Aggregate Production Function

Production

Mark Huggett

Georgetown University

January 23, 2018

Aggregate Production Function

Aggregate Production Function

1. Many growth theories assume an aggregateproduction function.

2. Thus, there is a technological relationshipbetween GDP, denoted Yt, and aggregatequantities of inputs of capital Kt and labor Lt

all measured at time t.

Yt = AtF (Kt, Lt)

3. The variable At denotes technology at time t.

Aggregate Production Function

Aggregate Production Function1. A simple theoretical foundation: N firms

(i = 1, 2, ..., N) all have the same (constant returns)production technology Y i

t = AtF (Kit , L

it)

2. All firms minimize the cost of production and face thesame input prices for capital and labor.

3. Claim: the entire economy behaves “as if” there is asingle firm with technology Yt = AtF (Kt, Lt) facing thesame input prices where aggregate inputs and outputare

Kt = K1t +K2

t + · · ·KNt and Lt = L1

t + L2t + · · ·LNt

Yt = Y 1t + Y 2

t + · · ·Y Nt

Aggregate Production Function

Properties of Production FunctionsConstant Returns to Scale

I A production function has constant returns to scale ifwhenever all factor inputs are scaled by a commonfactor λ > 0, then output is also scaled by the samefactor λ.

I This can be restated mathematically as the property

Y = F (K,L)⇒ λY = F (λK, λL) holds for all λ > 0

I Example: when λ = 2 a doubling of all inputs leads toa doubling of output or when λ = 1/2 a halving of allinputs leads to a halving of output.

Aggregate Production Function

Properties of Production FunctionsConstant Returns to Scale

A geometric description of constant returns to scale issometimes useful.

I An isoquant of a production function Y = F (K,L)displays all the different combinations of inputs (K,L)that produce the same level of output.

I If F (K,L) is constant returns then all possibleisoquants can be determined from the “unit” isoquant(the isoquant associated with output Y = 1).

I Specifically, the isoquant for any level of output Y > 0is determined by expanding or contracting each pointon the unit isoquant (radially) from the origin by thefactor multiple Y > 0.

Aggregate Production Function

Properties of Production FunctionsConstant Returns to Scale

An economic implication of constant returns

I If two countries have the same constant returnstechnology, then there is no economic advantage to size.

I Under constant returns a countries labor productivity(i.e. the ratio Y/L) is determined only by itscapital-labor ratio K/L.

Y = F (K,L)⇒ Y/L = F (K/L,L/L) = F (K/L, 1)

Aggregate Production Function

Properties of Production FunctionsDiminishing Marginal Products

the marginal product of a factor of production is the extraoutput caused by increasing the factor by one (small) unit,other things equal. Mathematically, marginal products arepartial derivatives. We will assume that all marginalproducts are diminishing (i.e. decreasing) as the relevantfactor input increases, other things equal.

Notation:

I FK(K,L) denotes the marginal product of capital atthe input levels (K,L)

I FL(K,L) denotes the marginal product of labor at theinput levels (K,L)

Aggregate Production Function

Properties of Production Functions

If F (K,L) has constant returns and well-definedderivatives, then the following holds for all (K,L) values:

(∗) Y = F (K,L) = FL(K,L)L+ FK(K,L)K

Interpretation:

I This is Euler’s Theorem for homogeneous functions(i.e. functions displaying constant returns).

I A competitive firm makes zero economic profit. Thefirm produces Y units of output. It pays laborFL(K,L)L and pays capital FK(K,L)K. Nothing isleft after paying all factors with constant returns.

Aggregate Production Function

Implications of Profit Maximization

Profit = F (K,L)−WL−RK

If a firm with technology F (K,L) maximzes profit takinginput prices (W,R) as given, then

I W = FL(K,L) and R = FK(K,L)

I Profit = F (K,L)−WL−RK =F (K,L)− FL(K,L)L− FK(K,L)K = 0 when F hasconstant returns.

