IS LM Dynamics

IS-LM Model 

-Siddha Raj Bhatta 

The IS-LM framework developed by J.R. Hicks in 1937, just one year after Keynes published his general theory to show at least theoretically how the real market(goods market) and money market attain their equilibrium simultaneously at some level of income(Y) and rate on interest(r).

Before establishing the general equilibrium condition, we first derive the IS and LM curve individually.

Derivation of IS Curve:

IS curve or Investment –Savings equality curve is the locus of combinations of rate of interest and level of income that leads to the equilibrium in the real market or product market. The product market is in equilibrium when aggregate demand equals aggregate supply.i.e.

AS = AD

In a two-sector economy in Keynesian model, AD consists of planned consumption expenditure(C) and planned investment expenditure (I).i.e.

AD=C+I

The total aggregate output is either consumed or saved. Thus

AS=C+S

Thus for equilibrium

AS=AD

Or, C+I=C+S

Or, I=S

This implies that equality of saving and investment gives the goods market equilibrium.

Using the goods market equilibrium framework, we derive the IS curve using the following four-panel figure (fig.1) where:

Panel A shows investment as an inverse function of rate of interest,

Panel B shows equality between savings and investment,

Panel C shows savings function as a positive function of income level and

Panel D shows the derivation of the IS curve.

To start with, assume a rate of interest r₁ at which the level of investment is I₁. For goods market to be in equilibrium, I₁=S₁ and for S₁ level of savings, income must be maintained at Y₁ level of income in panel C. Thus taking r₁ rate of interest and Y₁ level of income, we get a combination G of Y and r at which goods market is in equilibrium. Again, consider a lower rate of interest r₂ (r₂<r₁) at which investment would be higher. i.e. I₂>I₁. For equilibrium, S₂ = I₂ and for  S₂ level of savings, income must be maintained at Y₂ level(Y₂>Y₁). Thus, we get another combination H which leads to goods market equilibrium. Joining the points like G and H, we get a downward sloping IS curve.

Slope of IS Curve

The IS curve is negatively sloped. This is so because a lower rate of interest means a higher investment and for S=I, a higher level of savings must be maintained for keeping the goods market in equilibrium and vice versa. To show this, consider point E in fig. 2.1 panel D at which rate of interest is r₂ at which investment is I₂. But income level at point E is only Y₁, which gives S₁ level of savings only. Thus, obviously at point E, S<I, which is a disequilibrium point. Again, at point F, income is Y₂ which gives S₂ level of savings but the rate of interest is r₁ which means I₁ level of investment, thus clearly, at point F, S>I, which is also a disequilibrium point.

Therefore, if we had drawn the IS curves through the points like E and F, G and H or E and H, it would not passed through the points out of which at least one point would have been disequilibrium points. Thus, we cannot draw a horizontal IS curve, a vertical IS curve or a negatively sloped IS curve. This leads to the conclusion that IS curve is always negatively sloped.

 

Derivation of LM Curve

  The LM curve is the locus of combinations of rate of interest and level of income that leads to money market equilibrium.

Money market is in equilibrium when demand for money (M_d) equals supply of money (M_s).i.e.

L=

Hicks has taken the Keynesian money demand function, which is:

L=L_t + L_sp

Where,

 L_t=transaction and precautionary demand for money and is an increasing function of level of income. i.e.=f(Y); f’_y>0

L_sp= speculative demand for moneywhich is an inverse function of rate of interest.i.e.

L_sp=f(r); f’_r<0

Money supply is taken an exogenously determined by the central monetary authority. i.e.

M_s =

Money market equilibrium is given by

=L_t+L_sp

Following this equilibrium condition, we can derive the LM curve by using the following four-panel fig. (fig.2.2) where:

Panel A shows the speculative demand for money curve as an inverse function of the rate of interest;

Panel B shows the money market equilibrium where given the money supply curve ab with slope 1 (oa=ob), total money supply is divided between L_t and L_sp;

 Panel C shows the transaction demand for money as an increasing function of the level of income and

Panel D shows the derivation of LM curve.

Consider, initially, a given rate of interest as r₀ where speculative demand for money is L_sp0 and given the money supply , in panel B, transaction demand should be L_t0 so that money market remains in equilibrium. For transaction demand to be L_t0, the level of income must be maintained at the level of Y₀. Thus, we get a combination E(Y₀,r₀) in panel D that leads to the equilibrium of the money market.

