Introduction¶
- In Part 1, we looked at different ways of accounting for GDP and comparing GDP per capita and its growth rate across countries.
- The Solow growth model made dynamic predictions about income per worker given assumptions about the nature of production, capital, and investment.
- In that model, we focused on the effects of capital accumulation, technological growth, and the savings rate on income per worker over the long run.
- While $Y = C + I$, the growth model was about explaining investment, not consumption.
- In Part 2, we will abstract from capital accumulation and turn our attention to the short run, first focusing on household consumption.
Q: Why Consumption?
- The consumption share of GDP is the largest component, larger than investment, government spending, and net exports.
- The consumption share of GDP has increased over time.
- Of course this is quite different between high and low income countries, but high income countries generally have a large share of consumption to GDP.
Consumption Function¶
- Assume that aggregate consumption is a function of disposable income
$$ C = c_0 + c_1 \times (Y - T) $$
- $Y$ is income, while $T$ is net tax revenue. Thus, $Y-T$ is disposable income available to households.
- Note: $T$ is net tax revenue because some households receive transfers from the government, e.g., Social Security payments and medical expenses paid via Medicare
Parameters
- $c_1$ quantifies the fraction of current disposable income consumed by households
- In the language of micro, we might call it the "consumption elasticity of disposable income"
- In the language of macro, most people call it the "marginal propensity to consume" (MPC)
- If there were low and high income households in the model, which do you think would have a higher average MPC?
- $c_0$ is all consumption that is independent of current disposable income. Imagine if $Y-T = 0$, then
- households could dissave and consume out of their wealth, or
- households could borrow against their wealth or future income to consume.
Aggregate Expenditure¶
- In Ch. 2, the expenditure approach to GDP is $Y = C + I + G + NX$
- In the Solow growth model, we abstract from $G$ and $NX$ and just had $Y = C + I$
- In Part 2, we'll work with a closed economy but consider changes in government spending, so $Y = C + I + G$
- But we need to separate the expenditure (i.e., demand) side apart from the supply side, so let $Y$ be aggregate supply/output and aggregate expenditure is
$$ Z = C + I + G $$
- In equilibrium, $Y = Z$, but there are disequilibria where $Y \neq Z$
Disequilibrium $\rightarrow$ Equilibrium
If $Y > Z$, there is excess supply
- Inventories would accumulate
- Firms would probably decrease production $\downarrow Y$
If $Y < Z$, there is excess expenditure
- Inventories would deplete
- Firms would probably increase production $\uparrow Y$
Either way, there is a tendency for $Y$ to move toward $Z$, and thus $Y = Z$ is an equilibrium.
Endogneous Variables¶
- So far, the model has variables $Y, C, I, G, T$
- We've assumed $C$ is a function of $Y$ and in equilibrium, $Y = C + I + G$, i.e., $Y$ is a function of $C$. These are known as endogenous variables.
- However, we have not assumed functions for $I,G,T$ (yet). They are variables and could change, but they do not explicitly depend on $Y$ or $C$ (yet). These are known as exogenous variables.
- We can also treat parameters as exogenous variables, e.g., in the growth model we considered the effects of an increasing in the savings rate (a parameter).
- For example, we could consider a financial crisis that reduces asset prices a reduction in househould wealth that is reflected by a decrease in $c_0$. Another way to think of that is falling consumer confidence.
The Multiplier¶
- The goods market in equilibrium has 3 equations
- $C = c_0 + c_1 \times (Y - T)$ (a behavioral equation)
- $Z = C + I + G$ (an accounting identity)
- $Y = Z$ (an equilibrium condition)
- The main prediction this model makes is that an exogenous change in expenditure is multiplied into a larger endogenous change in equilibrium income/output.
- That result is known as Income Multiplication (or the Keynesian Multiplier).
