Determination of Income and Employment

Given/Concept:
Marginal Propensity to Consume (MPC) measures the responsiveness of consumption to a change in income.

Definition:
Marginal Propensity to Consume (MPC) is defined as the ratio of the change in ex ante consumption ($\Delta C$) to the change in income ($\Delta Y$). Mathematically:
$$MPC = \frac{\Delta C}{\Delta Y}$$
It shows how much of every additional rupee of income is spent on consumption. Since consumption increases with income but by less than the increase in income, 0 < MPC < 1.

Relationship between MPC and MPS:
We know that income ($Y$) is either consumed ($C$) or saved ($S$):
$$Y = C + S$$
Taking the change on both sides:
$$\Delta Y = \Delta C + \Delta S$$
Dividing throughout by $\Delta Y$:
$$1 = \frac{\Delta C}{\Delta Y} + \frac{\Delta S}{\Delta Y}$$
$$1 = MPC + MPS$$
$$\therefore MPS = 1 - MPC$$

Conclusion: MPC and MPS are complementary — they always sum to 1. If MPC = 0.8, then MPS = 0.2. A higher MPC implies a lower MPS and vice versa.

Ex Ante Investment:
- 'Ex ante' means 'planned' or 'intended'.
- Ex ante investment refers to the amount of investment that firms plan or intend to make during a given period at the beginning of that period.
- It is a desired/planned magnitude and may or may not be realised.
- Example: A firm plans to invest Rs 100 crores in new machinery at the start of the year.

Ex Post Investment:
- 'Ex post' means 'actual' or 'realised'.
- Ex post investment refers to the investment that has actually been made during a given period, including both planned investment and unintended changes in inventories.
- It is always realised at the end of the period.
- Mathematically: $\text{Ex post investment} = \text{Planned investment} + \text{Unintended inventory accumulation/decumulation}$

Key Difference:

| Basis | Ex Ante Investment | Ex Post Investment |
|---|---|---|
| Nature | Planned/Intended | Actual/Realised |
| Timing | Beginning of period | End of period |
| Includes unintended inventory changes | No | Yes |

Equilibrium Condition: The economy is in equilibrium when ex ante investment equals ex ante saving, i.e., when there are no unintended changes in inventories, so ex ante and ex post investment coincide.

Parametric Shift of a Line:
A parametric shift refers to a change in the position or slope of a line caused by a change in one of its parameters (i.e., the constants that define the line — its slope or intercept), while the variable on the horizontal axis remains unchanged.

Consider a general linear equation:
$$Y = a + bX$$
where $a$ is the intercept and $b$ is the slope. A change in $a$ or $b$ causes a parametric shift.

(i) When the slope decreases:
- The slope $b$ represents the steepness of the line.
- If the slope decreases (i.e., $b$ falls), the line becomes flatter — it rotates downward (clockwise) around the intercept point on the Y-axis.
- The intercept remains the same, but for any given value of $X$, the value of $Y$ is now lower (except at the intercept).
- Example: If MPC falls, the consumption function $C = \bar{C} + bY$ becomes flatter.

(ii) When the intercept increases:
- The intercept $a$ is the value of $Y$ when $X = 0$.
- If the intercept increases (i.e., $a$ rises), the entire line shifts upward in a parallel fashion — the slope remains unchanged but the line moves up.
- Every point on the new line is higher than the corresponding point on the old line by the same amount.
- Example: If autonomous consumption $\bar{C}$ increases, the consumption function shifts upward parallelly.

Summary:
- Decrease in slope → line becomes flatter (rotates clockwise around the Y-intercept).
- Increase in intercept → line shifts upward parallelly (no change in slope).

Effective Demand:
Effective demand refers to the level of aggregate demand for final goods and services that actually determines the equilibrium level of output and employment in the economy. Under the Keynesian framework, with a given price level and interest rate, the aggregate supply is assumed to be perfectly elastic. Therefore, output is determined solely by aggregate demand. The level of aggregate demand at which the product market is in equilibrium (i.e., aggregate demand = aggregate supply) is called effective demand.

Derivation of the Autonomous Expenditure Multiplier:

Given:
- Price of final goods is constant.
- Rate of interest is given (constant).
- Consumption function: $C = \bar{C} + cY$, where $\bar{C}$ is autonomous consumption and $c = MPC$.
- Investment is autonomous: $I = \bar{I}$.
- Let total autonomous expenditure: $A = \bar{C} + \bar{I}$.

