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Unformatted text preview: The Production & Distribution of
National Income
The Production Function,
Productivity,
Marginal Products of Capital & Labor,
Household Demand,
Equilibrium,
Growth Accounting.
Topic 4 (Text: 3.1, 3.2, 4.2, 4.3) 1 Key Concepts
• A macro description of production
Combine Capital, labor
Combine Capital labor
A lot of reality is subsumed in this
lot Productivity tells you how well K and L work together
Productivity • How decisions to rent capital & labor are made
(MPK & MPL & how they turn into “demand”)
MPL
demand”
• The demand for investment & the supply of
savings
• Equilibrium analysis & “shifts”
shifts”
• Growth accounting
9 The Production Function
The economy’s productive capacity is the
economy’
most fundamental determinant of economic
wellbeing
well• The production function is a mathematical
description of the economy’s productive
economy’
capacity 10 1
1 The Production Function (cnt.)
cnt.)
• A production function relates the amount of
output that can be produced to the amount
of inputs and the level of productivity
General form: Y = A F(K, L)
• The inputs are called factors of production.
factors
e.g. Capital, Labor, possibly Land, Human
Capital, & Organization
• A commonly used specification of the
production function is the CobbDouglas
Cobb(CD) function
(C11 The Production Function (cnt.)
cnt.)
• Example of a CD production function:
CY = A K0.3L0.7
K ↑ by 1%
L ↑ by 1% Y ↑ by 0.3% (ceteris paribus)
Y ↑ by 0.7% (ceteris paribus) • If doubling all inputs doubles output, the
all
production function is said to exhibit constant
constant
returns to scale (CRS)
If output less than doubles, decreasing returns
If
decreasing
If it more than doubles, increasing returns
If
increasing • The CD function exhibits constant returns to
Cconstant
scale but decreasing returns to each factor
decreasing
12 The Production Function (cnt.)
cnt.)
• K, L, and Y can be in any convenient units,
A will have the right units to connect the three • L can be raw labor hours, especially if A
includes labor quality, or
• L can be effective labor hours, adjusted for
labor quality / human capital
If K, L, and A are fixed, Y is determined
If
is
Y = A K0.3L0.7 13 2
2 Productivity
• “A” in the production function is called the
productivity or technology of the economy
• Higher productivity means that the same K
and L can produce more Y
• It is common to take K and L as raw
quantities and include quality in A
quality
• Changes in A are called productivity or
productivity
supply shocks
supply
14 Productivity (cnt.)
cnt.)
• We are concerned with how A varies
over time for a specific country, and
over
across countries at a given point in time
across • Increases in A over the long run are related
to economic growth
• Short run variations in A (productivity
shocks) are related to business cycles 15 Productivity (cnt.)
cnt.)
U.S. TF Productivity Growth
49  59 = 1.76%
60  69 = 2.03%
70  79 = 0.37%
80  89 = 0.99%
90  99 = 1.45%
00  05 = 0.92% 6.0% 4.0% 2.0% 0.0%
19
2.0% 49 19 53 19 57 19 61 19 65 19 69 19 73 19 77 19 81 19 85 19 89 19 93 19 97 20 01 20 05 4.0%
17 3
3 Productivity (concl.)
concl.)
• “What determines productivity growth?” is
critical to economic development and
prosperity
We’ll return to this issue
We’ 18 Marginal Products of
Capital & Labor
• So far we learned what makes production
happen
• The next step is to understand how factor
inputs are paid
i.e., what do K and L receive, or
i.e.,
what do K and L cost?
what
And WHY?
And 19 Marginal Products of
Capital & Labor
• We are led to the concept of marginal
product
MPs will give us the demand for factor inputs
MP
demand
MPL↓ with L (diminishing marginal product), and ↑
MPL
with K and A
MPK↓ with K (diminishing marginal product), and
MPK
↑ with L and A 20 4
4 Marginal Product of Labor
• The marginal product of labor (MPL) is the
extra amount of output which can be
extra
produced with an extra unit of labor
extra
MPL = ΔY/ΔL = slope of the production
MPL
function in the direction of L
A, K are fixed
In the CD production function
In
Cα ⎛K⎞
MPL = (1 − α ) A⎜ ⎟
⎝L⎠ 21 Marginal Product of Capital
• The marginal product of capital (MPK) is
the extra amount of output which can be
extra
produced with an extra unit of capital
extra
MPK = ΔY/ΔK = slope of the production
MPK
function in the direction of K
A, L are fixed
In the CD production function
In
C1−α ⎛L⎞
MPK = αA⎜ ⎟
⎝K⎠ 22 Factor Employment Decision
• Assume the typical firm that produces the
economy’s output is competitive
economy’
competitive
• It takes the prices of its output (P) and price
of its 2 inputs as given
W is the wage rate
R is the rental rate of capital • Individual firms’ actions do not affect prices
firms’
23 5
5 Optimal Labor Employed
The firm maximizes profits (revenue  costs):
Π = PY  WL – RK = P F(K,L)  WL – RK
PY
• How can the firm maximize profits?