Aggregate Production Function

Cobb-Douglas Production Function

This production function displays (i) diminishing marginalproducts, (ii) constant returns and (iii) constant factorshares.

Y = F (K,L) = AKβL1−β

I diminishing marginal productsFL(K,L) = (1− β)AKβL−β andFK(K,L) = βAKβ−1L1−β

I constant returns: note that the exponents add up to 1!

I labor′s share = FL(K,L)LF (K,L)

= (1−β)AKβL−βLF (K,L)

= (1− β)

I capital′s share = FK(K,L)KF (K,L)

= βAKβ−1L1−βKF (K,L)

= β

Aggregate Production Function

Why did the US college-wage premium increase from1950-2005 despite increases in ratio of total college labor(LC) to high school labor (LH)?

Figure 1 College Graduate and High School Graduate Wage Premiums: 1915 to 2005

1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010

0.2

0.3

0.4

0.5

0.6

College graduate wage premiumHigh school graduate wage premium

Sources and Notes: College Graduate Wage Premium: The plotted series is based on the log college/high school wage differential series in Appendix Table A8.1. We use the 1915 Iowa estimate and the 1940 to 1980 census estimates for the United States. We extend the series to 1990, 2000, and 2005 by adding the changes in the log (college/high school) wage differentials for 1980 to 1990 for the CPS, 1990 to 2000 from the census, and 2000 to 2005 from the CPS to maintain consistency in the coding of education across pairs of samples used for changes in the college wage premium. High School Graduate Wage Premium: The plotted series is based on the log (high school/eighth grade) wage differential series in Appendix Table A8.1. We use the 1940 to 1980 Census estimates for the United States. To maintain data consistency, we then extend this series backwards to 1915 using the1915 to 1940 change for Iowa and forward to 2005 using the 1980 to 1990 change from the CPS, the 1990 to 2000 change from the February 1990 CPS to the 2000 CPS, and the 2000 to 2005 change from the CPS.

Aggregate Production Function

Table 1 Changes in the College Wage Premium and the Supply and Demand for College Educated

Workers: 1915 to 2005 (100 × Annual Log Changes)

Relative Wage

Relative Supply

Relative Demand

(σSU = 1.4)

Relative Demand

(σSU = 1.64)

Relative Demand

(σSU = 1.84) 1915-40 -0.56 3.19 2.41 2.27 2.16 1940-50 -1.86 2.35 -0.25 -0.69 -1.06 1950-60 0.83 2.91 4.08 4.28 4.45 1960-70 0.69 2.55 3.52 3.69 3.83 1970-80 -0.74 4.99 3.95 3.77 3.62 1980-90 1.51 2.53 4.65 5.01 5.32 1990-2000 0.58 2.03 2.84 2.98 3.09 1990-2005 0.50 1.65 2.34 2.46 2.56 1940-60 -0.51 2.63 1.92 1.79 1.69 1960-80 -0.02 3.77 3.74 3.73 3.73 1980-2005 0.90 2.00 3.27 3.48 3.66 1915-2005 -0.02 2.87 2.83 2.83 2.82 Sources: The underlying data are presented in Appendix Table A8.1 and are derived from the 1915 Iowa State Census, 1940 to 2000 Census IPUMS, and 1980 to 2005 CPS MORG samples. Notes: The “relative wage” is the log (college/high school) wage differential, which is the college wage premium. The underlying college wage premium series is plotted in Figure 1. The relative supply and demand measures are for college “equivalents” (college graduates plus half of those with some college) relative to high school “equivalents” (those with 12 or fewer years of schooling and half of those with some college). The log relative supply measure is given by the log relative wage bill share of college equivalents minus the log relative wage series:

log log logSU

w Sw U

ww

S

U

S

U

⎛⎝⎜

⎞⎠⎟ =

⎛⎝⎜

⎞⎠⎟ −

⎛⎝⎜

⎞⎠⎟

where S is efficiency units of employed skilled labor (college equivalents), U is efficiency units of employed unskilled labor (high school equivalents), and and are the (composition-adjusted) wages of skilled and unskilled labor. The log relative wage bill is based on the series for the wage bill share of college equivalents in Appendix Table A8.1. The relative demand measure depends on

