IS-LM model

The IS/LM model is a mathematical model developed by John Hick and Alvin Hansen in order to describe macroeconomic ideas proposed by John Maynard Keynes in its General Theory of Employment, Interest and Money.

The model presents simultaneously two curves, IS and LM, in a graph in which the horizontal axis represents national income (Y) and the vertical axis represents the interest rate (i).

• IS illustrates the relationship between the level of income (Y) and the interest rate (i) in the real sector (goods and services). Lower is the interest rate higher is investment and thus the income.
• LM describes the relationship between the level of income and the interest rate in the monetary market. Higher is the level of income, higher is the need for money and thus the interest rate.

The intersection of the curves represents the general equilibrium for which both relationships are verified, that is to say the actual equilibrium of the economy.

The IS/LM model is the most common macroeconomic model. It enables to understand easily the relationships between the real sector and the monetary market as well as the effects of monetary and fiscal policies.

The model

Equilibriums in the real market : IS curve

“IS” stands for “investment and saving”. The IS curve represents all equilibriums for which total spending (consumption and investment) equals total output (income). This equation is equivalent to the equality between investment and saving, which is a basic accounting equation in a closed economy.

$Y=C+I$

Investment is a decreasing function of interest rate. Indeed, the interest rate represents the opportunity cost of an investment. Higher is the interest rate, higher must be the return on investment. It implies that when the interest rate is high many investments whose profitability would be low are not undertaken.

$Y=C+I(i)$

I is a decreasing function of i, and so is Y. Then IS is a decreasing curve.

Equilibriums in the monetary market : LM curve

“LM” refers to “Liquidity preference and money supply”. The LM curve represents all equilibriums where demand equals supply on the monetary market.

According to Keynes, people hold liquidities for two reasons.

• Citizens need cash for every day life transactions and because of uncertainty (they are cautious). Keynes calls those needs “transactions demand for money”. It is positively related to income (Y).
• When people want to save they must choose between securities (financial markets) and money (bank). Money is considered as an asset among others and whose price is the interest rate. When the interest rate seems too low, speculators expect it to increase and thus buy money (by selling financial assets), when they think interest rate are too high, speculators expect it to drop and sell money (by buying financial assets).

So according to Keynes, demand for money includes transactions demand (L1) and speculative demand (L2). In equilibrium we have the following relationship.

${\frac {M_{o}}{p}}=L_{1}(Y)+L_{2}(i)\Leftrightarrow L_{2}(i)={\frac {M_{o}}{p}}-L_{1}(Y)$

And since

${\begin{cases}L_{2}(i){\text{ is a decreasing function of }}L_{1}(Y)\\L_{2}{\text{ is a decreasing function of }}i\\L_{1}{\text{ is a increasing function of }}Y\end{cases}}\Leftrightarrow i{\text{ is positively related to }}Y$

then LM is a increasing curve.

General equilibrium

The general equilibrium is given by the intersection of IS and LM.