Keynesian Income-Expenditure Model Notes (Two- and Four-Sector)

The Keynesian Theory of Output: A Simple Model

• Core focus: how output is determined in an economy using the income–expenditure framework

• Key components: consumption function, planned expenditure (PAE), induced vs. autonomous expenditure, disequilibrium adjustment, and the roles of saving and investment

• Major ideas introduced in the material: the multiplier (the principle of effective demand), the paradox of thrift, and extensions to a four-sector model (including government and external sectors)

• Framing note: the common interpretation treats the model as a fixed-price, short-run framework, though the material questions whether that view fully captures Keynes’s original perspective

The Keynesian Income-Expenditure Model: Two-Sector, Closed Economy (No Government)

• Equilibrium analysis in a simple closed economy with two sectors: households (consumption) and firms (investment)

• Demand-determined output: output is driven by aggregate demand, not by an automatic tendency toward full employment

• Saved and investment link: saving is determined by income, while investment is exogenous

• Disequilibrium adjustment principle: when planned aggregate expenditure (PAE) does not equal actual output Y, unplanned inventory changes push Y toward equilibrium

• Key equilibrium condition (basic):

• $Y^* = PAE^*$ where $PAE = C<em>d + I</em>p$

• This equilibrium is graphically represented by the intersection of the PAE line with the 45-degree line, where actual output equals planned expenditure.

• In the simplest closed two-sector model with a linear consumption function, the equilibrium income is a multiple of the exogenous components of demand

The Consumption Function

• Planned current real consumption, denoted $C<em>d$, is a function of current real disposable income $Y</em>d = Y - T$, with taxes exogenous (T)

• Functional form (standard Keynesian form):

• $C<em>d = C</em>d^e + c(Y - T)$- where $C_d^e$ is autonomous (exogenous) consumption and $c$ is the marginal propensity to consume (MPC)

• Definitions:

  • Marginal propensity to consume (MPC): $c = \frac{\text{d}C_d}{\text{d}Y} \text{ (with T fixed)}$

  • Autonomous consumption: $C<em>d^e = a - cT \text{ (often written as } a \text{ or } \overline{C}</em>0 \text{ depending on notation)}$

  • Induced consumption: the part that varies with income: $c(Y - T)$

  • Average propensity to consume (APC): $APC = \frac{C_d}{Y - T}$

• Important relationships:

  • APC > MPC in the simple model (the average fraction of disposable income spent is higher than the marginal fraction, because some consumption is fixed as income rises)

The Consumption–Expenditure Depiction (Graphical Intuition)

• The consumption function can be depicted either as a function of disposable income $Y - T$ or of income $Y$ directly.

• Graphical Representation of the Consumption Function:

  • Axes: The horizontal axis represents disposable income ($Y - T$) or total income ($Y$), and the vertical axis represents planned consumption ($C_d$).

  • Plotting: It is typically an upward-sloping straight line.

  • Intercept: The vertical intercept represents autonomous consumption ($C_d^e$), the amount consumed even if disposable income were zero. This intercept can be also expressed as $(a - cT)$.

  • Slope: The slope of the consumption function is the marginal propensity to consume (MPC), $c$. This means for every dollar increase in income, consumption increases by $c$ cents.

• Two equivalent depictions:

  • For disposable income axis: $C<em>d = C</em>d^e + c((Y - T))$ with slope equal to MPC $c$

  • For income axis: $C<em>d = cY + (C</em>d^e - cT)$ with slope equal to MPC $c$

• In both depictions, the slope of the consumption function equals the MPC, and the APC is generally above the MPC when T is fixed.

• Graphical Representation of Equilibrium (PAE-Y Diagram):

  • Axes: The horizontal axis represents actual output ($Y$), and the vertical axis represents Planned Aggregate Expenditure (PAE).

  • 45-Degree Line: A reference line from the origin with a slope of 1. All points on this line represent instances where actual output ($Y$) perfectly matches planned aggregate expenditure (PAE). Equilibrium must lie on this line, where $Y = PAE$.

  • PAE Line: This line is derived by vertically summing the consumption function ($C<em>d$) and planned investment ($I</em>p$). Its vertical intercept is the total autonomous expenditure $(C<em>d^e + I</em>p)$. Its slope is the MPC ($c$) in the two-sector model.