wS wU

log( )DSU σ SU and follows from equation (3) in the text:

log( ) log logDSU

w Sw USU SU

S

U=

⎛⎝⎜

⎞⎠⎟ +

⎛⎝⎜

⎞⎠⎟σ

To maximize data consistency across samples in the measurement of education changes from 1980 to 1990 use the CPS, changes from 1990 to 2000 use the census, and changes from 2000 to 2005 use the CPS. The changes for 1915 to 1940 are for Iowa. See Autor, Katz, and Krueger (1998) for details on the methodology for measuring relative skill supply and demand changes.

The Race between Education and Technology 34

Wage column = (scaled) change in logWCt

WHt. Supply column =

(scaled) change in logLCtLHt

over the period. The premium logWCt

WHt

is

U-shaped over time whereas supply logLCtLHt

increases over time.

Aggregate Production Function

An Unsuccessful Explanation

I Yt = AtF (Kt, LHt , L

Ct ) = AtK

βt (LHt )αH (LCt )αC

I Firm minimizes cost of production taking (WHt ,W

Ct )

as given impliesWCt

WHt

=AtFC(Kt,L

Ht ,L

Ct )

AtFH(Kt,LHt LCt )

=αCAtK

βt (L

Ht )αH ,(LCt )

αC−1

αHAtKβt (L

Ht )αH−1(LCt )

αC

logWCt

WHt

= log αCαH

+ logLHtLCt

= log αCαH− log

LCtLHt

I Prediction: logWCt

WHt

decreases when logLCtLHt

increases.

I US Data: logLCtLHt

increases over time while logWCt

WHt

is

U-shaped.

Aggregate Production Function

An More Successful Explanation: Goldin and Katz (2007)The Race between Education and Technology

I Yt = AtF (LHt , LCt , λt) = At[λt(L

Ct )ρ + (1− λt)(LHt )ρ]1/ρ

I Firm minimizes cost of production taking (WHt ,W

Ct )

as given implies

(∗) WCt

WHt

=AtFC(L

Ht ,L

Ct ,λt)

AtFH(LHt ,LCt ,λt)

=λt(LCt )

ρ−1

(1−λt)(LHt )ρ−1

(∗∗) logWCt

WHt

= log λt(1−λt) + (ρ− 1) log

LCtLHt

I Goldin and Katz estimate (**) yields (ρ− 1) = −0.61

I Interpretation: US college wage premium is a race

between the depressing effects of increased supplyLCtLHt

and the increasing effects of technological change λtwhich complements skilled labor.

Aggregate Production Function

Technological Change

Economists believe that technological change is the keyreason why people in developed economies live vastly betterlives than people 100 years ago.

I Embodied technological change: technologicalimprovements are embodied in new capital (e.g. a newcar, new cell phone, new computer, new light bulb, newairplane, new tractor, new seeds, ...).

I Disembodied technological change: you can takeadvantage of a technological change (largely) by usingexisting inputs (e.g. Adam Smith’s pin factory,assembly line, double entry book keeping, just-in-timeinventory, faster computer algorithms, new medicalprocedures, ...)

Aggregate Production Function

Technological ChangeEmbodied technological change is highly relevant butdifficult to model. Need to keep track of many vintages ofcapital.Many influential growth theories (e.g Solow growth model)analyze disembodied technical change

I Neutral (disembodied) technical change:Yt = AtF (Kt, Lt) or Yt = F (KtAt, LtAt)

I Labor augmenting (disembodied) technical change:Yt = F (Kt, LtAt)

I Capital augmenting (disembodied) technical change:Yt = F (KtAt, Lt)

I Goldin and Katz employ neutral and skilled-laboraugmenting technical change