Lecture 2: National Income: Where It ComesFrom and Where It Goes

February 3, 2009

We begin with firms and how they determine output levels, and thereforenational income collectively. Then we look at how the markets for the differentproduction factors distribute the national income to households. Then we lookat how this income is divided between consumption demand and savings onthe consumer side, and we also look at investment demand and governmentpurchases. Then finally we close the model by putting together the demand andthe supply sides of the goods market:

C + I +G = Y

(consumption+investment+government purchases equal total production).

1 The aggregate production function

Y = F (K,L),

where K is aggregate capital stock and L is aggregate labor supply.

Factors of production: K and LProduction function: F

Aggregate output: Y

We assume that:

1. F is increasing in K and L

2. F is concave in K and L; in other words, MPL decreases with L andMPK decreases with K, where:

MPL =F

L(K,L) = F (K,L+ 1) F (K,L)

and

MPK =F

K(K,L) = F (K + 1, L) F (K,L)

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The production function is often assumed to display constant returns toscale:

F (2K, 2L) = 2F (K,L) = 2Y

F (zK, zL) = zF (K,L) = zY.

Example: Cobb-Douglas

F (K,L) = KL1,

where0 < < 1.

For given supply of labor L = L and given supply of capital K = K, wethus obtain the aggregate supply of goods and services:

Y = F (K,L).

Here we take the production technology to be invariant; however, we shallremove this assumption when we analyze economic growth.

2 How is national income distributed to the fac-tors of production?

Factor prices are the amounts paid to the factors of production. How arethey determined?

by the intersection between supply and demand for these factors.

Supplies of capital and labor are fixed, but demands result from profit-maximization decisions by firm.

When deciding how much services from the factors of production to pur-chase, each individual firm takes the price of the good it sells and theprices of the factor services it purchases, as given.

= unit price of good p, wage w (unit price of labor) and rental price ofcapital r (unit price of capital) are taken as given.

Note that we implicitly assume that capital is owned by households, andthat firms rent that capital.

Representative firm solves:

maxK,L

{pF (K,L) wL rK}

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For given K, firm increases L until

Profit = RevenueCost= p.MPL w= 0

=MPL =

F

L(K,L) = w/p = real wage.

Similarly, for given L, firm increases K until

Profit = RevenueCost= p.MPK r= 0

=

MPK =F

K(K,L) = r/p = real rental price of capital.

Therefore each factor is paid its marginal productivity.

Cobb-Douglas case:

MPL = (1 )(K/L) = w/p;MPK = (L/K)1 = r/p.

this system determines K = Kd and L = Ld as decreasing functionsof w/p and r/p respectively.

Eulers Theorem: if the production has the property of constant returnsto scale then economic profit is equal to zero; in other words, once eachfactor has been paid its marginal product, there is nothing left, the outputis fully exhausted.

F (K,L) = (MPK).K + (MPL).L.

Proof in Cobb-Douglas case:

(MPK)K + (MPL)L = K1(1)L1

+(1 )KL1

= KL1

Note that this is economic profit, not necessarily accounting profit: namely,if firms own the capital they use instead of renting it, then:

Accounting profit = Economic Profit+ (MPK).K.

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3 The demand for goods and services

3.1 Consumption

Disposable income:Y T.

This disposable income is divided between consumption and savings.

Consumption function:C = C(Y T ),

where C is increasing. We often assume that this function is linear, of theform:

C = c0 + c1(Y T ).

Marginal propensity to consume:

MPC =C

(Y T ) = c1.

and (1 c1) is the marginal propensity to save.

3.2 Investment

Firms buy investment goods to add to their stock of capital or to replaceobsolete machines.

Investment demand depends negatively upon the interest rate, which mea-sures the cost of funds to finance investment.

Opportunity cost of investing own funds is increasing in r because the firm could instead decide to invest its funds in bankdeposits at interest rate r

the higher r, the smaller the number of investments projects that aremore attractive than investing in bank deposits

Cost of borrowing is increasing in r because lenders have the alternative of putting the funds in bankdeposits and the firm must give them at least as much as what they couldget from the bank

the higher r, the more firms must pay their lenders and thereforethe smaller the number of investors who will be able to afford the cost ofinvestment

Nominal interest rate: the rate that investors pay to borrow money

Real interest rate: nominal interest rate corrected for the effects of infla-tion.

