Home > Aggregate Demand, Macro > The Keynesian model of AD

## The Keynesian model of AD

If you’ve done A-level you will probably have seen this, it’s pretty much the cornerstone of macro. In the short run, demand determines output. The goods market basically captures the interaction between demand, production and income. In the Keynesian model, demand is made up of consumption, investment, government spending, and net exports:

$Y=C(Y-T)+I(Y,r)+G+(X(Y*,\epsilon )-M(Y, \epsilon)$.

This is a fuller version of the Y=C+I+G+(X-M) that you will have seen at A-level, it just expresses what the components of AD are functions of:

Consumption – depends on disposable income which is basically income minus taxation. There will also be a level of autonomous consumption, which is the level of spending that will take place regardless of disposable income. Consumers will decide to spend a portion of their disposable income and save the rest. The extent to which they spend their disposable income is the marginal propensity to consume. So if we represent autonomous consumption by a and the marginal propensity to consume as b, we can write consumption as $C=a+b(Y-T)$. Here, b is the marginal propensity to consume and (1-b) would be the marginal propensity to save.

Investment – depends positively on income and negatively on the real interest rate, r. The greater the income in the economy, the more firms invest, and the higher the interest rate, the less they invest. They invest less at higher interest rates for two reasons, one because it means if they take out a loan to finance a project, the project will need to earn higher returns to cover the repayments, and two because if they are financing a project out of their own savings, a higher interest rate means a higher opportunity cost for the firm (it could simply save the money and earn the high interest rate).

Government Expenditure – generally treated as exogenous in these models. Remember with G that it doesn’t include transfer payments (eg benefits). In the Keynesian model G basically illustrates the state of fiscal policy.

Exports – depends positively on foreign income, Y*, and negatively on the real exchange rate, $\epsilon$. The richer our trade partners are, the more of our exports they will buy. The stronger the real exchange rate is (ie the stronger our currency is) the less they will buy because our goods will be more expensive for them.

Imports – depends positively on income, and positively on the real exchange rate. The richer our consumers and firms are, the more they import; the stronger our currency is, the more they import because it means foreign goods are relatively cheaper.

To keep things simpler at first we will think about a closed economy so leave net exports out of it…for now. And we will treat I as being fixed and exogenous, because we haven’t yet combined it with a money market equation that will bring r into play.

Our simple closed economy model of Y=C+I+G is $Y=a+b(Y-T)+I+G$

This expands to $Y=a+bY-bT+I+G \Rightarrow Y-bY=a-bT+I+G \Rightarrow Y(1-b)=a-bT+I+G$ so expressing this in terms of Y, $Y=\frac{1}{1-b}(a-bT+I+G)$

Here $\frac{1}{1-b}$ is the Keynesian multiplier. This means that every unit you increase something in the brackets (like a, or G for instance) will increase income by $\frac{1}{1-b}$.

Lets say we increase G by $x$. In the first round of spending, the government has x more to spend on goods in the economy, so this will stimulate production and stimulate output. The x goes into the system as extra income. But the story does not end there, because that x finds its way into the hands of consumers (who will have been workers paid by their firms, who in turn were paid by government for providing the extra goods). Those consumers now have more money to spend, they will save $(1-b)x$ of it and spend the rest, putting $bx$ back into the system. That bx available to spend by consumers, stimulates production and stimulates output, and then it finds its way back into the hands of consumers again. So they save $(1-b)bx$ and spend $b^2 x$ which stimulates more output.

So our first round of spending increased income by $x$, the second round increased it by $bx$, the third round by $b^2 x$. This is basically a geometric series: $\Delta Y=x + bx+b^2 x+b^3 x+...b^n x$

Take out the factor of x, $\Delta Y = x(1 + b+b^2 +b^3 +...b^n)$

Divide by x and multiply by (1-b), $\frac{\Delta Y}{x}(1-b)= (1 + b+b^2 +b^3 +...b^n)-(b+b^2 +b^3 +b^4+...b^{n+1})$ which cancels down to $\frac{\Delta Y}{x}(1-b)= (1 -b^{n+1}) \Rightarrow \Delta Y=\frac{(1 -b^{n+1})}{(1-b)}x$

There won’t be a fixed number of rounds of spending, the injection of money into the economy will just mean there are more and more rounds of spending with gradually less money being put back into the system each time, so this is like making n tend to infinity. As b, the marginal propensity to consume, is a figure between 0 and 1, this will mean as $n \rightarrow \infty, (b^{n+1}) \rightarrow 0$ so $\Delta Y \rightarrow \frac{1}{(1-b)}x$.

That is the multiplier.