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## The Keynesian Cross in the Open Economy

January 25, 2012 Comments off

The Keynesian Cross showed the relationship between planned expenditure and actual expenditure.

It was based on the Keynesian model of aggregate demand:

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:

In the context of the open economy we can expand our understanding of the Keynesian Cross by considering specifically the effect of exports and imports. Increasing exports adds to aggregate demand, whilst increasing imports decreases aggregate demand. So aggregate demand can be best thought of as the demand for domestic goods rather than the domestic demand for goods. That is, aggregate demand in an economy depends on the demand (both at home and abroad) for domestic produced goods, rather than the domestic demand for goods, some of which will be spent on buying imports that counts towards another economy’s aggregate demand.

We can therefore define three forms of demand:
DD – Domestic demand for goods (C + I + G + M)
AA – Domestic demand for domestic goods (C + I + G)
ZZ – Demand for domestic goods (C + I + G + Z)

Here we start with purple line DD, the domestic demand for goods, and subtract imports to get AA, the domestic demand for goods. Then we add exports, the equivalent of a vertical shift upwards because the demand for exports is exogenous and does not depend on domestic income like the other components of AD that increase with domestic income do. This gives us our ZZ line, the demand for domestic goods.

Now we can consider the implications of the DD and ZZ line, and whether it means we have a trade deficit or trade surplus. The ‘planned expenditure’ line in the Keynesian Cross corresponds to the ZZ line, really it is the ‘planned expenditure on domestic goods’.

The equilibrium comes at Y1, where the ZZ line intersects the 45 degree line. This is where demand for goods, Z, equals income, Y. But at this point, the DD curve is above the ZZ curve. This means the domestic demand for goods is higher than the demand for domestic goods, ie our economy’s citizens are demanding more goods than the demand for goods they produce. This means they must be importing more than they are exporting, and we have a trade deficit.

Again the equilibrium comes at Y1, where the ZZ line intersects the 45 degree line. This is where demand for goods, Z, equals income, Y. But at this point, the DD curve is below the ZZ curve. This means the domestic demand for goods is lower than the demand for domestic goods, ie our economy’s citizens are demanding fewer goods than the demand for goods they produce. This means they must be exporting more than they are exporting, and we have a trade surplus.

Categories: Aggregate Demand, Macro

## Monetary policy and the AD relation

The AD relation showed a downward sloping relationship between output and prices. The mechanism was really working through real money balances $\frac{M}{P}$, ie when prices rose, if the nominal money stock, M, stayed constant, then real money balances fell.

The opposite of that is what happens if (in the short run) prices stay constant and the money stock rises. Then real money balances will rise, so the analysis used for the AD relation would suggest that output would rise. If prices are constant then increasing the money stock represents a shift up of the LM curve, while decreasing the money stock represents a shift down of the LM curve. This is a simple way to think about monetary policy, a monetary expansion is increasing the money stock, a monetary contraction is decreasing the money stock.

Remember what influences the AD curve:

The AD curve tells you the equilibrium level of output in the goods and money markets, for any given price level. Anything which will change the equilibrium level of output at all price levels, will shift the whole AD curve. So you can basically think of this as anything ‘happening’ in the ISLM model which will cause the equilibrium level of output to change apart from a change in the price level, which is captured in the downward slope of the AD line.

So the AD curve shifts out to the right, if you have a fiscal expansion or a monetary expansion, or anything that will increase consumer confidence to encourage consumers to spend more (which means the IS curve shifts out just like the fiscal expansion)

The AD curve shifts in to the left, if you have a fiscal contraction or a monetary contraction, or anything that will decrease consumer confidence to cause consumers to spend less (which means the IS curve shifts in just like the fiscal contraction)

So we can express a relationship between output and monetary and fiscal policy like this: $Y_t = \gamma[\frac{M_t}{P_t}, G_t, T_t]$.

This is saying output is equal to a parameter multiplied by a function of monetary and fiscal policy. Now if we want to isolate the way monetary policy influences output, keep fiscal policy constant and forget about G and T.

$Y_t = \gamma[\frac{M_t}{P_t}]$.

In terms of growth you can use an approximation that if $Y_t = \gamma \frac{M_t}{P_t}$ then $g_{yt} \approx g_{mt} - g_{pt}$. Note that growth in prices in year t is inflation in year t, so $g_{yt} \approx g_{mt} - \pi_t$.

We can rearrange this to say $\pi_t \approx g_{mt} - g_{yt}$. This is a useful way of thinking about the relationship between inflation and money growth, it says that inflation in year 1 will be approximately equal to the rate of money growth minus the rate of output growth. This is basically because if the economy grows, there will be a growing level of transactions, and so a growing demand for real money. The real money stock always grows at the same rate that output grows. As the real money stock is $\frac{M}{P}$ then if M grows at a different rate to output, the difference will be made up by changes in P. If the rate of money growth matches the rate of output growth you will have stable prices and no inflation. This sounds deceptively easy and in reality inflation is easy to stop, if you turn the taps of money growth off, inflation will soon judder to a halt, the reason that isn’t done in practice is that that approach would have a seriously detrimental effect on output as it would mean there was not enough money around to support the level of transactions demanded, so the lack of money would trigger a recession.

## The AD relation

The AD relation basically shows us how the price level affects output. It is downward sloping, which tells us when prices are higher, output is lower, when prices are lower, output his higher.

This process can basically be explained through the ISLM model, particularly because the price changes take effect through the LM curve.

Start by going back to the model of money supply and money demand.

