Chapter 4 Determination Of Income And Employment

Aggregate Demand And Its Components

Macroeconomics uses theoretical frameworks, known as models, to explain the factors that determine key economic variables like national income, price levels, employment, and interest rates. When studying the determination of one variable, we often assume others remain constant, a principle called ceteris paribus ('other things remaining equal'). This chapter focuses on determining national income and employment assuming fixed prices and a constant interest rate.

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Consumption

In macroeconomics, terms like consumption and investment can refer to both actual, observed values (ex post) and planned, intended values (ex ante). For understanding how income is determined, we need to focus on the ex ante or planned components of aggregate demand.

Consumption demand (C) by households is primarily determined by their income (Y). The relationship between planned consumption and income is described by the consumption function. A simple linear consumption function is represented as:

$ \text{C} = \bar{\text{C}} + c\text{Y} $

• $\bar{\text{C}}$ (read as C-bar) is autonomous consumption. This is the level of consumption that occurs even when income is zero. It is independent of income.
• $c\text{Y}$ is induced consumption. This part of consumption depends directly on the level of income.
• $c$ is the marginal propensity to consume (MPC). It represents the change in consumption for a one-unit change in income.

  $ c = \frac{\Delta \text{C}}{\Delta \text{Y}} $

  MPC typically lies between 0 and 1 ($0 \le c \le 1$), meaning that as income changes, consumption changes by a fraction of that income change.

The part of income that is not consumed is saved (S). $S = Y - C$.

The marginal propensity to save (MPS) is the change in saving for a one-unit change in income. It is denoted by $s$.

$ s = \frac{\Delta \text{S}}{\Delta \text{Y}} $

Since $\Delta \text{Y} = \Delta \text{C} + \Delta \text{S}$, dividing by $\Delta \text{Y}$ gives $1 = \frac{\Delta \text{C}}{\Delta \text{Y}} + \frac{\Delta \text{S}}{\Delta \text{Y}}$, which means $1 = c + s$. Thus, MPC + MPS = 1.

Average Propensity to Consume (APC) is total consumption divided by total income ($APC = \frac{C}{Y}$).

Average Propensity to Save (APS) is total saving divided by total income ($APS = \frac{S}{Y}$). Note that $APC + APS = 1$.

Graphically, the consumption function $C = \bar{C} + cY$ is an upward-sloping line with intercept $\bar{C}$ on the vertical axis and slope $c$ (MPC).

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Investment

Investment (I) refers to the addition to the stock of physical capital (machines, buildings) and changes in inventories (stock of unsold goods). Investment goods are final goods used for future production, not intermediate goods.

While actual investment decisions are influenced by factors like the interest rate, in this simple model, we assume that planned or ex ante investment demand is autonomous, meaning it does not depend on the level of income. It is treated as a fixed amount determined outside the model.

$ \text{I} = \bar{\text{I}} $

Where $\bar{I}$ (read as I-bar) is autonomous investment, a positive constant.

Graphically, the investment function is a horizontal line at the level of $\bar{I}$, parallel to the income axis.

Determination Of Income In Two-Sector Model

In a simple economy consisting only of households and firms (a two-sector model, ignoring government and foreign trade), the total planned spending or ex ante aggregate demand (AD) for final goods is the sum of planned consumption and planned investment.

$ \text{AD} = \text{C} + \text{I} $

Substituting the consumption and investment functions:

$ \text{AD} = (\bar{\text{C}} + c\text{Y}) + \bar{\text{I}} $

$ \text{AD} = (\bar{\text{C}} + \bar{\text{I}}) + c\text{Y} $

Let $\text{A} = \bar{\text{C}} + \bar{\text{I}}$ be the total autonomous expenditure in the economy (expenditure that does not depend on income).

$ \text{AD} = \text{A} + c\text{Y} $

Graphically, the AD function is obtained by vertically adding the consumption and investment functions. It is a line parallel to the consumption function, with an intercept equal to the sum of autonomous consumption and autonomous investment (A).