  • Equilibrium: The intersection of the PAE line and the 45-degree line indicates the equilibrium level of income ($Y^*$). At this point, the amount firms plan to sell ($Y$) equals the amount households and firms plan to buy (PAE).

Equilibrium in a Two-Sector Model (Closed Economy, No Government)

• Equilibrium condition arises from equating output with the planned expenditure:

• $Y^* = PAE^* = C<em>d^e + cY^* + I</em>p$

• Rearranging (assuming constant taxes T):

• $Y^* = \frac{C<em>d^e + I</em>p}{1 - c}$

• If we explicitly write autonomous consumption as $C_d^e = a - cT$, then:

• $Y^* = \frac{a - cT + I_p}{1 - c}$

• Alternatively, defining the exogenous demand component as $A = C<em>d^e + I</em>p$, we have $Y^* = \frac{A}{1 - c}$

• The quantity $\frac{1}{1 - c}$ is the simple Keynesian multiplier in this closed-two-sector model

The Multiplier, the Principle of Effective Demand, and Saving–Investment Adjustment

• The multiplier concept: a change in autonomous (exogenous) spending propagates through the economy via induced consumption

• If autonomous spending rises from A to A + $\Delta A$, the total change in output is amplified by the multiplier:

• $\Delta Y^* = \frac{\Delta A}{1 - c} = k \Delta A \quad \text{where} \quad k = \frac{1}{1 - c}$

• Mechanism: a rise in exogenous demand increases income, which raises consumption by an amount proportional to MPC, causing further increases in income and so on, until the cycle converges

• Investment–Saving link (via the multiplier):

• A change in investment $\Delta I$ leads to a change in income $\Delta Y = \frac{\Delta I}{1 - c}$ and a corresponding change in saving $\Delta S = (1 - c)\Delta Y = \Delta I$

• Principle of effective demand: the rise in income just suffices to generate an equal rise in saving, given the marginal propensity to save; i.e., saving adjusts to investment through income changes

• Practical interpretation: in equilibrium, saving and investment are equal, but the path to equilibrium is governed by the induced-demand mechanism, not by automatic price adjustments

Paradox of Thrift

• Core idea: macro-level savings behavior can differ from micro-level intentions

• Keynesian insight: if people attempt to save more at a given income (or if autonomous saving rises while investment is unchanged), aggregate demand falls, leading to a lower equilibrium income and a fall in total saving

• Result: the aggregate saving in the economy may end up unchanged or even decrease if income falls enough

• Comparison: micro-level intuition (saving is virtuous) clashes with macro-level outcome (saving can reduce overall saving in equilibrium due to reduced income and demand)

• Pedagogical point: this paradox challenges the idea that saving always raises national welfare in the short run and underscores the role of aggregate demand in determining output

The Four-Sector Income-Expenditure Model (Add Government and External Sectors)

• Extends the two-sector model by adding government sector (G, taxes T) and external sector (exports X and imports M)

• Equilibrium condition becomes PAE = Y with leakage terms from taxation and imports

• Structure of expenditure: $PAE = C + I_p + G + X - M$

• Consumption with government and external links:

• $C = C_d = a + c(Y - T) = a + cY - cT$

• Endogenous tax and import equations (as opposed to exogenous constants):

• Taxes: $T = tY \text{ (tax rate on income)}$

• Imports: $M = mY \text{ (propensity to import; import function is proportional to income)}$

• Substituting gives: $C = a + cY - c(tY) = a + c(1 - t)Y$ and $M = mY$

• The open-economy PAE becomes: $PAE = a + c(1 - t)Y + I_p + G + X - mY$

• Equilibrium income (solving Y = PAE):

• $Y^* = \frac{a + I_p + G + X}{1 - c(1 - t) - m}$

• Denominator interpretation: the multiplier is reduced relative to the closed-economy case due to leakages from taxation (t) and imports (m)

• Graphical implications of endogenous leakages: In the PAE-Y diagram, the slope of the PAE line is now $c(1 - t) - m$. Since t > 0 and m > 0, this slope is flatter than in the two-sector model (where the slope was just $c$). A flatter PAE line means a smaller multiplier, as income changes less for a given change in autonomous expenditure.

