Income and Spending

Posted on April 2, 2015

Agreegate demand is the total amount of goods demanded in the economy. One can write aggregate demand as \(AD = C+I+G+NX\), where \(C\) is consumption, \(I\) is investment, \(G\) is government spending and \(NX\) is net exports.
At equilibrium level, the quantity of output produces is equal to the aggregate demand, i.e., \(Y=AD\).
When aggregate demand—the amount people want to buy—is not equal to the output, there is unplanned inventory investment or dis-investment. We call \(IU = Y-AD\) is the unplanned additions to the inventory. Clearly, if output is greater than aggregate demand the value of \(IU\) is greater than zero and vice-versa. If excess inventories accumulates, the firms cuts down on production until the aggregate demand and output is equal.

Consumption function and Aggregate demand

The relationship between consumption and income is described by consumption function.
We assume that the consumption demand increases with the level of income and hence the equation \(C=\bar{C} + cY\), \(\bar{C}>0\) and \(0<c<1\).
The variable \(\bar{C}\) is the intercept and \(c\) is the slope. The remaining fraction, i.e., \(1-c\) is saved.
The coefficient \(c\) is called marginal propensity to consume and \(1-c\) is called marginal propensity to save.

We assume that investment is \(\bar{I}\), government spending is \(\bar{G}\), taxes are \(\bar{TA}\), transfers are \(\bar{TR}\), and net exports are \(\bar{NX}\). The consumption now depends upon disposable income. Thus

\[ \begin{align} YD & = Y - TA + TR \\ C & = \bar{C} + cYD = \bar{C}+c(Y+TR-TA) \end{align}\]

Aggregate demand is the sum of consumption function \(C\), Investment, government spending and net exports.

\[ \begin{align} AD & = C+ I + G + NX \\ &= [\bar{C}-c(\bar{TA}-\bar{TR}) + \bar{I} + \bar{G} + \bar{NX}] + cY \\ &= \bar{A}+cY \end{align}\]

Here \(\bar{A}\) is independent of the level of income, i.e., autonomous. We also refer this as autonomous spending. (Note: here \(I\) does not represent income, rather, it is investment.)

At equilibrium, the output is equal to the aggregate demand, i.e., \(Y= AD\) and we denote the value of \(Y\) at equilibrium as \(Y_0\). One can easily show that \(Y_0 = \frac{1}{1-c} \bar{A}\), where \(c\) is the marginal propensity to consume.
We see that the equilibrium level of output is higher, when the marginal propensity to consume is higher and also when the autonomous spending is higher.
The book says that the change in autonomous spending and the change in output can be equated by the following relation \(\Delta Y = \frac{1}{1-c} \Delta{\bar{A}}\).

When there is no government and no foreign trade, in equilibrium, planned investment is equal to savings.
The above statement can also be phrased in the following way, when there is no government or foreign investment, income is either spent or saved, \(Y = C + S\). Also the aggregate demand is equal to Consumption plus Investment, \(Y = C + I\), which implies that \(I = S\).
If we include government and foreign trade, then income can either be saved, spent or used to pay taxes and hence \(Y = C + S + TA - TR\) and complete aggregate demand is \(Y = C + I + G + NX\) (the National Income accounting). Thus, we see that:

\[\begin{align} C + I + G + NX & = C + S + TA - TR \\ I & = S + (TA-TR-G) + NX \\ \end{align}\]

The term \((TA-TR-G)\) is called budget surplus and it’s negative is called budget deficit.

One can show that the equilibrium output changes when the autonomous aggregate demand increases by \(1\) unit and that \(\Delta AD = \frac1{1-c} \Delta \bar{A} = \Delta Y_0\).
The multiplier is the amount by which equilibrium output changes when autonomous aggregate demand increases by \(1\) unit.
We use the Greek letter \(\alpha\) to denote multiplier, where \(\alpha = \frac{1}{1-c}\), one can see that the larger the marginal propensity to consume, larger the value of multiplier.
Multiplier is useful is developing an explanation of fluctuations of output.
The multiplier suggests that output changes when autonomous spending (including investment) changes and the change in output can be larger than the change in autonomous spending.
The textbook mentions that multiplier is necessarily greater than one in a very simplified model (the model we are studying), but there do exist cases where the multiplier is less than one.