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Investment function:I = I(r),

where r = i denotes the real interest rate.

3.3 Government purchases

Subject to political economy considerations that we abstract from here= we fix:

G = G;

T = T .

Balanced budget if and only if:

G = T.

Budget deficit ifG > T ;

Budget surplus ifG < T.

4 Market equilibrium The condition for equilibrium on the market for goods and services issimply:

Y = C + I +G, (1)

where

1. the left-hand side of (1) is the aggregate supply of goods and servicesdetermined by the production function, that is by:

Y = F (K,L);

2. consumption demand is determined by:

C = c0 + c1(Y T );

3. investment demand is determined by:

I = I(r);

4. and G is the amount of government purchases, which we fix at G.

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We can reexpress equation (1) as:

Y =1

1 c1[c0 + I(r) +G c1T ],

where:

1. the term in brackets represents the demand for goods if output wereequal to zero; this term is called autonomous spending, to capturethe idea that it is the component of demand that does not dependon the level of output;

2. the factor 11c1 , which is a number greater than one, is called themultiplier; it captures the fact that in the short run, that is forfixed price, an increase in autonomous spending, e.g from increasesin government spending or from tax reductions, induces an increasein output which is bigger than the increase in autonomous spend-ing. The idea is that the increase in autonomous demand leads to anincrease in production and income, which in turn increases consump-tion demand and therefore income... but remember that here we areprimarily interested by the long run where prices are flexible andtherefore income supply is ultimately determined by factor supply,fixed at

Y = F (K,L).

In this case, the only effect of an increase in autonomous demand isto increase the interest rate r.

To better understand the role of the interest rate in clearing the marketfor goods and services, let us rewrite equation (1) as:

Y C G = I. (2)

The left-hand side of (2) is the output that remains after the demands ofconsumers and government have been satisfied; it is called national savingsor simply savings. This equation is thus of the form:

Savings = Investment,

or the IS relation.

Going one step further, let us rewrite (2) as:

(Y T C) + (T G) = I,

where the first term on the left-hand side is disposable income minusconsumption, which is private savings, and the second term on the lefthand side is government revenue minus government spending, which ispublic savings. The equation thus says that the flows into the financialmarkets (private and public savings) is equal to the flow out of the financialmarkets (investment).

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Equilibrium interest rate when capital stock and labor supply are fixed:then we have

Y = Y = F (K,L),

and, from (2):Y c0 c1(Y T )G = S = I(r)

= Figure 3-8.

Comparative statics:

1. An increase in government purchases G shifts the Savings line tothe left (it reduces total savings) which in turn reduces the supplyof funds, and thus pushes the interest rate upward so as to reduceinvestment demand accordingly. Thus the increase in governmentpurchase causes the interest rate to rise and private investment de-mand to decrease. We say that government purchases crowd outinvestment demand. See Figure 3-9

2. A reduction in taxes raises disposable income and therefore consump-tion demand, namely by the reduction in taxes times the marginalpropensity to consume c1. Again, this leads to a lower level of totalsavings, which therefore must be met by lower investment demandand therefore by a higher interest rate.

3. An increase in investment demand (for example because of a newtechnological innovation or because government encourages invest-ment through tax credits), shifts the investment curve upward, andtherefore simply raises the equilibrium interest rate.= Figure 3-11

4. A simple variant of this latter case, is to assume that savings also de-pends on the interest rate (a higher interest rate encourages savings);then an increase in investment demand raises both, the interest rateand the equilibrium investment.= Figure 3-12.

Short-run equilibrium: in the short run prices are rigid and therefore out-put Y is not determined by factor supply but instead by aggregate demand.In this case, equilibrium on the goods market is expressed by the equation:

S(Y ) = I(r), (3)

whereS(Y ) = Y co c1Y G.

Equation (2) is referred to as the IS curve.

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