market equilibrium
r

• At r, everyone who wants to make a loan has made
  one (SI)
• At r, the marginal investment is producing
  exactly what it costs to get the loan
• Always stay on Id
• Perfect competition assumed to always hold

SIs
r
Id
I
I