Let us now assume that interest rate reduces to r₁ level where speculative demand for money is higher (L_sp1) and given the money supply, transaction demand should be L_t1 for maintaining money market equilibrium. The corresponding level of income for this is Y₁, which provides us another combination F that maintains money market equilibrium.

 Generating the points like E and F and by joining them, we get an upward sloping LM curve as shown in fig. 2.2.

It is worth noting that at r_m level of interest rate, the LM curve is horizontal. This is due to the liquidity trap phenomenon present in the speculative demand for money curve in panel D. The perfectly elastic part of L_sp curve makes the LM curve in that range to have such a shape.

Shape of LM Curve

LM curve is positively sloped because at a lower interest rate, there is higher speculative demand for money. This implies that given the money supply, transaction demand for money at a lower interest rate should be lower and accordingly income should be low and vice versa.

To illustrate this argument, consider a point G in panel D fig. 2.2 where rate of interest is r₀ and level of income is OY₁. Here, higher interest rate means lower speculative demand for money or lower income level means lower transaction demand for money. Thus, at point G, money demand is less than money supply, which is clearly a disequilibrium point.

Again considering the point H, where a lower interest rate r₁ implies a higher speculative demand for money and higher level of income means a higher transaction demand for money. Thus, at point H, money demand is greater than money supply, which is again a disequilibrium point.

From this, we can argue that LM curve cannot be a vertical, a horizontal or a downward sloping line/curve. Also,

·         At any point to the left of the LM curve, money supply is greater than money demand and

·         At any point to the right of the LM curve, money demand is greater than money supply.

 

 General Equilibrium

  When we introduce the IS and LM curves together, we obtain a rate of interest and level of income that leads to equilibrium in both money and goods market.

 

In fig 2.3, The IS curve and LM curve have intersected at point e, which has determined the equilibrium rate of interest as r₀ and equilibrium level of income as Y₀.

Is Point ‘e’ a Stable Equilibrium Point?

The general equilibrium point ‘e’ is always stable. If the economy is at a point off the equilibrium point e, there are automatic forces of adjustment, which eventually lead the economy to the equilibrium point e.

To examine this, consider fig. 2.4  below.

At disequilibrium situation A, since it lies to the left of the LM curve M_d<M_s and since it lies to the right of IS curve, S>I. Therefore, excess money supply pulls the interest rate downwards and low investment pulls income level downwards. In such a situation, the economy takes the middle path and reaches point B. At point B, real(goods) market is in equilibrium as point B lies on the IS curve but money market is not in equilibrium as point B lies to the left of the LM curve. At point B, M_s>M_d and thus excess money supply pulls interest rate downwards and the economy reaches point C.

At point C, M_s>M_d since it lies to the left of the Lm curve and S<I as it lies to the right of the IS curve. There are thus forces pulling down the interest rate and pushing up the level of income. In such a situation, the economy takes a middle path and reaches the general equilibrium point ‘e’.

In this way, whenever the economy is at a point of disequilibrium, either in the goods market or in the money market or both it ultimately reaches the equilibrium point ‘e’. Thus, the equilibrium point ‘e’ is always stable.

The path to be followed by this mechanism can be shown in fig. 2.5.

The dotted line shows the trend line that the economy is likely to take from disequilibrium to equilibrium.

The disequilibrium spaces are divided into four segments where:

Space/segment	Goods Market	Money Market
I	S>I	Ms>Md
II	S<I	Ms>Md
III	S<I	Ms<Md
IV	S>I	Ms<Md

 Thus, from any of these disequilibrium conditions, the economy moves to the equilibrium level of income and interest rate that leads to the general equilibrium in the economy.

Elasticity of LM Curve and Effectiveness of Fiscal Policy

Actually, it is the elasticity of LM curve that determines to what degree the fiscal policy is effective to increase the level of output. In general, LM curve is made up of three segments: the horizontal part, upward-sloping part and vertical part as shown in fig. 2.6.

