# IS–LM Model

## Introduction: The IS–LM Model in Two-Sector Economy

It's interesting that Keynes had developed his product market theories in isolation of the product market, and money market theories in isolation of the product market, whereas the activities and variables of the two sectors are interrelated, interdependent and interactive.

Therefore, changes in the variables of one sector affect the activities of the other sector. In simple words, changes in product market affect the money market equilibrium and *vice versa*.

The Keynesian theory ignores the effect of changes in the money market on the product market and the effect of changes in the product market on the money market.

Therefore, his theories related to the product and money markets are considered to be partial and incomplete. It was J.R. Hicks who highlighted this fact and developed his own model in 1937—just one year after the publication of Keynes’s *The General Theory.* Opens in new window

He integrated Keynesian theories of product and money markets to show how equilibrium of both the sectors coincide at the same level of income and interest rate. His model is widely known as the *IS–LM* model.

In this model, the term *IS* represents the product sector equilibrium condition (*I* = *S*) and the term *LM* represents the money market equilibrium condition (*L* = *M*), where *L* stands for *liquidity preference* or money demand (*M*_{D}) and *M* stands for money supply (*M*_{S}).

It is important to note here that Hicks has developed his *IS–LM* model in the framework of a simple, two-sector economy. The economists have, however, extended his model to three-sector and four-sector models also.

For our purpose, we elaborate on the Hicksian *IS–LM* model in a simple, two-sector economy, including household and firm sectors only. The three-sector and four-sector *IS–LM* models are subsequent entries.

We begin by showing the interdependence of the product and the money markets.

## The Interdependence of Product and Money Markets

As mentioned above, working of the product and the money markets is interlinked and interdependent. The two most important variables that interlink the working of the two sectors are investment and interest rate.

The investment (*I* ) is a money-market variable determined by the demand for and supply of money.

Let us now look at the interdependence of the product and money markets in a simple economy model.

### Dependence of Product Market on Money Market

The product market attains its equilibrium at the level where *Y* = *C* + *I*. Referring back to the Keynesian analysis of the product-market equilibrium, *I* was assumed to be a constant factor or an autonomously or exogenously determined variable.

In reality, however, *I* is not only autonomously or exogenously determined: it is determined within the system also by the level of income and the interest rate. More importantly, given the income, investment (*I* ) depends on the rate of interest.

Assuming a constant ∆*I*/∆*i*, the inverse relationship between the investment (*I*) and the interest rate (*i*) is stated by a linear investment function^{2} of the following form.

*I* = *Ī* – *hi,* (*h* > *0*) --- (eqn. 1)

where, *Ī* = ‘autonomous investment,’ *i* = interest rate and *h* = ∆*I*/∆*i*.

The implication of the investment function in the interdependence of the product and money markets can be shown as follows.

Recall that the product market is in equilibrium where *Y* = *C* + *I*

Here, *C* (consumption) is the function of income, and *I* (investment) is the function of interest.

For the sake of brevity, let us denote consumption function as *C*(*Y*) and investment function as *I*(*i* ). By substitution, the product market equilibrium condition can be rewritten as *Y* = *C* (*Y* ) + *I*(*i* ) --- (eqn. 2)

Eqn. 2 implies that unless *i* (interest) is determined, *I* cannot be determined and unless *I* is determined *Y* cannot be determined. It means, unless *i* is determined, the equilibrium level of *Y* cannot be determined.

Also, the interest rate (*i*) is determined in the money market and equilibrium rate of interest is determined where *M*_{d} = *M*_{s}. For interest rate to remain stable, money market must be in a stable equilibrium. It may thus be concluded that

*Unless money market reaches its equilibrium and interest rate (i) is determined, product market cannot attain its equilibrium.*

This shows the dependence of the product market on the money market.

### Dependence of Money Market on Product Market

Let us now look at the dependence of the money market on the product market.

