Lecture 3 · Slides L3 · Mankiw ch. 3 and 6 · about 60 min

National Income: Where It Comes From and Where It Goes

A complete model of a closed economy in the long run: how output is produced, how it is paid out to workers and capital owners, and how the interest rate balances saving and investment.

By the end you can

Explain why output is fixed in the long run and what determines it
Derive factor prices from marginal products and show that Cobb-Douglas income shares are constant
Set up C, I, G and find the equilibrium interest rate in the loanable funds market
Predict the effects of fiscal policy and investment demand shifts on r, I and C
Use the open-economy identity S − I = NX and read exchange rates the way the lecture quotes them

The circular flow

Every lira spent on goods is a lira of income for someone. Households sell labour and capital to firms and receive wages and rent. They spend part of that income on goods, pay part in taxes, and save the rest. Firms borrow those savings to invest.

This lecture builds a model that answers three questions about this flow:

Supply: what determines how much the economy produces?
Distribution: how is that income divided between workers and owners of capital?
Demand: who buys the output, and what makes demand equal supply?

The model is classical: prices, wages and the interest rate are flexible and adjust until every market clears. That is a good description of the long run, which is what this semester is about.

Output is fixed by factors and technology

Output depends on the factors of production and on technology:

Y = F(\bar{K}, \bar{L})

The bars mean that in this model capital and labour are fixed: the economy has what it has, and all of it is used. With $K$ , $L$ and the technology in $F$ all fixed, output $\bar{Y}$ is fixed too.

That single fact drives the rest of the lecture. Because $Y$ cannot change, anything that raises one component of spending must lower another.

Returns to scale

Scale every input by the same factor $z$ (for example $z = 1.2$ means 20% more of everything) and compare output:

Returns to scaleFormula sheet →

F(zK, zL) \;\begin{cases} = zF(K,L) & \text{constant} \\ > zF(K,L) & \text{increasing} \\ < zF(K,L) & \text{decreasing} \end{cases}

$z$: common scaling factor, z > 1

Use it when a question gives a production function and asks what kind of returns it has. Substitute zK and zL and factor out z.

$F(K, L) = K + L$ also has constant returns: $zK + zL = z(K + L)$ . You used constant returns in the growth lecture to write $y = f(k)$ .

How income is distributed

Assume firms are competitive. A firm hires labour until the extra output from one more worker, the marginal product of labour, equals the real wage. It rents capital until the marginal product of capital equals the real rental price:

Factor pricesCompetitive firmsFormula sheet →

\frac{W}{P} = MPL \qquad \frac{R}{P} = MPK

$W/P$: real wage
$R/P$: real rental price of capital
$MPL$: marginal product of labour
$MPK$: marginal product of capital

Use it when you need a wage or rental price from a production function.

The marginal product of labour is the extra output from one more unit of labour, holding capital fixed: $MPL = F(K, L+1) - F(K, L)$ . The firm’s extra revenue from that worker is $P \times MPL$ and the extra cost is $W$ , so it keeps hiring while $P \times MPL > W$ and stops where $MPL = W/P$ . The same logic for capital gives $MPK = R/P$ .

Both marginal products diminish. If $L$ rises while $K$ is fixed, there are fewer machines per worker and each extra worker adds less. If $K$ rises while $L$ is fixed, there are fewer workers per machine.

Pay each factor its marginal product and add it up. With constant returns, Euler’s theorem says the payments exactly exhaust output:

Y = MPL \times L + MPK \times K

So economic profit is zero. Everything the economy produces is paid out as labour income or capital income.

Cobb-Douglas: shares that do not move

Paul Douglas noticed that labour’s share of US income was roughly constant for decades, even as capital per worker grew enormously. Charles Cobb found the production function that delivers exactly that:

Cobb-Douglas marginal productsCobb-DouglasFormula sheet →

Y = A K^{\alpha} L^{1-\alpha} \;\Rightarrow\; MPK = \alpha\frac{Y}{K}, \quad MPL = (1-\alpha)\frac{Y}{L}

$A$: total factor productivity
$\alpha$: capital's share of income
$1-\alpha$: labour's share of income

Use it when a question gives Y, K or L and asks for a factor price or an income share.

