# IS-LM: Concept and Derivations

## Introduction

The IS-LM model is a macroeconomic framework used to analyze the interactions between the goods market and the financial market in an economy. It helps to understand how changes in the interest rate and aggregate demand affect output and employment in the short run.

IS Curve: The IS (Investment-Savings) curve shows the relationship between the interest rate and output (real GDP) in the goods market. The IS curve slopes downward because lower interest rates stimulate investment and increase aggregate demand, leading to higher output and employment. Conversely, higher interest rates reduce investment and aggregate demand, causing a decrease in output and employment.

LM Curve: The LM (Liquidity-Money) curve represents the relationship between the interest rate and output in the financial market. The LM curve slopes upward because higher interest rates lead to an increase in the demand for money, reducing the supply of loanable funds and causing a rise in interest rates. Conversely, lower interest rates increase the supply of loanable funds and reduce the demand for money, causing a fall in interest rates.

## Some underlying assumptions

Fixed price level: The model assumes a fixed price level in the short run.
Closed economy: The model assumes a closed economy where there are no international trade or capital flows.
Simplified fiscal policy: The model assumes that changes in government spending or taxes are the only fiscal policy tools.
Fixed money supply: The model assumes that the money supply is fixed by the central bank.
Liquidity trap: The model assumes that interest rates cannot fall below a certain level, known as the "zero lower bound," due to the possibility of individuals holding onto cash instead of investing.

## Mathematical derivation

The IS Curve:

Investment function: $I = \alpha - \beta r---(i)$

Saving function: $S = -\gamma + \lambda Y---(ii)$

Good market equilibrium: I = S

Here, I = Investment; S = Saving; Y = Income; $\beta$=Interest elasticity of investment; $\lambda$=MPS; $\alpha$ and $\gamma$ are intercepts of the investment function and saving function respectively.

Equation (i) represents an investment function. Investment and rate of interest have a negative relationship, so $\beta$ has a negative sign. Equation (ii) represents a saving function. Savings have a positive relationship with the level of income, so $\lambda$ has a positive sign. Consumers save after a certain level of income only, so $\gamma$ is negative.

Now,

I = S

$\alpha - \beta r$ = $-\gamma + \lambda Y$

$r=\frac{\gamma+\alpha}{\beta}-\frac{\lambda}{\beta}Y---(1)$

Equation (1) represents the downward sloping IS curve or function as $-\frac{\lambda}{\beta}<0$.

The slope of IS curve is $\frac{\lambda}{\beta}$.

The intercept of IS curve is $\frac{\gamma+\alpha}{\beta}$.

The slope of IS Curve: The slope of IS curve depends upon MPS and the Interest elasticity of Investment. As MPS increases, the slope of IS curve increases eventually steepening the IS curve. Likewise, as the interest elasticity of investment increases, the slope of IS curve decreases eventually flattening the IS curve.

The shift in IS Curve: The shift in IS curve depends upon the intercept and slope of the Investment function and the intercept of the saving function as the intercept depends upon $\alpha$,$\beta$, and $\gamma$.

The LM Curve:

Speculative demand for money: $M_{SP}=\alpha-\beta r---(iii)$

Transaction and precautionary demand for money: $M_{TP}=\gamma+\lambda Y---(iv)$

Money demand function ($M_D$) = $M_{SP}+M_{TP}$

Money supply function ($M_S$)= $\bar{M_S}$

Money market equilibrium: $M_S = M_D$

Here, r = Rate of interest; Y = Income; $\beta$ = Interest elasticity of money; $\lambda$ = Income elasticity of money

Equation (iii) is speculative demand for money. Speculative demand for money and rate of interest have a negative relationship, so $\beta$ has a negative sign. Why is the relationship negative? The answer is simple. The value of bonds and market interest rates have a negative relationship, that is, $\text{Value of Bond} = \frac{\text{Coupon Interest}}{\text{Market interest rate}}$. When market interest rates are high, the value of bonds is low implying low speculative demand for money. Conversely, low market interest rates suggest higher bond value implying high speculative demand for money.

Equation (iv) is Transaction and Precautionary demand for money. As the level of income increases, the transaction as well as the precautionary demand for money increases. So, $\lambda$ has a positive sign.

Likewise, the money supply is exogenously determined by the central bank or monetary authority. So, we assumed it as constant.

Now,

$M_S = M_D$

$\bar{M_S}=M_{SP}+M_{TP}$

$\bar{M_S}=\alpha-\beta r + \gamma+\lambda Y$

$r = \frac{\alpha + \gamma - \bar{M_S}}{\beta}+\frac{\lambda}{\beta} Y---(2)$

Equation (2) is the LM curve or function. The LM curve slopes upward as $\frac{\lambda}{\beta} >0$.

The slope of LM curve is $\frac{\lambda}{\beta}$.

The intercept of LM curve is $\frac{\alpha + \gamma - \bar{M_S}}{\beta}$.

The slope of LM curve: The slope of LM curve depends upon income elasticity and interest elasticity of money.

The shift in LM curve: LM curve shifts due to a change in the intercept of either speculative demand for money or transaction and precautionary demand for money or both.

## The intersection of IS and LM Curves

The intersection of the IS and LM curves represents the combination of interest rates and output that is consistent with both market clearing in the goods market and the money market. This combination of interest rates and output represents the short-run equilibrium in the economy.

Equating equations (1) and (2), we get the equilibrium level of the rate of interest and national income.

$\frac{\gamma+\alpha}{\beta}-\frac{\lambda}{\beta}Y= \frac{\alpha + \gamma - \bar{M_S}}{\beta}+\frac{\lambda}{\beta} Y$

$Y^{*} = \frac{\bar{M_S}}{\lambda}$

Substituting the value of $Y^{*}$ in Equation (2), we get

$r^{*}=\frac{\alpha+\gamma-2\bar{M_S}}{\beta}$

$Y^*$ and $r^*$ are the combinations of national income and rate of interest, respectively, that are consistent with both market clearing in the goods market and the money market.

## Conclusion

The IS-LM model is a useful tool for analyzing the impact of monetary and fiscal policy on the economy. For example, an expansionary monetary policy that increases the money supply will reduce interest rates and shift the LM curve to the right. This will lead to an increase in output and employment, as the intersection of the IS and LM curves shifts to a higher output level. Similarly, an expansionary fiscal policy that increases government spending will shift the IS curve to the right, leading to higher output and employment.

It's important to note that the IS-LM model has its limitations and assumes that prices are fixed in the short run. It also ignores important interactions between the goods and financial markets, as well as the role of expectations and the behavior of individuals and firms in the economy. Despite these limitations, the IS-LM model remains a useful starting point for understanding the interactions between the goods market and the financial market in the short run.