Determination of Income in Two-Sector Model

4.2 Determination of Income in Two-Sector Model

The Basic Framework: Ex Ante Aggregate Demand

In an economy without a government, the planned (ex ante) aggregate demand for final goods comes from exactly two sources: households who consume, and firms who invest. There is no government spending, no taxes, and no foreign sector. The aggregate demand function is therefore:

where $C$ is ex ante consumption expenditure and $I$ is ex ante investment expenditure.

Both $C$ and $I$ have specific behavioural content. Consumption depends on income according to the Keynesian consumption function:

where $\overline{C}$ is autonomous consumption (the minimum consumption that occurs even when income is zero) and $c$ is the marginal propensity to consume ($0 < c < 1$). Investment, in this simple model, is treated as entirely autonomous — it does not depend on current income. So we write:

Substituting these into the aggregate demand expression gives:

The two autonomous terms — $\overline{C}$ and $\overline{I}$ — are often lumped together into a single term called total autonomous expenditure, denoted $\overline{A}$. So:

and the aggregate demand function becomes:

Equilibrium in the Goods Market

The equilibrium condition for the final goods market is that planned output (ex ante supply) equals planned aggregate demand (ex ante demand). Let $Y$ stand for the ex ante output of final goods that producers plan to produce. Then equilibrium requires:

Substituting the expression for $AD$:

This is equation (4.3) in the textbook. It is not an identity — it is an equilibrium condition. It holds only when the goods market is in balance.

What Happens When Plans Differ from Outcomes?

Suppose producers plan to produce an output $Y$, but planned aggregate demand $\overline{A} + cY$ falls short of that output. Then goods remain unsold. These unsold goods pile up as inventories — stocks of output that firms have produced but not yet sold.

Inventory changes can be of two kinds:

• Planned inventory investment: when a firm deliberately decides to hold a certain level of stocks (for smooth production, to meet unexpected orders, etc.)
• Unplanned inventory investment: when actual sales differ from planned sales, forcing the firm to add to or run down its existing inventories involuntarily

When ex ante output exceeds ex ante demand, the difference shows up as unplanned accumulation of inventories. In the ex post accounting identity, this unintended inventory build-up is counted as part of actual investment $I$, so that actual output always equals actual $C + I$ — but the $I$ now includes an undesired component.

Conversely, if ex ante demand exceeds ex ante output, inventories are depleted unintentionally (negative unplanned inventory investment), signalling that producers need to raise output.

A Brief Detour: Introducing the Government

The textbook mentions that a government can be introduced through two fiscal variables: Tax ($T$) and Government Expenditure ($G$). Both are treated as autonomous — they do not depend on income in this simple framework.

Government expenditure $G$ adds directly to aggregate demand, just like consumption and investment. Taxes, however, reduce the disposable income of households:

Households consume only a fraction of their disposable income, so the consumption function becomes:

The equilibrium condition then becomes:

or

$Y = (\overline{C} + \overline{I} + G - cT) + cY$ …