# The Keynesian Cross and the IS curve

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In this post, we build the IS curve from the equilibrium positions on the goods market. The curve is later used in several models (see this post and that one). The approach presented here is deliberately simplified, as a first step toward building the IS LM model and integrating different macroeconomic effects.

We start from a demand function made of consumption, investment and public expenditures:

$\begin{equation} Y^d = c(y) + I(y,r) + g \end{equation}$

Keynes specified the consumption function by talking of a “fundamental psychological law” and the famous marginal propensity to consume. Instead of this approach, we would like rather to underline that consumption can be linked to the distribution of income. To understand this, we distinguish two categories of agents in our simplified model:

• the “workers”, who receive income on the form of a salary
• the “capitalists”, who receive the returns from capital, the profits

We make the further hypothesis that the workers will consume all their earnings, while capitalists only consume a fixed part of their profits (this last hypothesis is relaxed in this post where capitalists’ consumption depends on their real wealth). This allows us to write the following consumption function:

$c(y) = \alpha y + C_{\pi}$

Where $$\alpha$$ is the share of income going to workers ($$\alpha y = \frac{W N}{P}$$), while $$C_{\pi}$$ is the fixed consumption of capitalists. This model underlines that a large part of current consumption is dependent on current income, while part of this consumption will be untouched by variations in income. We see that $$\alpha$$ will necessarily be inferior to one (the share of income accrued by workers is inferior to one), a good property to obtain a stable equilibrium. In this simplified model, income is equal to consumption plus savings, thus we have that:

$S = y - c(y) = (1-\alpha) y - C_\pi$

Our investment function is dependent on income and the real interest rate. We suppose that the real interest rate will have a negative impact on investment by increasing the cost of borrowing, while current income will have a positive effect on investment through positive expectations. We assume for now a linear relationship between those variables, which include an autonomous term accounting for the investment that is independent of both income and the real interest rate:

$I(y,r) = I_y y + I_r r + \bar{I}$

Where $$I_y > 0$$, $$I_r < 0$$, that is, the coefficient of sensibility of investment to income is positive, and the coefficient of sensibility of investment to the real interest rate is negative.

Inserting into the first equation, we obtain

$Y^d = \alpha y + C_{\pi} + I_y y + I_r r + \bar{I} + g$

From the equilibrium condition that income is equal to aggregate demand, we can derive the expression of the equilibrium income given a real interest rate (supposed here to be exogenous):

$y^* = \frac{C_\pi + I_r \bar{r} + \bar{I} + g}{1-\alpha-I_y}$

Where $$\bar{r}$$ is a fixed level of the real interest rate, supposed to be exogenous in this model of the goods market. The equation for IS can be derived similarly by isolating $$r$$:

$r = \frac{(1-\alpha-I_y)y-C_\pi-\bar{I}-g}{I_r}$

The condition to obtain a downward sloping IS curve is that $$1-\alpha-I_y>0$$ or alternatively $$1-\alpha>I_y$$, the propensity to save is higher than the propensity to invest, which can be easily visualized on the diagram representing income and investment - savings.

The IS curve can help us visualize the effect of a disequilibrium in the economy. On the right of this curve, we are in a situation of excess supply of goods, and the equilibrium will be reestablished by a fall in the real interest rate, which shifts the demand curve upward on the Keynesian Cross. Symmetrically, a situation of excess demand of goods on the left will be ended with an increase in the interest rate, shifting the demand curve downward because of the fall in investment.

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