• The rule is very simple:
• Hire labor as long as:
MPL ≥ W/P (the real rental rate of labor)
MPL
labor)
i.e. as your profits increase after you pay the
real wage
24 Labor Demand
Y MPL K Fixed
2.5 2.0 K is fixed 1.5 Given Wages 1.0 0.5 0.0
0 5 10 15 20 25 L
25 Maximizing Behavior
Profits
MPL > W/P
Profits ⇑ as L ⇑ MPL < W/P
Profits ⇓ as L ⇑ Max Profits K is fixed
A is fixed
L
ProfitMaximizing L
26 6
6 Optimal Capital Employed
The firm maximizes profits (revenue  costs):
Π = PY  WL – RK = P F(K,L)  WL – RK
PY
• How can the firm maximize profits?
• The rule is very simple:
• Hire Capital as long as:
MPK ≥ R/P (the real rental rate of capital)
MPK
capital)
i.e. as your profits increase after you pay the
real rental rate
27 Optimization (cnt.)
cnt.)
• How much of each factor does the firm
hire?
Π = PY  WL – RK = P F(K,L)  WL – RK
PY
To find optimal amount of L:
To ΔΠ = P ΔY  W ΔL = 0
ΔY/ΔL = W/P = MPL
MPL
To find optimal amount of K:
To ΔΠ = P ΔY  R ΔK = 0
ΔY/ΔK = R/P = MPK
MPK
29 Labor Demand (cnt.)
cnt.)
• W/P is called the real wage since it
real
measures the amount of output the wage can
purchase, rather than $s
• Since the firm hires up to the point where
MPL equals the real wage, the MPL
MPL
the
schedule is the firm’s labor demand curve
firm’
or demand schedule
or demand 30 7
7 Labor Demand (cnt.)
cnt.)
• Two critical assumptions allow us to call it
the firm’s demand for labor:
firm’
The firm is a pricetaker in the product and
The
pricefactor markets
The firm maximizes profits
The 31 Capital Demand
• Follow the same logic for capital to get the
firm’s demand for capital:
firm’
Firm rents capital to the point where
Firm
MPK = R/P, the real rental price of capital
MPK
real
This is not the cost of buying capital!
This not MPK schedule is the firm’s capital demand
MPK
firm’ • Real rental price of unit of capital =
Interest rate + Depreciation rate 32 Capital Demand
• The macro model we are using has one good
and therefore one price
• If there are more goods (as in real life), then
the real rental price also includes the price
real
appreciation of the capital
• Real rental price of unit of capital (R/P) =
(R/P
Interest rate + Depreciation rate – Capital Gains
Think of housing
Think 33 8
8 Capital Demand
Y MPK L Fixed 4.0
3.5
3.0 L is fixed 2.5
2.0 CoC 1.5
1.0
0.5
0.0
0 5 10 15 20 25 K
34 Marginal Products Again
• Economic Π/P = Y – (W/P)* L  (R/P) * K,
All quantities are “real” here
All
real” In a competitive equilibrium
In • Economic Π/P = Y  (MPL * L)  (MPK * K), or
Y = (MPL * L) + (MPK * K) + Π/P
Payments to labor + Payments to capital + Economic profit • For CRS production functions, Euler’s theorem
Euler’
implies:
• Y = F(K,L) = (MPK * K) + (MPL * L), which in
F(
turn implies,
Economic profit = 0
Economic
35 Come Again! Zero Profits?
• If so, how to explain “profits” that we see
profits”
all around?
• With CRS, payments to factor inputs
exhaust output
This condition is just saying that all the output
This
all
is paid to the factors that produce it
Firm’s “profits” are returns to equity
Firm’ profits”
Entrepreneurial profit?
