Chapter 17 Notes: A Simple Keynesian Model

The Simple Keynesian Model: A Closed Economy (No Government)

Learning framework and core questions

• Equilibrium concept: total income (Y) is at its equilibrium level when it equals aggregate spending (A). In symbols: $Y = A$.

• Equilibrium level of income is the point where spending determines production; production adjusts passively to demand.

• Key symbols:

  • $Y$ = total production, income or output in the economy.

  • $A$ = total spending or aggregate demand (consumption + investment).

  • $Y_f$ = full-employment level of income.

• Historical debate:

  • Say’s Law (supply creates its own demand): aggregate spending always equals aggregate income, and there can never be insufficient demand.

  • Keynes’s view: aggregate demand (A) is the force that determines total production or income (Y).

Basic setup and assumptions (the simple model)

• The economy consists of only two sectors: households and firms.

• Government is absent: no government spending or taxes.

• Foreign sector is absent: no exports, imports, or monetary/exchange effects.

• Prices are given: no inflation analysis in this model.

• Wages are given: no labour market dynamics.

• The money stock and interest rates are given: no monetary policy or financial market dynamics.

• Spending (demand) is the driving force; production (supply) adjusts passively to changes in demand.

• Aggregate demand is the sum of:

  • Consumption spending, $C$

  • Investment spending, $I$

• The model often uses C and I to construct the aggregate demand: $A = C + I$.

Consumption spending

• The consumption function links household consumption expenditure to total income.

• Key properties of consumption:

  • Consumption rises with income (positive relationship).

  • There is autonomous consumption even when income is zero (consumption floor).

  • The increase in consumption is smaller than the increase in income (diminishing marginal propensity to consume).

• The marginal propensity to consume (MPC) is the slope of the consumption function, denoted by $c$, and is the ratio of the change in consumption to the change in income:

  • $c = \dfrac{\Delta C}{\Delta Y}$

• Non-income determinants of consumption (from Box 17-3):

  • Interest rates

  • Expectations

  • Wealth

• The consumption function (as presented) is written as:

  • $C = C + cY$

  • Here, the first term (often denoted 'a' in standard notation) is autonomous consumption; the second term is induced consumption $cY$.

Investment spending

• Aggregate spending in the economy is the sum of consumption and investment: $A = C + I$.

• The level of investment is considered exogenous in this simple model: the investment function is given and does not depend on current income in the basic setup.

• Investment decisions are discussed in context of how investment changes affect overall income (later in the multiplier section).

• A typical illustration shows the investment function as a separate relation, often depicted as a vertical or flat line in the simple model, indicating independence from $Y$.

The simple Keynesian model: closed economy without government

• Two key graphical elements:

  • The aggregate spending (demand) function, $A = C + I$, which shows how spending depends on the income level through C.

  • The 45-degree line, which represents all points where income equals expenditure: $Y = A$.

• Equilibrium occurs at the point where the aggregate spending function intersects the 45-degree line. At this point, $Y = A$ and there is no inherent pressure for the economy to expand or contract.

• In words: equilibrium income is reached where planned spending equals actual output; if spending exceeds output, production/income will rise; if spending is less than output, production/income will fall.

The algebraic (analytical) version of the model

• General expression for the simple closed economy:

  • $Y = C + I$

  • Substituting the consumption function: $Y = a + cY + I$ where $a$ is autonomous consumption and $c$ is the marginal propensity to consume.

  • Solve for Y:

  • $Y - cY = a + I$

  • $(1 - c)Y = a + I$

  • $Y = \dfrac{a + I}{1 - c}$

• Numerical example from the transcript (Box/Illustration):

  • Consumption: $C = 50 + 0.8Y$

  • Investment: $I = 150$

  • Aggregate demand: $A = C + I = 200 + 0.8Y$

  • Equilibrium condition: $Y = A$ gives

  • $Y = 200 + 0.8Y$

  • $0.2Y = 200$

  • $Y = 1000$

• Interpretation: at the equilibrium level, total income equals total spending, and the 45-degree line intersects the aggregate spending function at that income level.

The 45-degree line and equilibrium income (visual intuition)

• The 45-degree line shows all points where income equals expenditure (Y = A).

• Equilibrium occurs where the aggregate spending curve crosses the 45-degree line, confirming that production equals spending.

• The location of the intersection determines the equilibrium income level.

The algebraic version (summary)

• The algebraic form of the simple Keynesian model is expressed as:

  • $Y = a + cY + I$

  • Rearranged: $(1 - c)Y = a + I$

  • Therefore: $Y = \dfrac{a + I}{1 - c}$

• This equation highlights how the equilibrium income depends on autonomous spending (a + I) and the marginal propensity to consume (c).

The investment multiplier and its effects

• Concept: a change in investment spending (ΔI) affects the equilibrium income by more than the initial change due to successive rounds of spending.

• Mechanism (illustrative chain):

  • An investment project injects income into the economy (wages, profits) that is partially spent by recipients, a portion determined by the MPC (c).

  • For example, if MPC (c) = 0.8, the initial dose of spending from a new factory might generate 0.8 of a unit of additional spending per unit of initial investment.

  • Recursively, incomes and spending continue to circulate, creating a multiplied effect on total income.

• Example from the transcript (investment of R1 billion, MPC = 0.8):

  • Initial spending to workers and suppliers leads to an initial income increase of 0.8 × R1 billion = R0.8 billion.