In the Keynesian system, money market reaches its equilibrium where*M*_{s} = *M*_{d}

and interest rate (*i*) is determined where *M*_{S} = *M*_{D}.*M*_{d} = *M*_{t} + *M*_{sp}, where *M*_{t} = *kY* and *M*_{sp} = ƒ(*i* ).

Therefore, *M*_{d} = *kY* + ƒ(*i* ) --- (eqn. 3)

Eqn. 3 implies that unless *Y* is determined, *kY* cannot be determined and, therefore, *M*_{d} cannot be determined.

And, unless *M*_{d} is determines, money market equilibrium cannot be determined and interest rate (*i* ) would not be determined. It may thus be concluded that

*Unless product market reaches its equilibrium and Y is known, money market cannot reach its equilibrium. *

This shows the dependence of the money market on the product market.

It needs to be emphasized here that *unless both product and money markets reach equilibrium simultaneously, the economy Opens in new window cannot attain its general equilibrium nor can any of the two sectors be in equilibrium.*

## The IS–LM Model: An Elementary Exposition

After having shown theoretically the interdependence of the product and money markets, we move on to the to present Hick’s *IS–LM* model and show how product and money markets interact to reach their equilibrium simultaneously and also the same level of income and interest rate.

The *IS–LM* model combines the equilibrium conditions of the product and money markets to arrive at the general equilibrium. In order to show general equilibrium, Hicks had derived two curves, namely *IS* and *LM* curves.

Note that the term ‘general equilibrium’ is used to denote the simultaneous equilibrium of all elements of the economy including individual products, individual decision-makers (households, firms, labor, etc.), money market, at both micro and macro levels.

However, here the term ‘general equilibrium’ has been used to denote the simultaneous equilibrium of the product and money markets.

The *IS* curve (meaning *I* = *S*) represents the money market equilibrium, both at different rates of interest and level of the aggregate product or national income.

In deriving his

*IS*curve, Hicks made an important deviation from the Keynesian approach.

- While Keynes assumed investment (
*I*) to be autonomous and determined exogenously, Hicks assumes that*I*is determined endogenously and is the function of the rate of interest i.e.,*I*= ƒ(*i*). - Likewise, his
*LM*-schedule shows the equilibrium path of the money market at different rates of interest and levels of income.

He has then combined the two schedules to show the general equilibrium of the economy. This has come to be widely known as the *IS-LM* model.

## The Basic Model

In his *IS-LM* model, in its simplest form, Hicks integrates the equilibrium conditions of the product and money markets and produces conditions for the general equilibrium.

He incorporates a money market variable, interest (*i*), into the income determination model by replacing Keynes’ constant *I* with the investment function.

With these modifications in the Keynesian model Opens in new window, Hicks defines the equilibrium level of income as:

*Y* = *C* (*Y* ) + *I* (*i* )

This function yields the *IS*-curve. It shows the relationship between *Y* and *i* at different equilibrium levels of saving and investment (*I* = *S* ) and the product market equilibrium at different levels of *Y* and *i*.

Similarly, the *IS-LM* model incorporates income, *Y,* the main product-market variable, in the money market model by linking total demand for money to *Y*.

This is done by using an *M*_{t}-function of the form *M*_{t} = *kY,* and an *M*_{sp}-function as *M*_{sp} = *L* (*i* ) into the money market model. The money-market equilibrium condition is then written as *M*_{s} = *M*_{d} = *kY* + *L* (*i* ) --- (eqn. 4)

Eqn. 4 yields the *LM*-schedule which shows the relationship between *Y* and *i* at different equilibrium levels of *M*_{d} and *M*_{s}. It shows also the money market equilibrium at different levels of *Y* and *i*.

Finally, the *IS-LM* model brings the *IS* and *LM* functions together and lays down the condition for the general equilibrium as:

*IS* = *LM*

or *C* (*Y* ) + *I* (*i* ) = *kY* + *L* (*i* ) ---- (eqn. 5)

If *C* (*Y* ), *I* (*i* ) functions are known, the equilibrium values of *Y* and *i* can be easily obtained.