Entrepreneurial 36 9
9 Example #1
• ACME has the following production function:
No. of No. of Widgets
MPL
$MPL
Workers Produced
2
15
3
21
4
26
5
30
• What is the MPL for each level of employment?
MPL
• If ACME gets $5/widget, how many workers will it
hire at $27/worker?
37 Example #2
• Below is GDP, K, L (N), for the US economy
GDP
Year
1960
1970
1980
1990 Y
2,263
3,398
4,615
6,136 K
2,390
3,525
5,032
6,650 N
65.8
78.7
99.3
118.8 • Assume the Prod. Function is Y = AK0.3L0.7
AK • By what % did the U.S. TFP grow in each
decade?
39 Markets
• To figure out the equilibrium we need to
study
The demand
The
The supply
The
And find out at what point they are equal • Prices are determined by the requirement
that “markets clear”
i.e., Demand = Supply
i.e., 48 10
10 Markets (cnt.)
cnt.)
• In this economy there are markets for goods,
goods
labor, and capital
labor
capital
We’ll show that all the key economic
We’
quantities are determined! 49 Markets (cnt.)
cnt.)
EQUILIBRIUM:
• Labor market equilibrium determines wage
rate and quantity of labor employed
Labor Demanded = Labor Supplied
Labor • Capital market equilibrium determines rental
rate and quantity of capital and investment
Investment = Savings
Investment • Goods market equilibrium ensures
Goods Produced = Goods Demanded
Goods
If 2 markets are in equilibrium the 3rd is as well!
If
50 Markets (cnt.)
cnt.)
Labor Markets:
• Labor demand is the MPL
MPL
• Any upwardsloping supply for labor will
upwardresult in an equilibrium wage 51 11
11 Markets (cnt.)
cnt.)
Capital Markets:
• How are the equilibrium capital stock and
investment determined?
The MPK function determines the amount of
The MPK
desired capital, given the real rental rate R/P
desired
To achieve their desired capital stock, firms
To
have to invest 52 Demand For Investment
• Firm’s desired capital level is given by the real
Firm’
rental price (R/P) and MPK
MPK
• To get to the desired K, K, the firm invests
the
Desired I =
K + Depreciation – K0
Depreciation
(Desired)
(Current)
where
Depreciation = δ K0
K0 is Current K
δ is the depreciation rate
rate
53 Demand For Investment (cnt.)
cnt.)
• I is the gross addition to K in each period
• The investment vs. interest rate curve
(MPK) is downward sloping
Recall
Recall R/P = r + δ A higher real interest rate increases the user cost
higher
of capital and thus reduces desired investment
Desired I = (Desired) K + δ K0 – K0 > 0
Since
K
decreases as R/P increases,
Since
I does the same thing
54 12
12 Figure 4.6 Gross and net investment, 192919292005 55 Supply of Savings
• Think about the right hand components of
the incomeexpenditure identity:
incomeY = C + I + G + NX
NX
Assume closed economy initially, so NX = 0
Assume
NX
Also assume that govt. purchases are
Also
exogenously given at G, instead of modeling
the political process • By definition,
Households consume and firms invest
Households
56 Supply of Savings (cnt.)
cnt.)
• Household’s decision to consume is based
Household’
on its income and the interest rate
Savings = Income  Consumption
The consumption and savings decisions are two
The
sides of the same coin • For now assume that savings are either fixed
or ↑ with the interest rate (holding Y constant)
Theory of the consumptionsavings decision
Theory
consumptionwill come later
Supply of Savings is upward sloping
Supply
57 13
13 Equilibrium
• The capital market clearing condition is:
Sd = Id
desired savings = desired investment • The real interest rate adjusts until desired
savings equal desired investment
The equilibrium real interest rate is where the
The
Savings and Investment curves intersect
Savings
Investment 58 Equilibrium (cnt.)
cnt.)
• Important insight on “equilibrium”:
equilibrium”
From an individual firm’s point of view,
From
firm’
prices are given, the firm determines its demand
(partial equilibrium) From the economywide point of view,
From
economythe confrontation of the sum of the demands with
the sum of the supplies determines prices
(general equilibrium) 59 Equilibrium (cnt.)
cnt.)