  • The transcript states: "Total spending in the economy will therefore increase by R800 million (ie 0,8 x R1 billion), hence it will 1,8 billion." [Note: This reflects the text’s phrasing; the standard multiplier result yields a larger total income impact than the initial injection.]

• Key formula for the multiplier (in the standard framework):

  • The multiplier is defined as the ratio of the total change in income to the initial change in investment:

  • $k = \dfrac{\Delta Y}{\Delta I} = \dfrac{1}{1 - c}$

  • Therefore, the total change in income is: $\Delta Y = k \Delta I = \dfrac{\Delta I}{1 - c}$

• Worked example from the transcript (alternative numeric illustration):

  • If the economy has the consumption function form $C = a + cY$ with a = 2, I = 2, and a proposed investment change; the equilibrium would be determined by the same structural logic, yielding a higher final level of Y than the initial injection due to the multiplier effect.

• Alternative numerical example in the transcript for a given setting: {

  • Given: $C = 2\text{ million} + 0.6Y, \quad I = 2\text{ million}$

  • Equilibrium would be: via $Y = a + cY + I$ with a = 2, c = 0.6, I = 2 → (1 - 0.6)Y = 4 → 0.4Y = 4 → Y = 10\text{ million}$</p></li><li><p>If investment increases by$\Delta I = 12\text{ million}$, the new equilibrium would adjust by the multiplier amount.</p></li></ul></li><li><p>The remainder of the multiplier discussion is summarized in Table 17-2 and Box 17-8 (multiplier as a geometric series), and Figure 17-10 (multiplier summary).</p></li><li><p>Paradox of thrift is introduced as a related concept in Box 17-9 (how attempts to save more can reduce overall income in the short run).</p></li></ul><h3 collapsed="false" seolevelmigrated="true">Quick recap: relationships and dependencies</h3><ul><li><p>Equilibrium condition:$Y = A = C + I$</p></li><li><p>The consumption function links$C$to income via autonomous consumption and MPC:</p><ul><li><p>$C = a + cY$(as represented in the transcript)</p></li></ul></li><li><p>The investment function is treated as exogenous in the basic model:$I = \text{constant}$(subject to change to study the multiplier).</p></li><li><p>Equilibrium income depends on both autonomous spending (a) and investment (I) and on the propensity to consume (c):</p><ul><li><p>$Y = \dfrac{a + I}{1 - c}$</p></li></ul></li><li><p>The multiplier shows how an initial change in investment propagates through the economy to produce a larger change in total income:$k = \dfrac{1}{1 - c}$.</p></li></ul><h3 collapsed="false" seolevelmigrated="true">Boxed concepts and cross-links</h3><ul><li><p>Say’s Law vs Keynesian view: supply creates its own demand vs aggregate demand determines output.</p></li><li><p>The 45-degree line as a graphical tool to identify equilibrium points.</p></li><li><p>The paradox of thrift: increased saving during a downturn can reduce overall income.</p></li><li><p>Equilibrium in terms of saving and investment is available in Box 17-6 (conceptual link to the saving-investment balance).</p></li></ul><h3 collapsed="false" seolevelmigrated="true">Important macro concepts (as listed in the chapter)</h3><ul><li><p>Macroeconomics</p></li><li><p>Consumption spending</p></li><li><p>Investment spending</p></li><li><p>Aggregate spending (demand)</p></li><li><p>Total production or income</p></li><li><p>Keynesian model</p></li><li><p>Equilibrium</p></li><li><p>Inventories (stocks)</p></li><li><p>Say’s law</p></li><li><p>Consumption function</p></li><li><p>Autonomous consumption</p></li><li><p>Induced consumption</p></li><li><p>Marginal propensity to consume</p></li><li><p>Saving</p></li><li><p>Investment function</p></li><li><p>Excess demand</p></li><li><p>Excess supply</p></li><li><p>45-degree line</p></li><li><p>Equilibrium level of income</p></li><li><p>Multiplier</p></li><li><p>Paradox of thrift</p></li></ul><h3 collapsed="false" seolevelmigrated="true">References to textual figures and boxes</h3><ul><li><p>Figure 17-3: The investment function</p></li><li><p>Figure 17-4: Investment and the level of income</p></li><li><p>Figure 17-5: The aggregate spending function</p></li><li><p>Figure 17-6: The 45-degree line</p></li><li><p>Figure 17-7: The equilibrium level of income</p></li><li><p>Box 17-6: Equilibrium in terms of saving and investment</p></li><li><p>Box 17-8: The multiplier as the sum of a geometric series</p></li><li><p>Box 17-9: The paradox of thrift</p></li></ul><h3 collapsed="false" seolevelmigrated="true">Appendix: compact equations for quick review</h3><ul><li><p>Consumption function (standard form):$C = a + cY$</p></li><li><p>Aggregate demand:$A = C + I$</p></li><li><p>Equilibrium condition:$Y = A$, implies$Y = a + cY + I$</p></li><li><p>Solve for equilibrium income:$ (1 - c)Y = a + I \ Y = \dfrac{a + I}{1 - c} $</p></li><li><p>Investment multiplier:$ k = \dfrac{1}{1 - c},\quad \Delta Y = k \Delta I $</p></li><li><p>In the numeric example: with$a = 50, c = 0.8, I = 150$, the equilibrium is$ Y = 1000 $$.