We analyse the effect of prices, through the vertical money supply curve. Remember that we are really thinking of money supply and money demand in terms of real money balances, $\frac{M}{P}$. The Central Bank controls the amount of money in the economy, M, but to know what this is worth in terms of real money balances, you have to take into account the price level as well. When prices rise, (M/P) falls so the vertical money supply curve shifts to the left. When prices fall, (M/P) rises so the vertical money supply curve shifts to the right.

If the amount of money in the economy, M, is changing at the same time as prices, then the directon the money supply curve will shift depends on what happens to the overall value of (M/P), for instance if M is rising, but P is rising faster, then (M/P) will be getting smaller, so the supply curve goes left.

Lets think of the effect of higher prices on output. P rising means the money supply curve shifts left (assuming M is fixed or P is rising faster than M). Shifting the money supply curve left while money demand is held constant, means the equilibrium interest rate will be higher.

The LM curve tells us how output affects the interest rate that will bring the money market into equilibrium. If the money supply curve has shifted left in the money supply/money demand diagram, this means that we have a higher equilibrium interest rate full stop, ie at all levels of output the equilibrium interest rate will be higher than it was before. This is basically shifting up the LM curve.

So an increase in prices means a rise in the equilibrium interest rate at all levels of output (as shown by the money supply/ money demand diagram), and translates into shifting up the LM curve.

Here is the effect in the ISLM model:

All other things being equal, the rise in prices has shifted up the LM curve, which has meant the equilibrium interest rate that brings the money market into equilibrium is higher at all levels of output. Given that the IS curve is unchanged, it means the money and goods markets will come into equilibrium at a higher interest rate. On the IS curve, a higher interest rate means that we will have a lower equilibrium level of output, because higher interest rates mean firms invest less, investment falls, so output falls.

When prices rise and everything else stays the same, output falls.

This is basically the downward sloping AD relation, it says rising prices = falling output.

The AD curve tells you the equilibrium level of output in the goods and money markets, for any given price level. Anything which will change the equilibrium level of output at all price levels, will shift the whole AD curve. So you can basically think of this as anything ‘happening’ in the ISLM model which will cause the equilibrium level of output to change apart from a change in the price level, which is captured in the downward slope of the AD line.

So the AD curve shifts out to the right, if you have a fiscal expansion or a monetary expansion, or anything that will increase consumer confidence to encourage consumers to spend more (which means the IS curve shifts out just like the fiscal expansion)

The AD curve shifts in to the left, if you have a fiscal contraction or a monetary contraction, or anything that will decrease consumer confidence to cause consumers to spend less (which means the IS curve shifts in just like the fiscal contraction)

Categories: Aggregate Demand, Macro

## The Keynesian Cross

You may have seen the Keynesian cross at A-level. This can be a fiddly one to understand at first. The idea behind this is there will be a level of planned expenditure in the economy which is the amount consumers (households), firms, and government plan to spend on goods and services. This is basically the demand for goods and services. When this planned expenditure is higher than firms producing goods expected, then will start depleting their inventory stocks of goods, and this will stimulate more production, they will start hiring more workers and produce more, so income will rise. When the planned expenditure is lower than the firms producing goods expected, then they start building up their inventory stocks with unsold goods, which will cause them to step back on production, lay workers off, and income will fall.

Either way, the goods market comes into equilibrium because firms producing goods will end up adjusting to the overall level of planned expenditure in the economy.

We can express this by expressing the closed economy Keynesian model of AD in terms of ‘demand’ which we will call Z.
$Z=a+b(Y-T)+I+G$. In this equation, C has been replaced by its components, a (autonomous consumpton), and b (marginal propensity to consume). The Keynesian Cross takes I and G and T to be exogenous, they are fixed amounts when you draw the line of planned expenditure (demand), if you change them you have to shift the whole line up or down.

The Keynesian Cross model has demand, Z on the vertical axis and income, Y on the horizontal axis, so the planned expenditure line will be upward sloping (Z increases as Y increases because when incomes are higher, consumers have more disposable income, and so demand more goods, hence Y appears in the equation above). As Y is multiplied by b, then b gives us the slope of the planned expenditure curve. Remember that b is the marginal propensity to consume, so we can say that the marginal propensity to consume will determine how steep the planned expenditure curve is.

The fixed components of the planned expenditure curve, a, T, I and G, determine the height of the curve (where it intercepts the vertical axis. If you increase a, I or G, then you push the curve up. If you increase T (taxation) then you push the curve down – because T has a minus sign in the equation so it affects Z negatively.

On the Keynesian Cross you will also always see the 45 degree line drawn in. This is because of the fact that the goods market always comes into equilibrium, like mentioned above, if planned expenditure (demand) is running ahead of firms’ production, then their existing inventory stocks get depleted and it encourages them to produce more as they know the demand is there to sell their goods. If demand is below firms’ production then they are just accumulating unsold stocks in their inventories, so they scale their production levels down. Eventually we have a situation where planned expenditure (demand) = actual production (output), so demand equals output (or income). So the 45 degree line is just because we have drawn Z on one axis and Y on the other, we know that in equilibrium Z=Y which will be somewhere on this 45 degree line. The actual point where it comes into equilibrium will be where the planned expenditure line crosses the 45 degree line.

Here we have an equilibrium at Y1, where the planned expenditure curve intersects the 45 degree line.

So the key points from the Keynesian Cross model are:

The marginal propensity to consume gives the slope of the planned expenditure curve

The planned expenditure curve shifts up if we increase autonomous consumption, investment or government spending, or if we decrease taxation. If this happens we get a higher level of income at equilibrium.

The planned expenditure curve shifts down if we decrease autonomous consumption, investment or government spending, or if we increase taxation. If this happens we get a lower level of income at equilibrium.

Categories: Aggregate Demand, Macro

## 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.

Categories: Aggregate Demand, Macro