Multiply through: capital income is $MPK \times K = \alpha Y$ and labour income is $MPL \times L = (1-\alpha)Y$ . The parameter $\alpha$ is capital’s share of income.

Push capital up. Wages rise and the return per unit of capital falls, yet the split bar stays still. Only $\alpha$ moves it.

Notice what the formulas say: $MPL = (1-\alpha)\,Y/L$ is proportional to average labour productivity $Y/L$ . So the neoclassical theory predicts that real wages grow with labour productivity. The lecture shows Turkish data on hourly labour productivity and real wages to test that prediction: when the two drift apart, something outside the simple competitive model (bargaining power, informality, measurement) is at work. A rise in technology $A$ raises both marginal products in the same proportion.

Who buys the output

In a closed economy, output is bought for consumption, investment or government purchases:

Y = C + I + G

Each piece gets its own simple theory.

Consumption depends on disposable income $Y - T$ :

C = C(Y - T)

The marginal propensity to consume $MPC$ is the share of an extra lira of disposable income that is spent. It lies between 0 and 1.

Investment depends negatively on the real interest rate $r$ , the true cost of borrowing:

I = I(r)

Net exports depend on foreign income $Y_F$ (which drives exports), domestic income $Y_D$ (which drives imports) and the real exchange rate: $NX = f(Y_F, Y_D, RER)$ . We return to exchange rates at the end of this lecture.

Government purchases $G$ and taxes $T$ are set by policy. They are exogenous: $G = \bar{G}$ , $T = \bar{T}$ . If $T > G$ the government runs a surplus, and if $T \lt G$ it runs a deficit.

The interest rate balances it all

Put the pieces into $Y = C + I + G$ :

\bar{Y} = C(\bar{Y} - \bar{T}) + I(r) + \bar{G}

Everything here is fixed except $r$ . The interest rate is the one variable that can adjust to make demand equal supply.

Rearrange to see it as a market. Output not consumed by households or bought by the government is national saving:

National savingFormula sheet →

S = \underbrace{(Y - T - C)}_{\text{private}} + \underbrace{(T - G)}_{\text{public}} = Y - C - G

$Y - T - C$: private saving
$T - G$: public saving (budget surplus)

Use it when you need the supply of loanable funds. Compute C first, then subtract.

Equilibrium requires

Loanable funds equilibriumConditionFormula sheet →

Y - C(Y - T) - G = I(r)

$r$: real interest rate, the variable that adjusts

Use it when you are asked for the equilibrium interest rate. Solve the saving side for a number, then solve I(r) for r.

Saving is the supply of loanable funds and does not depend on $r$ in this model, so it is a vertical line. Investment is the demand for loanable funds and slopes down. The real interest rate is their price.

Run these three experiments:

Raise $G$ to 1250. How far does investment fall?
Reset, then cut $T$ to 900. Compare with the $G$ experiment.
Reset, then shift investment demand right. Does investment rise?

Fiscal policy crowds out investment

When $G$ rises by $\Delta G$ with taxes unchanged, public saving falls by $\Delta G$ and so does national saving. $S$ shifts left, $r$ rises, and investment falls by exactly $\Delta G$ . Output cannot rise, so extra government spending is paid for entirely by lower investment. This is crowding out.

A tax cut works through consumers. Disposable income rises by $\Delta T$ , consumption rises by $MPC \times \Delta T$ , so national saving falls by $MPC \times \Delta T$ . Crowding out is smaller than for an equal rise in $G$ , because part of the tax cut is saved.