• Special cases and intuition:

  • If taxation is fully exogenous and non-endogenous, the denominator reduces to $1 - c - m$, recovering a simpler fixed-tax intuition

  • If both taxes and imports are endogenous (T and M depend on Y), the multiplier is smaller because more income leaks out of the expenditure loop at higher Y

• Endogenous vs exogenous leakage discussion:

  • Endogenous leakage ($T = tY$, $M = mY$) makes the multiplier depend on the tax rate and import propensity, dampening the stimulus impact of autonomous spending

  • The expression above demonstrates the adjusted multiplier: $k = \frac{1}{1 - c(1 - t) - m}$

Path to Equilibrium and Practical Implications

• The disequilibrium process remains: if PAE > Y, unplanned inventories fall (or stocks $\downarrow$), firms hire more, output increases; if PAE < Y, stocks rise, output falls toward equilibrium

• In the four-sector model with endogenous leakages, the speed and magnitude of the adjustment depend on the size of the multiplier and the intensity of leakages

• Conceptual takeaway: adding government and foreign sectors reduces the multiplier due to additional leakages, but the mechanism of effective demand still operates—output adjusts until planned expenditure equals actual output

Key Formulas and Numerical Illustrations

• Simple two-sector model (exogenous taxes):

  • $C<em>d = C</em>d^e + c(Y - T)$

  • $C_d^e = a - cT \text{ (autonomous component)}$

  • $Y^* = \frac{C<em>d^e + I</em>p}{1 - c} = \frac{a - cT + I_p}{1 - c}$

  • Multiplier (two-sector): $k = \frac{1}{1 - c}$

  • Change in income from a change in autonomous demand: $\Delta Y^* = \frac{\Delta A}{1 - c}$

  • Change in saving corresponding to a change in investment (identity in equilibrium):

  • $\Delta S = (1 - c) \Delta Y = \Delta I \quad\Rightarrow\quad \Delta Y = \frac{\Delta I}{1 - c}$

• Four-sector model (endogenous taxes and imports):

  • $Y^* = \frac{a + I_p + G + X}{1 - c(1 - t) - m}$

  • Where: $T = tY, \quad M = mY, \quad C = a + c(1 - t)Y$

• Example (illustrative numbers):

  • Suppose in the two-sector model: MPC $c = 0.8$, autonomous component $a = 20$, exogenous investment $I_p = 40$, and T = 0 (for simplicity)

  • Then $Y^* = \frac{20 + 40}{1 - 0.8} = \frac{60}{0.2} = 300$

  • Multiplier $k = \frac{1}{1 - 0.8} = 5$, so a rise in autonomous spending by 60 would yield $\Delta Y^* = 60 \times 5 = 300$

• Open economy example with endogenous leakages: suppose $t = 0.2, m = 0.1, c = 0.8$

  • Multiplier: $k = \frac{1}{1 - 0.8(1 - 0.2) - 0.1} = \frac{1}{1 - 0.64 - 0.1} = \frac{1}{0.26} \approx 3.85$

  • Endogenous tax/import leakages reduce the multiplier from 5 to about 3.85 in this setup

Conceptual and Practical Implications

• The Keynesian framework emphasizes demand-side explanations for output and employment fluctuations, especially in short-run, fixed-price contexts

• The multiplier shows how initial shifts in autonomous spending create broader income effects through induced consumption

• The paradox of thrift highlights potential macroeconomic instability from savings-seeking behavior when investment and demand are not sufficiently responsive

• The four-sector model illustrates how government policy and international trade interact with domestic demand, and how fiscal/foreign leaky channels can dampen the impact of fiscal stimulus

• Policy relevance: under fixed-price assumptions, expansionary fiscal policy can raise output and employment, but the size of the impact depends on the MPC and the extent of leakages (taxes and imports)

• Ethical and practical implications: in a recession, policy makers may rely on fiscal stimulus to boost demand; however, excessive emphasis on saving could unintentionally shrink aggregate demand and worsen unemployment (a macroeconomic “paradox”), underscoring the interdependence of private behavior and public policy

Connections to Foundations and Real-World Relevance

• Links to foundational macroeconomics concepts: demand-determined output, investment–saving identity, and the adjustment process through inventory changes

• Real-world relevance: the model underpins Keynesian-inspired fiscal stabilization policies, open-economy considerations (trading partners, exchange rates), and the study of how taxes and imports influence the effectiveness of policy

• Ethical/policy considerations: circular flow of income, distributional effects of fiscal policy, and the trade-offs between short-run stabilization and long-run growth