We present a simple IS LM model with two markets: the goods market and the “money” market. The goods market is describe in a previous post We linearise the model so that we have a very simple solution, graphically straightforward.

The equations are the following

\[Y^d = \alpha \cdot (y-t) + C_{\pi} + I_y \cdot y + I_{D} \cdot \frac{D}{p} + I_r \cdot r + \bar{I} + g\]With \(Y^d\) global demand, \(\alpha\) the share of income going to workers, \(y\) the income / production, \(t\) taxes, \(C_{\pi}\) an autonomous level of consumption (assumed to come from “capitalists” in our previous post), \(I_y\), \(I_D\) and \(I_r\) the sensibility of investment to income, real debts and the interest rate, \(\bar{I}\) the level of autonomous investment and \(g\) public expenditures. \(\frac{D}{p}\) is the level of real debts, nominal debts are fixed here to \(2000\). The introduction of real debts in our investment function allows us to capture at least some part of the argument made by Irving Fisher in 1933, that is, the adverse effect of deflation on investment (see Fisher, 1933).

The equilibrium condition is simply that

\[Y^d = y\]The demand for money is:

\[M^d = p(L_y \cdot y + L_r \cdot r)\]Where \(M^d\) is the demand for money,\(p\) is the price level, \(L_y\) and \(L_r\) are the sensibilities of the demand for money to income and interest rate. The money supply is \(M^s\) and real money supply is denoted \(\bar{M}\), our equilibrium condition is that \(\frac{M^d}{p} = \bar{M}\).

From these two equations and the equilibrium conditions we can deduce the equations of IS and LM.

\[\begin{equation} r = \frac{(1-\alpha - I_y) \cdot y + \alpha t - C_{\pi} - I_D \frac{D}{p} - \bar{I} -g}{I_r} \tag{IS} \end{equation}\] \[\begin{equation} r = \frac{1}{L_r}(\frac{\bar{M}}{p} - L_y y) \tag{LM} \end{equation}\]Because \(I_r\) is assumed negative (investment evolves inversely to the interest rate), the condition for IS to be downward sloping in the \((y,r)\) plane is that \(1-\alpha - I_y > 0\), that is \(1- \alpha > I_y\), the saving function has a higher slope than the investment function. For LM, \(L_y\) is assumed positive, a higher level of income increases the demand for money, while \(L_r\) is negative, because an increase in the interest rate will mean that bonds will have a greater yield and the opportunity cost of holding money goes up. This means that LM will be upward sloping, except that the interest rate cannot go below some value \(r_{min}\) (liquidity trap).

To find the equilibrium solution we can reorganize these two equations so that the two unknowns are on the left:

\[\begin{align} (1-\alpha-I_y) \cdot y - I_r \cdot r &= -\alpha t + C_{\pi} + I_{D} \cdot \frac{D}{p} + \bar{I} + g\\\\ L_y \cdot y + L_r \cdot r &= \frac{\bar{M}}{p} \end{align}\]The solutions for \(r\) and \(y\) will be:

\[\begin{align} y* &= \frac{(-\alpha t + C_{\pi} + I_{D} \cdot \frac{D}{p} + \bar{I} + g)L_r+I_r \frac{\bar{M}}{p}}{L_r(1-\alpha-I_y) + I_r L_y} \\\\ r* &= \frac{\frac{\bar{M}}{p}(1-\alpha-I_y) - L_y(-\alpha t + C_{\pi} + I_{D} \cdot \frac{D}{p} + \bar{I} + g)}{L_r(1-\alpha-I_y) + I_r L_y} \end{align}\]The following application allows the user to change parameter values and observe the impact of this change on the equilibrium of the economy. A “shock” can be added on one of the parameters to keep the original parameters on the graph.

For instance, a shock on \(g\), public expenditures, will increase income as well as the interest rate, creating an “eviction effect” of private investment. A second shock, an increase on \(M^s\) for instance, will mitigate the effect of the rise in the interest rate and a higher level of income will be obtained.

A shock on prices will impact both curves if \(I_D\) is different from 0 (if real debts have an effect on investment). At first, a decrease in prices will tend to move the equilibrium toward the right and bottom of the plane, but once the interest rate is at the level of the liquidity trap, income will rapidly decrease.