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ECONOMY IN THE LONG RUN

Chapter 3

National Income: Where it Comes from and Where it Goes

1 Chapter 3: National Income: Where it comes from and where it goes. ECON204 Fall 2012

QUESTIONS ABOUT THE SOURCES AND USES OF GDP

Here we develop a static classical model of the macroeconomy: prices are fully flexible and adjust to ensure the full use of all resources.

The model ‘explains’:

Determinants of the level of production/income —labour, capital, and (technology & land)o Aggregate supply

Who gets the income from production?o Distribution of income

Who buys the output of the economy? - allocation of output consumption, investment; o Aggregate demand

Equilibrium in the economy


1. Determinants of the total Production of Goods and Services

Factors of Production (FOP)

Capital (K): an aggregate measure of the stock of all machinery, buildings, equipment available for production.

Labour (L): an aggregate of all available labour. Hours worked.

Both variables are exogenous/fixed in this

model:K=K∧L=L


Full utilization of resources

Production Function

Shows how much output can be produced using the available technology and the inputs:

Y=F (K , L)

Example: If, Y=K12 . L

12

Then for K=400∧L=100 , Y=200

Constant returns to scale (CRS) is a property of some production functions


Constant Returns to Scale (CRS)

If all inputs are increased by a constant proportion then output increases by a constant proportion.

Mathematically: zY=F(zK , zL)

where z is the constant of proportionality.

Consider the previous example: Y=F (K ,L )=K12 . L

12

F ( zK , zL )=¿

Since K and L are assumed exogenous, the supply of output is determined as, Y=F (K , L)

This is the long run or natural rate of output.

2. Distribution of Income


Given output (Y) , how the income from production is distributed?

We assume competitive markets for the two factors: Labour and Capital.

Factor prices – wages and rent are determined by the equilibrium in factor markets

Figure: Factor Price Determination


The firm’s demand for factors

Firm’s goal is to maximize profit


Vertical Supply function indicates inputs are either

fixed or used fully (classical view)

Choose K and L given P

Output: Y=F (K ,L )

Profit={Revenue−Costs }

Costs: labour costs and capital costsπ=TR−TC

¿ PY−W . L−R . K

¿ PF (K ,L )−W .L−R .K

Where, R is the rental rate per unit of capital per period, and W is the wage rate per unit of labour per period

Determining firm’s demand for inputs

Define the marginal product (MP)

MP is the slope of the production function


Marginal product of labour: MPL= ∂Y

∂L=FL

Marginal product of capital:MPK=∂Y

∂ K=FK


Figure: Production Function


Diminishing Marginal Product

How does the firm decide whether to hire an additional worker?

The firm will hire an additional worker if the extra revenue generated by that worker exceeds the cost of hiring the worker

Value of Marginal Product of Labour is equal to the nominal wage: VMPL = W


To maximize profits, firms choose L and K so that marginal profit is equal to zero

∂π∂L

=0

P .MPL−W=0

MPL=WP

Marginal product of labour (MPL) = real wage∂ π∂K

=0

P .MPK−R=0

MPK=RP

Marginal product of capital (MPK) = real rent

Figure: Labour demand (MPL), labour supply and real wage


Who Gets What and How Much?

Total Labour income: WP ×L=MPL× L

Total Capital income: RP ×K=MPK×K

How about the firm? – economic profit

Income that remains after paying for factors of production (i.e., labour and capital)

Economic profit = Y−expenditures on FOP

¿Y−WP×L− R

P× K

¿Y−MPL×L−MPK× K


THE EQUILIBRIUM REAL WAGE

The real wage adjusts to equate

labor demand with supply.

equilibrium real wage

Economic Profit

Under CRS production function Economic Profit is

Zero

The same conclusion drawn from competitive market assumption

Economic profit=Y−MPL× L−MPK ×K=0

orF (K , L )=MPL×L+MPK ×K


If the production function has constant returns to scale, then

Labour’s share to the real Income: MPL×LY

Capital’s share to the real income:MPK× KY

For Canada, labour’s share to real income is 0.67 (i.e. 2/3) in the post war period and Capital’s share is 0.33 (i.e., 1/3). Empirically, these shares have been stable.


Y MPL L MPK K

laborincome

capitalincome

nationalincome

Example: what functional form of production produces constant factor shares if factors earn their marginal product? That is, if the following is true

MPK .K=αY

MPL .L=(1−α)Y

What does F (K ,L) look like?

è The Cobb-Douglas production functionY=A .L1−α . Kα

Note that this function satisfies constant returns to scale. So all output will be exhausted.


Calculating marginal products

For labour:

MPL= ∂Y∂L

=(1−α ) A L−αK α=(1−α )YL

For Capital:

MPK=∂Y∂ K

=αA L(1−α )K (α−1)=αYK

For Canadian economy, where, α=13, the production function

(Cobb-Douglus) is:Y=A K 1/3 L2/3

3. Demand for Goods and Services — Allocation of Output


Final production is allocated across four components, or sources of demand: Y = C + I + G + NX

C = consumer demand for g & s

I = demand for investment goods

G = government demand for g & s

For now, assume the economy is closed; no exports or imports. This simplifies the model (and gets included later).