Step 1: Write the Aggregate Demand (AD) equation.
$$AD = C + I = \bar{C} + cY + \bar{I} = A + cY$$

Step 2: Equilibrium condition.
At equilibrium, Aggregate Demand = Aggregate Supply (Output):
$$Y = AD$$
$$Y = A + cY$$

Step 3: Solve for equilibrium income $Y$.
$$Y - cY = A$$
$$Y(1 - c) = A$$
$$Y = \frac{1}{1-c} \cdot A$$

Step 4: Derive the multiplier.
If autonomous expenditure changes by $\Delta A$, the change in equilibrium income $\Delta Y$ is:
$$\Delta Y = \frac{1}{1-c} \cdot \Delta A$$

The Autonomous Expenditure Multiplier is:
$$\boxed{k = \frac{\Delta Y}{\Delta A} = \frac{1}{1-c} = \frac{1}{1-MPC} = \frac{1}{MPS}}$$

Interpretation: Since 0 < c < 1, the multiplier k > 1. This means a unit increase in autonomous expenditure leads to a more-than-proportionate increase in equilibrium income through the multiplier process (successive rounds of spending and income generation).

Given:
- Autonomous expenditure: $A = \bar{C} + \bar{I} =$ Rs 50 crores
- Marginal Propensity to Save: $MPS = 0.2$
- Therefore, $MPC = 1 - MPS = 1 - 0.2 = 0.8$
- Level of income: $Y =$ Rs 4000 crores

Step 1: Write the Aggregate Demand formula.
$$AD = A + MPC \times Y$$

Step 2: Calculate ex-ante Aggregate Demand.
$$AD = 50 + 0.8 \times 4000$$
$$AD = 50 + 3200$$
$$AD = \text{Rs } 3250 \text{ crores}$$

Step 3: Check equilibrium condition.
For equilibrium: $AD = AS = Y$

Here, $AD =$ Rs 3250 crores but $Y =$ Rs 4000 crores.

Since AD < Y, i.e., 3250 < 4000, the economy is NOT in equilibrium.

Step 4: Find equilibrium income (for reference).
$$Y^* = \frac{A}{1 - MPC} = \frac{50}{1 - 0.8} = \frac{50}{0.2} = \text{Rs } 250 \text{ crores}$$

Reason for disequilibrium:
Aggregate demand (Rs 3250 crores) is less than aggregate supply/income (Rs 4000 crores). This means there is excess supply in the economy. Firms will face unintended accumulation of inventories. As a result, producers will cut down production, and income will fall until it reaches the equilibrium level of Rs 250 crores.

Conclusion: The economy is not in equilibrium because ex-ante aggregate demand (Rs 3250 crores) $\neq$ income/output (Rs 4000 crores). There is excess supply of Rs 750 crores.

Paradox of Thrift:

Definition:
The Paradox of Thrift states that if all households in an economy increase their savings (i.e., become more thrifty) simultaneously, the total savings in the economy may not increase — and in fact, the aggregate income of the economy will fall. This is paradoxical because what seems rational and beneficial for an individual (saving more) turns out to be harmful for the economy as a whole.

Explanation with the Keynesian Model:

Suppose all households decide to save more (i.e., MPS increases or autonomous consumption $\bar{C}$ falls). This means the consumption function shifts downward.

- A fall in consumption → fall in aggregate demand.
- Fall in aggregate demand → fall in equilibrium output/income (via the multiplier process in reverse).
- As income falls, savings also fall (since $S = Y - C$).

Algebraically:
At equilibrium: $S = I$ (ex ante)

If investment $\bar{I}$ is autonomous (fixed), then equilibrium saving must also remain equal to $\bar{I}$. So even if households try to save more, the fall in income brings savings back to the original level.

Multiplier Effect:
Let autonomous consumption fall by $\Delta \bar{C}$ (i.e., people decide to save more).
$$\Delta Y = \frac{1}{1 - MPC} \times (-\Delta \bar{C})$$
Income falls by a multiplied amount. The rise in intended savings is offset by the fall in income.

Diagrammatic Summary:
- Saving function shifts upward (people want to save more at every income level).
- New equilibrium is at a lower level of income.
- Actual savings remain unchanged (equal to autonomous investment).

Conclusion:
The Paradox of Thrift shows that individual rationality does not always lead to social rationality. An individual's attempt to save more is virtuous at the personal level, but if everyone does it simultaneously, it reduces aggregate demand, output, employment, and income — leaving the economy worse off without any net increase in total savings. This is a classic example of the fallacy of composition in macroeconomics.