• Demand for output in the economy
depends on the:
1. Consumptionsaving decision of the
Consumptionhouseholds,
2. Investment decision of the firms, and
3. Purchases of the government • These demands jointly determine the
equilibrium
Wages, rental rate, output
Wages,
60 14
14 Equilibrium Analysis
• A typical definition of equilibrium is:
Households maximize utility
Households
lies behind consumptionsavings decision
lies
consumptiondetermines saving and labor supplied
determines Firms maximize profits
Firms
determines demand for labor and capital, and
determines
therefore investment Prices adjust such that all markets clear
Prices
supply = demand in all markets
supply
“prices” means wages & interest rates
prices”
61 How to Ask Questions: a Function “Shift”
Shift”
• All demand and supply functions are
multidimensional or multivariate
More than one argument
More
Univariate
y = f (x1)
Univariate
Multivariate y = f (x1, x2, x3, … xn)
Multivariate • A 2dimensional graph shows the relation of
2y to xi
• When xj changes, it is represented as a
“shift”
shift”
62 Shifts in the Equilibrium
• Some factors that shift the savings curve:
Changes in G (e.g. increases during wars)
Changes
A decrease in taxes
decrease • Some factors that shift the investment curve:
Technological innovation that increases MPK
Technological
MPK
Investment tax credit
Investment • In the next few classes, we will delve deeper
into how labor supply, consumptionsaving,
consumptionand investment decisions are made
63 15
15 Growth Accounting
• Very often productivity is discussed in terms of
growth rates rather than levels
• We first need to understand the mechanics of
productivity growth; that is, how it is calculated
We’ll see this again when we study economic growth
We’ • Consider the CD production function:
CY = A K0.3L0.7
• Relation between output growth rate and input &
productivity growth rates is:
ΔY/Y = ΔA/A + 0.3(ΔK/K) + 0.7(ΔL/L)
Gr(Y) means the same thing
65 Growth Accounting (cnt.)
cnt.)
ΔY/Y = ΔA/A + 0.3(ΔK/K) + 0.7(ΔL/L) or
gY = gA + 0.3gK + 0.7gL
• This is the growth accounting equation
growth
• Its purpose is to separate
The output growth due to the accumulation of
The
inputs, and
The output growth due to productivity increases
The
66 Growth Accounting (cnt.)
cnt.)
ΔA/A = ΔY/Y  0.3 (ΔK/K)  0.7 (ΔL/L)
• How is it used?
• ΔA/A is the Total Factor Productivity
(TFP) growth rate
• We have estimates of the growth rates of
Y, K, and L
We back out an estimate of TFP growth
We back 67 16
16 Growth Accounting (cnt.)
cnt.)
• Another concept is Labor Productivity:
AL = Y/L
• Therefore, ΔAL/AL = ΔY/Y  ΔL/L ≡ G(AL)
This is different from TFP, because it lumps
This
together increases that come from capital
capital
and those that come from TFP
TFP 68 Growth Accounting in United States
(percent per year)
19291948 19481973 19731982 19291982
Source of growth
labor growth 1.42 1.40 1.13 1.34 capital growth 0.11 0.77 0.69 0.56 total input growth 1.53 2.17 1.82 1.90 productivity growth 1.01 1.53 0.27 1.02 total output growth 2.54 3.70 1.55 2.92 70 Growth of Output and Productivity
Around 1973 (percent per year)
output growth productivity growth 19601973 19731990 19601973 19731990
United States 4.0 2.5 1.6 Japan 10.0 4.0 5.9 0.0
1.8 Europe 4.9 2.3 3.2 1.3 OECD 5.3 2.7 2.8 0.7 71 17
17 Example #4
• The observed growth rates in the
economy are:
Y = 8.2%, K = 10%, L = 1%
1. What is the growth rate of TFP?
2. What is the growth rate of AL? 72 Glossary of Terms
Y
A
K
L
P
W, W/P
R, R/P δ
Π
MPK
MPL
C, I, G, NX
NX Total output (connect to GDP)
GDP
Economywide productivity (technology)
EconomyPhysical capital
Labor
Output price (general price level)
Wage rate (nominal & real)
Capital rental rate (nominal & real)
Depreciation rate
Profits
Marginal product of capital
Marginal product of labor
As in topic 3 (superscript d stands for desired)
74 THE END 75 18
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This note was uploaded on 03/02/2010 for the course BUAD 350 taught by Professor Safarzadeh during the Spring '07 term at USC.
 Spring '07
 Safarzadeh
 Accounting, National Income

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