Equilibrium in the goods market occurs when planned aggregate demand equals planned aggregate supply (which is the planned output of final goods, Y). Producers plan their output (ex ante supply) based on expectations of demand. In equilibrium, these plans are realized.

$ \text{Ex ante Aggregate Demand} = \text{Ex ante Aggregate Supply} $

$ \text{AD} = \text{Y} $

If ex ante AD is not equal to planned output (Y), there will be unintended changes in inventories. If AD > Y, demand exceeds planned supply, leading to unplanned decumulation of inventories. If AD < Y, demand falls short of planned supply, leading to unplanned accumulation of inventories. These unintended inventory changes signal producers to adjust their output in the next period until equilibrium ($AD=Y$) is restored.

Note: In this simplified model without government taxes/subsidies, GDP is identical to National Income. We use Y to represent both equilibrium output and income.

Determination Of Equilibrium Income In The Short Run

In the short run, with unused resources available, producers can increase output without significant increases in marginal costs, justifying the assumption of a fixed price level. Under this fixed price assumption, the equilibrium level of income is determined solely by the level of aggregate demand, based on the effective demand principle.

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Macroeconomic Equilibrium With Price Level Fixed

Under the assumption of fixed prices and unused resources, the aggregate supply curve is effectively perfectly elastic up to the full employment level. This means that any level of output demanded at the going fixed price can be supplied. In diagrams, this situation is often represented by a 45-degree line from the origin, where every point on the line signifies that planned aggregate supply (on the vertical axis) equals the level of income/output (on the horizontal axis).

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Graphical Method

Equilibrium is found graphically where the ex ante Aggregate Demand (AD) line intersects the ex ante Aggregate Supply (AS) line (the 45-degree line).

The intersection point (E) represents the equilibrium level of income (Y*) and aggregate demand (AD*). At this point, planned spending by households and firms exactly equals the planned output, and there are no unintended changes in inventories.

If the economy is at an income level below Y*, AD > Y, leading to unintended inventory depletion. Producers increase output. If the economy is at an income level above Y*, AD < Y, leading to unintended inventory accumulation. Producers decrease output. This adjustment process moves the economy towards the equilibrium Y*.

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Algebraic Method

Equilibrium occurs where AD = Y.

$ \text{Y} = \text{A} + c\text{Y} $

To find the equilibrium income (Y*), rearrange the equation to solve for Y:

$ \text{Y} - c\text{Y} = \text{A} $

$ \text{Y}(1 - c) = \text{A} $

$ \text{Y}^* = \frac{\text{A}}{(1 - c)} $

Since $1-c = s$ (MPS), the equilibrium income can also be written as:

$ \text{Y}^* = \frac{\text{A}}{s} $

This equation shows that the equilibrium level of income is directly proportional to the total autonomous expenditure (A) and inversely proportional to the marginal propensity to save (s) or directly proportional to $1/(1-c)$.

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Effect Of An Autonomous Change In Aggregate Demand On Income And Output

When there is an autonomous change in aggregate demand (e.g., a change in $\bar{C}$ or $\bar{I}$), the AD curve shifts. This leads to a change in the equilibrium level of income. An increase in autonomous expenditure shifts the AD curve upwards, leading to a higher equilibrium income. A decrease shifts it downwards, leading to a lower equilibrium income.

The change in equilibrium income is not just equal to the initial change in autonomous expenditure; it is larger. This amplification is known as the multiplier effect.

Answer:

Graphically, the increase in investment from 10 to 20 shifts the AD curve upward from $AD_1$ to $AD_2$. The new equilibrium $E_2$ is at a higher income level (300) compared to the initial equilibrium $E_1$ (250). The increase in income ($\Delta Y = 50$) is five times the initial increase in investment ($\Delta I = 10$).