At the minimum rate of interest, LM curve is perfectly elastic to the interest rate. This range is called Keynesian Range or Liquidity trap range (ab segment). The bc part of the LM curve which is elastic is called intermediate range and the vertical part which is completely elastic to interest rate is called Classical Range.

a)      Keynesian Range:

In this part, an expansionary fiscal policy (increase in govt. expenditure or reduction in tax resulting in the rightward shift of IS curve from IS₁ to IS₂ leads to increase in output from Y₁ to Y₂ without a rise in interest rate. This is because the size of the fiscal multiplier will be equal to Keynesian multiplier and there is no crowding out effect in the economy. So, the fiscal policy is completely effective. This justifies the Keynesian prescription of govt. intervention in case of liquidity trap.

b)      The Classical Range

In this range, the LM curve is perfectly inelastic and is thus vertical. Here, an expansionary fiscal policy leading to a shift in IS curve from IS₇ to IS₈ leads to a rise in interest rate only without any increase in output. Thus, fiscal policy is all-ineffective in this range. This justifies the Classicists’ view of the non-intervention of the government.

c)       Intermediate Range

This range can be divided two segments:

·         When we are nearer to Keynesian range, a shift in IS curve from IS₃ to IS₄ due to an expansionary fiscal policy results in  a rise in both interest rate and income by Y₃Y₄ but the increase in income is greater than the increase in interest rate r₃r₄. Though, there will be the crowding out effect, the fiscal policy is said to be more effective.

·         When we are nearer to the classical range, a shift in IS curve from results in an increase in income as well as interest rate but the increase in interest rate in greater than the increase in income. Thus, in this part, fiscal policy is said to be less effective.

Thus, in summary:

If the economy is in the Keynesian range, fiscal policy is completely effective.

If the economy is in the classical range, fiscal policy is completely ineffective.

If the economy is in the intermediate range, the more effective will be the fiscal policy the more nearer it is to the Keynesian range and the less effective will be the fiscal policy the more nearer it is to the Classical range.

Elasticity of IS Curve and Policy Effectiveness

The elasticity of IS curve determines the effectiveness of monitory policy in obtaining its objective of increasing the level of output/income. The steeper the IS curve, the less effective is the monitory policy. The role of IS curve in determining the effectiveness of the monetary policy can be illustrated with the help of fig. 2.7 below.

 In fig. 2.7, LM₀ is the initial LM curve. When an expansionary monetary policy is introduced with an objective of increasing the level of output, the LM curve shifts from LM₀ to LM₁. In such a scenario, in the Keynesian range, where the LM curve is perfectly elastic, the monetary policy cannot bring any increase in income/output and is thus completely ineffective. Here, the elasticity of IS curve does not play any role in determining the effectiveness of monetary policy. In such a case, the additional money supply is kept as speculative balances such that transaction demand as well as the level of income remains the same.

 In the classical range where LM curve is vertical or perfectly inelastic, the monetary policy is completely effective because it raises the level of income. Here also, the elasticity of IS curve does not play any role in determining the effectiveness of monetary policy if we neglect the effect on interest rate.

Finally, in the intermediate range, if the economy is closer to Keynesian range, the less effective is the monetary policy and closer the economy to the classical range, the more effective is the monetary policy in raising the level of income. Further, the elasticity of the IS curve also does play a vital role in determining the increase in income level and the effectiveness of the monetary policy. As we clearly see in the figure, a flatter IS curve increases income by higher amount and increases the effectiveness of monetary policy in the intermediate range and vice versa.

 

Change in Fiscal Policy Instrument and IS LM Framework

Generally, the shift in IS curve is due to the fiscal policy factor because they are related to the real sector (production sector) of the economy. We can broadly divide the changes in fiscal policy instruments in two categories:

 Changes following expansionary fiscal policy and

 Changes following the contractionary fiscal policy

The former policy includes the policy actions like increase in govt expenditure, increasing the autonomous level of investment, lowering the rate of tax, increasing the subsidies, restricting imports, policy measures to increase the exports, policy measures to reduce compulsory and voluntary savings, early payment of public debt, etc. The latter includes  reduction in govt. expenditure, reduction in the autonomous level of investment, raising the rate of tax, cutting the subsidies, freeing imports, policy measures to increase in compulsory and voluntary savings, postponement of payment of public debt, etc. The overall impact of an expansionary fiscal policy is a rightward shift of the IS curve and the overall impact of the contractionary fiscal policy is a leftward shift in the IS curve.