When investment demand rises

Suppose firms become more optimistic, or a tax credit makes investment more attractive. The $I(r)$ curve shifts right. But saving is fixed, so the amount of investment cannot change. Only the interest rate rises.

The open economy

Now let goods and capital cross borders. Split each type of spending into domestic and foreign goods: $C = C^d + C^f$ , and so on. Output is spending on domestic goods, including exports:

Y = C^d + I^d + G^d + X = (C + I + G) - (C^f + I^f + G^f) + X = C + I + G + X - M

With $NX = X - M$ this is the familiar $Y = C + I + G + NX$ , or $NX = Y - (C + I + G)$ : net exports equal output minus domestic spending.

Subtract $C$ and $G$ from both sides of the identity. National saving $S = Y - C - G$ , so:

Net capital outflow identityIdentityFormula sheet →

S - I = NX

$S$: national saving
$I$: domestic investment
$S - I$: net capital outflow
$NX$: trade balance

Use it when any question links saving, investment and the trade balance. If S is less than I, the country borrows from abroad and runs a trade (current account) deficit.

Exchange rates

The nominal exchange rate is the relative price of two currencies. The real exchange rate is the relative price of the goods of two countries: the rate at which you can trade your goods for theirs.

Real exchange rateFormula sheet →

\varepsilon = e \times \frac{P}{P^*}

$\varepsilon$: real exchange rate
$e$: nominal exchange rate, foreign currency per unit of domestic currency
$P$: domestic price level
$P^*$: foreign price level

Use it when you compare the price of domestic goods with foreign goods in a common currency.

When $\varepsilon$ is high, domestic goods are expensive relative to foreign goods, so exports fall, imports rise and $NX$ falls. When $\varepsilon$ is low, $NX$ rises. In the long run the real exchange rate adjusts so that $NX(\varepsilon) = S - I(r^*)$ .

Change	Meaning	Export prices	Exports	Imports	$NX$
$e$ rises	lira appreciates	rise	fall	rise	falls
$e$ falls	lira depreciates	fall	rise	fall	rises
$P/P^*$ rises	domestic goods relatively dearer	rise	fall	rise	falls

Take percentage changes of $\varepsilon = e \cdot P/P^*$ :

Nominal exchange rate and inflationFormula sheet →

\%\Delta e = \%\Delta \varepsilon + (\pi^* - \pi)

$\pi$: domestic inflation
$\pi^*$: foreign inflation

Use it when a question asks what happens to the lira when Turkish inflation exceeds foreign inflation.

If Turkish inflation is higher than US inflation, $\pi^* - \pi$ is negative and $e$ falls: the lira depreciates. Over long periods this relationship between relative prices and the exchange rate is one of the most reliable in macroeconomics.

Exam practice

Summary and review

Long run: $Y = F(\bar{K}, \bar{L})$ is fixed by factors and technology.
Competitive factor prices: $W/P = MPL$ and $R/P = MPK$ . With constant returns, factor payments add up to $Y$ .
Cobb-Douglas: capital’s share is $\alpha$ , labour’s share is $1-\alpha$ , whatever $K$ and $L$ are.
Demand: $C(Y - T)$ , $I(r)$ , and exogenous $G$ , $T$ .
The real interest rate clears the loanable funds market: $S = Y - C - G = I(r)$ .
A rise in $G$ crowds out investment one for one. A tax cut of $\Delta T$ crowds out $MPC \times \Delta T$ .
A rise in investment demand raises $r$ but not $I$ .
Returns to scale: compare $F(zK, zL)$ with $zF(K,L)$ .
Accounting profit $=$ economic profit $+ MPK \times K$ .
Open economy: $S - I = NX$ . Small open economy: $r = r^*$ .
Lecture convention: $e$ = foreign currency per lira, so $e \uparrow$ = appreciation. $\varepsilon = e P/P^*$ ; $\%\Delta e = \%\Delta\varepsilon + \pi^* - \pi$ .