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The Multiplier Mechanism

The investment multiplier (or autonomous expenditure multiplier) is the ratio of the total change in equilibrium income to the initial change in autonomous expenditure.

$ \text{Multiplier} = \frac{\Delta \text{Y}}{\Delta \text{A}} $

The multiplier effect occurs because the initial increase in autonomous spending leads to an equivalent increase in income for factors of production. A portion of this increased income is then spent on consumption (determined by MPC), which becomes income for others, leading to further consumption, and so on. This creates a chain reaction of spending and income increases.

Let $\Delta A$ be the initial increase in autonomous expenditure. This leads to an equal increase in income in the first round ($\Delta Y_1 = \Delta A$).

• Round 1: Income increases by $\Delta A$. Consumption increases by $c \cdot \Delta A$.
• Round 2: The increase in consumption ($c \cdot \Delta A$) becomes income for others ($\Delta Y_2 = c \cdot \Delta A$). Consumption increases by $c \cdot (c \cdot \Delta A) = c^2 \cdot \Delta A$.
• Round 3: The increase in consumption ($c^2 \cdot \Delta A$) becomes income ($\Delta Y_3 = c^2 \cdot \Delta A$). Consumption increases by $c \cdot (c^2 \cdot \Delta A) = c^3 \cdot \Delta A$.
• This process continues infinitely, with each round's income increase being a fraction ($c$) of the previous round's income increase.

The total increase in income ($\Delta Y$) is the sum of income increases in all rounds:

$ \Delta \text{Y} = \Delta \text{A} + c \cdot \Delta \text{A} + c^2 \cdot \Delta \text{A} + c^3 \cdot \Delta \text{A} + \dots $

$ \Delta \text{Y} = \Delta \text{A} (1 + c + c^2 + c^3 + \dots ) $

Since $0 \le c < 1$, the series $(1 + c + c^2 + c^3 + \dots )$ is an infinite geometric series that converges to $\frac{1}{1-c}$.

$ \Delta \text{Y} = \Delta \text{A} \left( \frac{1}{1-c} \right) $

The multiplier is therefore:

$ \text{Multiplier} = \frac{\Delta \text{Y}}{\Delta \text{A}} = \frac{1}{1-c} = \frac{1}{s} $

The size of the multiplier depends on the MPC ($c$) or MPS ($s$). A higher MPC (lower MPS) leads to a larger multiplier, as a greater portion of each additional dollar of income is respent, generating more income in subsequent rounds.

Paradox of Thrift: This concept highlights that if everyone in the economy tries to save a larger proportion of their income (increase MPS or decrease MPC), the total amount of saving in the economy might not increase; it could even decrease or remain unchanged. This happens because the increased desire to save leads to reduced consumption and thus reduced aggregate demand. This fall in demand, through the multiplier effect, causes a significant decrease in the equilibrium level of income. With lower income, even a higher saving rate (MPS) applied to a smaller income might result in the same or lower total saving.

Some More Concepts

The equilibrium level of income determined in this model does not necessarily mean that all factors of production are fully employed. It simply means that, at that income level, planned aggregate demand equals planned output, and there is no tendency for income to change in the short run.

Full Employment Level of Income: This is the level of income achieved when all available factors of production (including labor) are fully utilized in the production process.

If the short-run equilibrium income (Y*) is less than the full employment level of income, it indicates a situation of deficient demand. The aggregate demand is not high enough to absorb the output that could be produced if all resources were fully employed. In the long run, deficient demand can lead to deflationary pressures (falling prices).

If the short-run equilibrium income (Y*) is greater than the full employment level of income, it indicates a situation of excess demand. The aggregate demand exceeds the economy's capacity to produce at full employment. Since output cannot increase further in the short run once full employment is reached, excess demand leads to inflationary pressures (rising prices).

The Effective Demand Principle states that, in the short run with fixed prices, the equilibrium level of output and employment is determined solely by the level of aggregate demand.