Let us examine what happens in our simple two-sector economy when autonomous investment is increased. (Though there is nothing like a fiscal policy in two-sector economy, we are limiting to this one to make our analysis simple. It can be generalized to three and four sector economies).

Consider fig. 2.8, where the investment curve in panel A has shifted rightwards parallelly due to increase in autonomous investment. Due to this shift, now, at r₀, rate of interest, investment will be higher equal to I₁’.Due to this higher investment, a higher level of savings should be maintained and for this purpose, a higher level of income must be maintained. This shows that in this new situation, a higher level of income is needed to restore equilibrium in the goods market at the same rate of interest r₀ as before. That is why new IS curve should pass through point F in fig 2.8 not through E. This shift in IS curve also shifts the general equilibrium point of the economy from E to G as such rate of interest in the economy increases from r₀ to r₁ and level of income increases from Y₀ to Y₁ only. This increase in interest rate is due to the fact that more money is devoted to autonomous investment by withdrawing from speculative balances and speculative balances and rate of interest rate are inversely related. Increasing interest rate discourages the investment in the economy and does not let the full impact of increase in investment to realize in the economy. This phenomenon is popularly termed as Crowding out effect in the sense that if govt. raises its expenditure, it raises the rate of interest and reduces the private sector investment. This happens only if the monetary policy does not respond to this phenomenon by increasing the level of money supply or when fiscal policy is  working alone.

 

Change in Monetary Policy Instruments and IS LM Framework

The shift in LM curve is due to the monetary policy factors because they are related to the monetary sector of the economy. We can broadly divide the changes in monetary policy instruments in two categories:

§  Changes following expansionary monetary policy and

§  Changes following the contractionary monetary policy

The former policy includes the policy actions like increase in the level of money supply, reducing the rate of interest, easing the availability of credit, etc. It is also called loose monetary policy. This objective is often achieved by reducing the CRR, bank rate, refinance rate, open market operations and any other indirect methods. The latter includes reduction in the level of money supply, raising the level of interest rate, tightening the availability of credit, etc by using the policy instruments. The overall impact of an expansionary monetary policy is a rightward shift of the LM curve and the overall impact of the contractionary monetary policy is a leftward shift in the LM curve.

To analyze what the implications of  change in monetary policy instruments do have in the economy, let us consider an increase in money supply in terms of fig.  2.9 below where the initial level of money supply curve ab (oa=ob) shifts now to a’b’ (ob’=oa’).

In fig. 2.9, the initial level of equilibrium is ay r₀ rate of interest and Y₀ level of income. Supposing that the level of money supply has increased from oa=ob to oa’=ob’.  The level of speculative demand for money L_sp is same at L_sp0. Now, to keep equilibrium in the money market, the transaction demand has to increase from L_t0 to L_t1 thus necessitating an increase in the level of income because transaction demand is positively related with the level of income. Thus, in this new situation, the r₀ level of interest rate is associated with a higher level of income Y₁’ to keep money market in equilibrium and the LM passes through the point F or it shifts rightwards. Due to this rightward shift of the LM curve, the general equilibrium point of the economy also shifts from E to G reducing the rate of interest and increasing the level of income. Though, there is no crowding out effect in this case, the increase in the level of income is rather small. This shows that if monetary policy works alone, it will not be very much effective in increasing the level of output/income.

This follows that an expansionary monetary policy leads to an increase in the level of income/output with a fall in the interest rate. The same logic applies in analyzing the implications of a contractionary monetary policy.   

Simultaneous Changes in the Instruments of Monetary and Fiscal Policy

When both fiscal policy and monetary policy instruments are changed, both IS and LM curves shift together. In this case, it can be seen that increase in income will be far larger than the case of change in the instruments of a single policy. To illustrate this argument, consider fig. 2.10 where expansionary fiscal policy has been followed by expansionary monetary policy.

Due to expansionary fiscal policy, the IS curve shifts from IS₁ to IS₂ generating the crowding out effect due to the increase in interest rate from r₁ to r₂. When expansionary monetary policy follows this situation, LM curve shifts from LM₁ to LM₂ reducing the interest rate to r₁ and increasing income further to Y₁. The total increase in income will be Y₀Y₁ with the stable interest rate and no crowding out effect. This shows that compatibility and coordination between monetary and fiscal policies is essential for stability and growth.