We want to describe the determinants of consumption, investment and government spending


Consumption Expenditure (C)

Assume that aggregate consumption is a function of only of disposable income (YD = Y-T):

C=C (Y−T ) - consumption function

Shows that (Y – T ) Þ C

For example,C=α+β (Y−T )=α+ βY D

Where, Y D=disposable income, and β=MPC

The Marginal Propensity to consume (MPC):

MPC= ∂C∂Y D

=β (¿ our exmaple)


The MPC is assumed to be less than one; for every dollar increase in the disposable income, a fraction is consumed (MPC) and a fraction is saved

As such, Sprivate=Y D−C

Figure: Consumption function and MPC


How does the Saving function look like?

Investment (I)


Firms and households both purchase investment goods (buildings, machinery, housing).

I t=K t+1−K t

Note that we do not keep track of changes in the stock of capital at this stage; it is taken as given. Here investment’s role is only as an expenditure component.

The purpose of investment expenditure is to earn a return in the future. For investment to be profitable, the return must exceed the cost — where both return and cost are measured in real terms.

The investment function: I = I(r)


r is the real interest rate (not the nominal interest rate); which includes compensation for inflation.

The real interest rate (r) is

the cost of borrowing

the opportunity cost of using one’s own funds to finance investment spending.

For the aggregate economy, as the real interest rate falls more investment expenditure becomes profitable.

So, I Þ ¯r

Figure ??

Government Purchases (G)


Government purchases (of consumption and investment goods) are denoted by G; government tax revenue (in real terms) is denoted by T.

Both G and T are assumed exogenous; G=G∧T=T

Useful definitions:

If T > G, budget surplus = (T – G ) = Positive public saving.

If T < G, budget deficit = (G – T )= Negative public saving

If T = G , “balanced budget,” public saving = 0.

4. Equilibrium in the Economy


We have described the Aggregate Demand, AD, side of our model; the Aggregate Supply, AS, side of our model; and the distribution of income. Bringing these together determines the equilibrium of our economy.

Our complete model of the economy is:Y=C+ I+G Aggregate Demand

C=C (Y−T )Consumption function

I=I (r ) Investment function

G=G

T=T

Y=F (K ,L ) Aggregate supply

AD=ASEquilibrium

Substituting various behavioural functions, we getY=C (Y−T )+ I (r )+G


Y=C+ I (r )+G

The real interest rate (r) must adjust to ensure that this condition is met.

è This can be motivated most easily as equilibrium in the market for loanable funds.

The loanable funds market

A simple supply-demand model of the financial system

One asset: “loanable funds”


o demand for funds: Investment

o supply of funds: Saving

o “price” of funds: real interest rate

S is the supply of loanable funds — what the economy is willing to forgo in current consumption.

I(r) is the demand for loanable funds (based upon available investment projects).

If r is too low, then the demand for loanable funds exceeds the supply, the real interest rate gets bid upwards.

If r is too high, then there is excess supply of loanable funds and the real interest rate gets bid downwards.


Define national savings as:S=Y−C−GS= [ (Y−T )−C ]+[T−G ]

¿ SPRIVATE+SPUBLIC

From our equilibrium condition,Y=C+ I (r )+G

¿ , [ (Y−T )−C ]+[T−G ]=I (r )

S=I (r )

Figure: Equilibrium in the market for loanable funds


Note that this simple equilibrium for loanable funds critically assumes a closed economy.

Special Role of r


r adjusts to equilibrate the goods market and the loanable funds market simultaneously:

If Lonable funds market is in equilibrium, thenY−C−G=I

Y=C+ I+GGoodsmarket equilibrium

Thus,Eq 'm∈L .F market⇔Eq ' m∈Goodsmarket

EXERCISE: Calculate the change in saving


Suppose MPC = 0.8 and MPL = 20.

For each of the following, compute DS :

a. DG = 100

b. DT = 100

c. DY = 100

d. DL = 10


Loanable Funds Market Equilibrium

The Effects of Fiscal Policy

Fiscal policy refers to taxes (T) and government spending (G).

In our model, these two variables are exogenous variables; so it is legitimate to ask what happens to the endogenous variables when one or both of these changes.

Question we can ask — what are the effects of an increase in government spending, holding taxes constant?


Recall the definition of national savings and its components:S=Y−C−G

S= [ (Y−T )−C ]+[T−G ]

¿ SPRIVATE+SPUBLIC

Clearly, an increase in G holding T constant lowers public saving (the government has increased its deficit or reduced its surplus).


An increase in investment demand when saving depends on r

Why might saving depend on r ?

How would the results of an increase in investment demand be different?

Would r rise as much? Would the equilibrium value of I change?


Chapter 3 – At a Glance

Total output is determined by

the economy’s quantities of capital and labour

the level of technology

Competitive firms hire each factor until its marginal product equals its price.

If the production function has CRS property, then labour income plus capital income equals total income (output).

A closed economy’s output is used for

Consumption, investment, government spending


The real interest rate adjusts to equate the demand for and supply of

goods and services

loanable funds

A decrease in national saving causes the interest rate to rise and investment to fall.

An increase in investment demand causes the interest rate to rise, but does not affect the equilibrium level of investment if the supply of loanable funds is fixed.


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