Production, Distribution and...

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Production, Distribution and Production, Distribution and AllocationAllocation

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ECN 101 ECN 101 -- MACROECONOMICSMACROECONOMICS slide 1

Roadmap to this LectureRoadmap to this Lecture1. The economy in 3 markets and 3 agents

2. Properties of a production function:1. Output is always positive2. More inputs more output3. Constant returns to scale4. Diminishing marginal returns

3. Determining factor demand and factor prices

4. Income distribution

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Overview:Overview:

We are interested in describing the production side of the economy

How are the prices of the factors of production determined?

How is total income distributed?

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The Economy in the LongThe Economy in the Long--RunRunBasic Assumption: in the long-run, prices are fully flexible and markets clear.

This assumption is a basic assumption of the Classical Paradigm

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A Macroeconomy in 3 MarketsA Macroeconomy in 3 MarketsThe Goods and Services Market: where goods and services are traded

The Factors of Production Market: where labor is hired and capital is rented

Financial Market: where household savings are channeled into investment

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A Macroeconomy in 3 AgentsA Macroeconomy in 3 AgentsHouseholds: consume goods and services (demand) and provide labor (supply) and savings (supply).

Firms: produced goods and services (supply), hire labor and capital (demand), and invest (demand)

Government: provides goods and services (supply) and also consumes them (demand). It also borrows (demand)

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A Flow MatrixA Flow Matrix

Investment

[Demand]

Deficit Financing

[Demand]

Savings

[Supply]

Financial

Employees

[Demand]

Public Employees

[Demand]

Work

[Supply]

Labor

Investment

[Demand]

Govt. Expenditures

[Demand]

Public Good

[Supply]

Consumption

[Demand]

Goods and Services

FirmsGovernmentHouseholds

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Factors of productionFactors of production

K = capital, tools, machines, and structures used in production

L = labor, the physical and mental efforts of workers

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The production functionThe production functionThe Production function: Y = F (K,L)

shows how much output (Y ) the economy can produce fromK units of capital and L units of labor.

reflects the economy’s level of technology.

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Returns to scale: a reviewReturns to scale: a reviewInitially Y1 = F (K1,L1 )

Scale all inputs by the same factor z:

K2 = zK1 and L2 = zL1

(If z = 1.25, then all inputs are increased by 25%)

What happens to output, Y2 = F (K2 ,L2 ) ?

If constant returns to scale, Y2 = zY1

If increasing returns to scale, Y2 > zY1

If decreasing returns to scale, Y2 < zY1

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The CobbThe Cobb--Douglas Production FunctionDouglas Production Function

F(K,L)=AKα Lβ

A, α, β >0

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Properties of Production FunctionsProperties of Production Functions

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1. Output is always Positive1. Output is always PositiveMathematically:

F(K,L) ≥ 0 for any K, L ≥ 0

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Marginal product of labor (Marginal product of labor (MPLMPL))

def:The extra output the firm can produce using an additional unit of labor (holding other inputs fixed):

MPL = F (K,L +1) – F (K,L)

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Exercise: Exercise: compute & graph MPLcompute & graph MPLa. Determine MPL at each

value of L

b. Graph the production function

c. Graph the MPL curve with MPL on the vertical axis and L on the horizontal axis

L Y MPL0 0 n.a.1 10 ?2 19 ?3 27 84 34 ?5 40 ?6 45 ?7 49 ?8 52 ?9 54 ?

10 55 ?

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answers:answers:

Production function

0

10

20

30

40

50

60

0 1 2 3 4 5 6 7 8 9 10

Labor (L)

Out

put (

Y)

Marginal Product of Labor

0

2

4

6

8

10

12

0 1 2 3 4 5 6 7 8 9 10

Labor (L)

MPL

(uni

ts o

f out

put)

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2. More inputs, more 2. More inputs, more ouputouputone extra unit of input (either L or K), everything else equal, delivers at worst nothing, at best, more output.

Mathematically:– FK=∂ F/∂ K ≥ 0, marginal productivity of

capital: How much more output if I increase K by 1 unit.

– FL=∂ F/∂ L ≥ 0, marginal productivity of labor: How much more output if I increase labor by 1 unit.

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Partial DerivativesPartial DerivativesExample: the Cobb-Douglas production function

F(K,L)=AKα Lβ

– FK=A( α Kα -1) Lβ = α Y/K≥ 0

– FL=AKα (β Lβ -1) = β Y/L ≥ 0

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3. Constant Returns to Scale3. Constant Returns to ScaleCobb-Douglas production function:

Y = AKα Lβ

Y* = A(zK)α (zL)β = Yz(α + β)

Hence:– If α + β > 1 then Y* > F(zK, zL), IRS– If α + β = 1 then Y* = F(zK, zL), CRS– If α + β < 1 then Y* < F(zK, zL), DRS

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4. Diminishing Marginal Returns4. Diminishing Marginal ReturnsThe marginal productivity of an extra unit of an input declines the more of that input is used in production

Mathematically:

– FKK ≤ 0– FLL ≤ 0

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Youtput

The MPL and the production functionThe MPL and the production function

Llabor

F K L( , )

1

MPL

1

MPL

1MPL

As more labor is added, MPL ↓

Slope of the production function equals MPL

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Diminishing marginal returnsDiminishing marginal returns

As a factor input is increased, its marginal product falls (other things equal).