Multiplier Example
- Suppose an Infrastructure Spending Bill is passed, or $G \uparrow$ by $\Delta G$
- Aggregate expenditure, $Z$, directly increases by $\Delta G$
- Firms see higher demand for their outputs, e.g., increased orders for steel, cement, and construction equipment and vehicles, so they increase production, $Y \uparrow$ also by $\Delta G$ (on impact)
- Households receive more income and they consume an additional $c_1 \Delta G$
- But $C$ is part of $Z$, so the loop repeats, i.e, the initial change in government spending multiplies into additional income over time.
| Multiplier Step | $\Delta Y$ | $\Delta C$ |
|---|---|---|
| 0 | $\Delta G$ | $c_1\Delta G$ |
| 1 | $c_1\Delta G$ | $c_1^2\Delta G$ |
| 2 | $c_1^2\Delta G$ | $c_1^3\Delta G$ |
| 3 | $c_1^3\Delta G$ | $c_1^4\Delta G$ |
| ... | ... | ... |
Since we've assumed $0 < c_1 < 1$, each additional step is adding a smaller amount to income/output.
Add up the $\Delta Y$ column and we get
$$ \Delta Y = \Delta G + c_1\Delta G +c_1^2\Delta G +c_1^3\Delta G + \cdots \\ = \Delta G (1 + c_1 +c_1^2 +c_1^3 + \cdots) \\ = \frac{\Delta G}{1-c_1} $$
In other words, the multiplication process is represented by a geometric series (see the Wiki on that).
E.g., suppose $\Delta G = \$1T$ and $c_1 = 0.25$, then
$$ \Delta Y = \frac{\Delta G}{1-c_1} = \frac{\$1T}{0.75} = \$1.33T $$
$1/(1-c_1)$ is known as the government spending multiplier in this model.
Equilibrium Output¶
The goods market makes predictions about output/income, $Y$.
Q: How do changes in exogenous variables affect $Y$?
Combining the goods markets equations (imposing that the model is in equilibrium)
$$ Y = c_0 + c_1(Y-T) + I + G $$
Solving for equilibrium output, $Y$
$$ Y = \frac{1}{1-c_1} \left(c_0 - c_1 T + I + G \right) $$
$c_1, c_0, T, I, G$ are all exogenous parameters/variables at the moment.
Other Multipliers
From the solution for equilbrium output, we can see the government spending multiplier (which is the same as the earlier example)
$$ \Delta Y = \frac{1}{1-c_1} \Delta G $$
An exogenous change in investment has the same multiplier
$$ \Delta Y = \frac{1}{1-c_1} \Delta I $$
However, an exogenous change in net taxes has a different multiplier because households first choose consumption vs. savings and a change in net taxes has the opposite effect on consumption (than a change in governement spending).
$$ \Delta Y = \frac{-c_1}{1-c_1} \Delta T $$
Goods Market Graph¶
- The goods market graph will show output, $Y$, vs expenditure, $Z$.
- Blanchard, Macroeconomics (9th Edition, 2025) introduces new notation for a graph of aggregate expenditure and calls it $ZZ$.
- $ZZ$ differentiates the aggregate expenditure function from particular realizations of aggregate expenditure that we label as just $Z$ on the vertical axis.
- If output is below equilibrium, then expenditure exceeds output, $Z_{low} > Y_{low}$.
- In micro language, we would call that a shortage. But in macro, excess expenditure could be met with inventories (i.e., past production).
- Excess expenditure creates an unplanned decrease in inventories, which is an incentive for firms to increase output.
Experiment¶
- Suppose $G\uparrow$ from $G_0$ to $G_1$.
- An exogenous increase in $G$ will shift up the $ZZ$ curve.
- That creates excess expenditure, which depletes inventories.
- Firms increase production until the new equilibrium is reached.
Savings¶
- Define private saving as disposable income not consumed, $S_{private} \equiv Y - T - C$.
- Define public saving as net tax revenue minus government spending, $S_{public} \equiv T - G$
- Note that in a goods market equilibrium $Y = C + I + G$.
- Subtract $T$ from both sides and move $C$ to the left hand side
$$ Y - T - C = I + G - T \\ \rightarrow S_{private} = I - (T - G) \\ \rightarrow I = S_{private} + S_{public} \equiv S_{national} $$
- Thus, in equilibrium investment equals saving.