In a previous post we constructed the IS-LM model from a demand for goods, a demand for money, and two equilibrium conditions. By adding a very simple liquidity trap, this model was already able to tell us some things about the Keynes effect and the Fisher effect in different situations.

In this model, we will make consumption depend on real balances, just as we did for investment before. This represents the Pigou effect (Pigou, 1943), which is assumed to run inverse to the evolution of prices: a fall in prices will increase real balances and consumption, shifting the IS curve to the right.

The Keynes effect also represents a negative relationship between prices and income, via the interest rate: a fall in prices increases the real money supply and shifts LM to the right. On the other hand, the Fisher effect (Fisher, 1933) impacts the demand for goods negatively through a rise in real debt when prices fall to a lower level. Thus, all other things equal, when prices fall to a lower level, investment decreases and the IS curve is shifted to the left.

When the Pigou effect and the Keynes effect dominate the Fisher effect, the relation between prices and income will be negative. On the contrary, when the Fisher effect dominates, this relation becomes positive and the aggregate demand curve is upward sloping! This situation will arise most likely when the economy is stuck in the liquidity trap: the Keynes effect will be inefficient, a shift of LM to the right after a fall in prices will not increase income.

This is largely in reaction to this idea that Pigou argued in the 1940s that Keynes had not taken into account the effect on real balances that bears his name. The real balance effect was much debated, but for our purposes here we can note that it can still be dominated by the Fisher effect when prices fall too low. This means that with a liquidity trap, our aggregate demand function can have a very odd shape in the (y,p) plane, decreasing for high enough prices, but increasing for a low level of prices.

To see this effect, we start from our previous model, adding in the IS equation a term making consumption depend on real balances (we will use the same \(\frac{D}{p}\) as in the investment function for the Fisher effect). We obtain the following equation for IS:

\[\begin{equation} r = \frac{(1-\alpha - I_y) \cdot y + \alpha t - C_{\pi} \frac{D}{p} - I_D \frac{D}{p} - \bar{I} -g}{I_r} \tag{IS} \end{equation}\]Where \(C_\pi\), previously an autonomous level of consumption, is now a coefficient of sensibility of consumption to real balances, assumed positive, and the other parameters and variables are the same as in the previous post.

We change our LM equation a bit more to obtain a good liquidity trap from which we will derive an aggregate demand curve with the properties explained above. Our equation takes the following form:

\[\begin{equation} r = r_{min} + \frac{r_{max}}{1+e^{\frac{L_y}{L_r}(y-\frac{M^s}{p})}} \tag{LM} \end{equation}\]We have introduced a maximum and a minimum level for the interest rate r and the other parameters remain unchanged. An increase in the real stock of money will shift LM to the right, a change in the coefficient of sensibility of the demand for money to the interest rate (\(L_r\)) or to income (\(L_y\)) will change the slope of the LM curve in its upward sloping part (this equation is a simple logistic curve).

With this equation for LM, it is not so easy to determine the equation for aggregate demand in the (y,p) plane, but this is where graphical techniques are really helpful, and we can easily visualize the form of aggregate demand in the following application, where the parameters will yield different forms for AD.

The strength of the Pigou effect can be adjusted with the value of \(C_\pi\): a higher value will increase its strength and we can see that it does not take much to have a Pigou effect always dominating the Fisher effect. The latter is controlled by \(I_D\): a lower value of this parameter will increase its strength.