National Income: Where It Comes From and Where It Goes

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:

1. Supply: what determines how much the economy produces?
2. Distribution: how is that income divided between workers and owners of capital?
3. 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.

Measuring economic growth

Growth compares output with the previous period:

The same data can be reported in three ways:

• Quarterly growth at an annual rate. The change from one quarter to the next is compounded for a year: a 0.3% quarterly change means an annual rate of about $4 \times 0.3 = 1.2\%$.
• Four-quarter (year-over-year) growth. Each quarter is compared with the same quarter of the previous year, which offsets seasonal variation.
• Full-year growth. The four quarterly changes of the year are added up.

Growth compounds over long horizons. Starting from the first quarter of 1998, the level 96 quarters later is $GDP_{2022} = GDP_{1998}(1+g)^{96}$, where $g$ is the average growth rate over those 96 quarters.

Output is fixed by factors and technology

Output depends on the factors of production and on technology:

The two factors are capital $K$, the tools, machines and structures used in production, and labour $L$, the physical and mental effort of workers, measured above all as the time people spend working.

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:

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

Distribution is determined by factor prices, the amounts paid per unit of each factor: the wage is the price of labour and the rental rate is the price of capital. 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:

The firm takes the product price and the factor prices as given and chooses $L$ and $K$ to maximise $\text{Profit} = P \times F(K, L) - W \times L - R \times K$.

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 $P \times MPL \gt W$ 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. The $MPL$ curve is the firm’s demand curve for labour, and the fixed labour force is a vertical supply line; their intersection sets the real wage. The $MPK$ curve is the firm’s demand curve for renting capital.

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

So economic profit is zero. Everything the economy produces is paid out as labour income or capital income. (The deck’s summary labels $MPL \times L$ as total capital income; it is the total wage bill, and $MPK \times K$ is the total return to capital.)

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:

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. An increase in labour has the mirror effect: $MPL$ falls and $MPK$ rises. Cobb-Douglas has constant returns: $A(zK)^{\alpha}(zL)^{1-\alpha} = z A K^{\alpha} L^{1-\alpha}$.

Notice what the formulas say: $MPL = (1-\alpha)\,Y/L$ is proportional to average labour productivity $Y/L$, and $MPK = \alpha\,Y/K$ is proportional to average capital productivity $Y/K$: each marginal product is proportional to its average product. The ratio of labour income to capital income is the constant $(1-\alpha)/\alpha$, whatever $K$, $L$ and $A$ are. 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.

The rule of 70 and the sources of growth

The rule of 70 converts a growth rate into a doubling time:

At 7% a year, GDP doubles in $70/7 = 10$ years. Some economists use 69 or 72 instead; the differences come from how compounding is treated. The rule is useful because small differences in growth rates lead to large differences in the size of economies over time.

Long-run growth must be sustainable, efficient and inclusive. The lecture lists four ways to create it:

1. More physical capital. Adding machines and structures raises labour productivity.
2. Technological improvement. Better technology lets the same workers and machines produce more.
3. A larger labour force. Expanding employment adds to output, if the other conditions are met.
4. More human capital. Education and skills, including social and institutional capital.

In the short run, expansionary policy can stimulate output, but it brings inflation and instability. The lasting solution is to shift long-run aggregate supply to the right, so that income is permanently higher; higher income then produces more saving and more tax revenue, which finance capital accumulation, human capital and technology.

Who buys the output

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

Each piece gets its own simple theory.

Consumption depends on disposable income $Y - T$:

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 real interest rate is the cost of borrowing, and also the opportunity cost of using your own funds to finance investment. Firms borrow to buy plant, equipment and new buildings; households borrow to buy new houses. Here $r$ is the real interest rate, not the rental rate of capital $R$.

Net exports follow $NX = f(Y_F, Y_D, RER)$. Foreign income $Y_F$ determines export demand. Domestic income $Y_D$ determines import demand, and it does so negatively. The real exchange rate $RER$ sets the relative price of export and import goods, with real effects through purchasing power, and it is linked to domestic and foreign inflation. 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 $T \gt G$ the government runs a surplus, and if $T \lt G$ it runs a deficit. Government purchases exclude transfer payments such as pensions and unemployment benefits, because transfers are not purchases of goods and services. In practice taxes are a function of income: total taxes are direct taxes, related directly to income, plus indirect taxes, related to income through consumption and prices. A deficit is financed by issuing Treasury bonds, that is by borrowing, or by printing money.

The interest rate balances it all

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

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:

Equilibrium requires

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. The demand comes from firms borrowing for plant and equipment and from households borrowing for houses. The supply comes from households making bank deposits and buying bonds; the government adds to it when it does not spend all of its tax revenue. The interest rate is both the cost of borrowing and the return to lending.

Run these three experiments:

1. Raise $G$ to 1250. How far does investment fall?
2. Reset, then cut $T$ to 900. Compare with the $G$ experiment.
3. 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:

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:

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.

The deck prints a price of 12000 TL for the shirt, but its own answer of 2 American shirts requires 1200 TL; the example uses 1200.

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 $NX(\varepsilon) = S - I(r^*)$.

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

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. $\varepsilon = e P/P^*$; $\%\Delta e = \%\Delta\varepsilon + \pi^* - \pi$.