Intuition:↑L while holding K fixed

⇒ fewer machines per worker

⇒ lower productivity

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5. Determinants of Output Growth5. Determinants of Output GrowthY growth =

(MPK x K share) x K growth +(MPL x L share) x L growth

Mathematically: Y = F(K,L)

dY = FK dK + FL dL hence

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CobbCobb--Douglas ExampleDouglas ExampleFK = αY/K; and FL = β Y/L . Hence:

or simply

α is the capital share; β is the labor share

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Factor Prices and Quantities of Factor Prices and Quantities of EquilibriumEquilibrium

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Assumptions of the modelAssumptions of the model

1. Technology is fixed.

2. The economy’s supplies of capital and labor are fixed at

and = =K K L L

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Determining GDPDetermining GDPOutput is determined by the fixed factor supplies and the fixed state of technology:

,= ( )Y F K L

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The distribution of national incomeThe distribution of national income

determined by factor prices, the prices per unit that firms pay for the factors of production.

The wage is the price of L ,the rental rate is the price of K.

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NotationNotation

W = nominal wage

R = nominal rental rate

P = price of output

W /P = real wage (measured in units of output)

R /P = real rental rate

W = nominal wage

R = nominal rental rate

P = price of output

W /P = real wage (measured in units of output)

R /P = real rental rate

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What Real Wage Really MeansWhat Real Wage Really MeansW is measured in $/unit of work

P is measured in $/unit of good

Hence

W/P = ($/unit of work)/($/unit of good)

= units of good/units of labor

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How factor prices are determinedHow factor prices are determined

Factor prices are determined by supply and demand in factor markets.

Recall: Supply of each factor is fixed.

What about demand?

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Survey: Grads finding hot jobs market Survey: Grads finding hot jobs market Employers set to hire nearly 15% more college

grads this spring, survey says; starting salaries up for many business, engineering majors.

[…]accounting degree graduates are receiving an average starting salary of $46,188, up 5.4 percent from a year ago. Right behind are economics/finance graduates, who are getting average offers of $45,058, up 5.3 percent, and business administration/management majors, who are seeing average offers 3.9 percent higher than a year ago at $40,976.

Source:

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Demand for laborDemand for laborAssume markets are competitive: each firm takes W, R, and P as given

Basic idea:A firm hires one extra unit of labor if the cost does not exceed the benefit.

cost = real wage (W/P)benefit = marginal product of labor

(MPL)

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Check your understanding:Check your understanding:

Which of these production functions have diminishing marginal returns to labor?

a) 2 15F K L K L= +( , )

b) F K L K L=( , )

c) 2 15F K L K L= +( , )

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Exercise (part 2)Exercise (part 2)

Suppose W/P = 6.

d. If L = 3, should firm hire more or less labor? Why?

e. If L = 7, should firm hire more or less labor? Why?

L Y MPL0 0 n.a.1 10 102 19 93 27 84 34 75 40 66 45 57 49 48 52 39 54 2

10 55 1

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MPLMPL and the demand for laborand the demand for labor

Each firm hires labor up to the point where

MPL = W/P

Each firm hires labor up to the point where

MPL = W/P

Units of output

Units of labor, L

MPL, Labor demand

Real wage

Quantity of labor demanded

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The equilibrium real wageThe equilibrium real wage

The real wage adjusts to equate labor demand with supply.The real wage adjusts to equate labor demand with supply.

Units of output

Units of labor, L

MPL, Labor demand

equilibrium equilibrium real wagereal wage

Labor supply

L

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Determining the rental rateDetermining the rental rate

We have just seen that MPL = W/P

The same logic shows that MPK = R/P :

diminishing returns to capital: MPK↓ as K ↑

The MPK curve is the firm’s demand curve for renting capital.

Firms maximize profits by choosing Ksuch that MPK = R/P .

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The equilibrium real rental rateThe equilibrium real rental rate

The real rental rate adjusts to equate demand for capital with supply.

The real rental rate adjusts to equate demand for capital with supply.

Units of output

Units of capital, K

MPK, demand for capital

equilibrium equilibrium R/PR/P

Supply of capital

K

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The Neoclassical Theory of Distribution

The Neoclassical Theory The Neoclassical Theory of Distributionof Distribution

states that each factor input is paid its marginal product

accepted by most economists

states that each factor input is states that each factor input is paid its marginal productpaid its marginal product

accepted by most economistsaccepted by most economists

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How income is distributed:How income is distributed:

total labor income =

If production function has constant returns to scale, then

total capital income =

W LP

MPL L= ×

R KP

MPK K= ×

Y MPL L MPK K= × + ×

laborincome

capitalincome

nationalincome

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EulerEuler’’s Theorem and Income s Theorem and Income DistributionDistribution

Euler’s theorem states that if F(X,Y) is a homogenous function of degree one such that F(zX, zY) = zF(X,Y). Then:

F(X,Y) = FX X + FY Y

An implication of Euler’s theorem is that for any production function with CRS, then

F(K,L) = FK K + FL L

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A CobbA Cobb--Douglas ExampleDouglas ExampleWhen α + β = 1, we know that the Cobb-Douglas production function has CRS. Hence, a direct application of Euler’s theorem or direct application of the economic arguments presented above:

MPK x K + MPL x L = α Y+(1-α )Y = Y

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Recap: Assumptions MadeRecap: Assumptions MadeCRS technology: even when some industries seem to violate this assumption, it holds true for entire economies failry well

Perfect Competition: is a good approximation to long-run behavior even if it is violated often

Profit Maximization: hardly needs justification…

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Outline of modelOutline of modelA closed economy, market-clearing modelSupply side

factor markets (supply, demand, price)determination of output/income

Demand sidedeterminants of C, I, and G

Equilibriumgoods marketloanable funds market

DONEDONE

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