IS - Product Market - Goods Market

Focus

Variation of $r$ and $y$ which equilibriates the goods market

Foundation

From the Keynesian Model, we have (assuming tax as a function of income)

$$\begin{array}{rl}y& =c(y-t(y))+i+g\\ y-c& =s(y-t(y))+t(y)=i+g\end{array}\text{\hspace{1em}(1)(2)}$$

Till now we didn’t discuss what is the investment (we were given, it was exogenous).

But, now we will see that it is not autonomous, rather induced and it is endogenous. Predominantly, it depends on the ROI

$$i=\varphi (r)\text{\hspace{1em}(3)}$$

If the cost of borrowing, $r$ is high

• It will discourage borrowing for investments
• Present Discount Value, PDV for our returns will reduce.

Investors compare $C$ (cost of borrowing) and $\rho$ (rate of returns)

$C$	$\rho$	Decision
108	104	Not economical to invest
104	104	No real difference, worthwhile investment
This means, there is a negative relation between

$$r\downarrow ⟹i\uparrow$$

and thus interest rates are dependent on $r$

$$\begin{array}{c}y=c(y-t(y))+g+i(r)\\ \\ i(r)=s(y)\end{array}\text{\hspace{1em}(4)(5)}$$

Equation (4) describes the pairs of $y$ and $r$ that will maintain the product market equilibrium. $(y,r)$

Equation (5) forms the basis for forming the IS curve. (equilibrium condition)

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Explanation: For every rise in $r$, the level of income needs to fall.¹

• For every rise in $r$, investment falls leading to fall in the level of income for which the saving needs to fall in order to generate equilibrium in the goods market.
• Thus for every rise in $r$, the $y$ needs to fall.
• Thus, the equilibrium pairs of $r$ and $y$ must be negatively sloped.
• Thus, the combinations of $r$ and $y$ which maintain the equilibrium between $i+g$ and $s+t$. Thus,

Definition

The $IS$ curve represents the pairs of $r$ and $y$ which would keep the product market in equilibrium, in the sense that

$$i+g=s+t$$

Derivation

All the relationships discussed so far, that contribute the location of equilibrium pairs of $r$ and $y$, in the product market are summarized in the Four Quadrant (FQ)¹ diagram.

Quadrant	Contents
SE	It reflects the upside-down version of $s+t$ as a function of level of income $y$.
NW	It reflects government spending and investment which is a decreasing function of rate of interest.
SW	It reflects the $45°$ which is used to equate $s+t$ (from the SE Quad.) and $i+g$ (from the NW Quad.) and this is the Static Equilibrium condition for the product market. savings and investments
NE	It reflects the $IS$ curve representing the equilibrium values of $r$ and $y$

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Situation $s\uparrow$ or $r\uparrow$

Savings Increase

When savings increase, then there is a reduce in the demand for goods and services. This leads to a decrease in output.² With lower income, consumers have less disposable income to spend, leading them to spend less on goods and services. This lowers the confidence of companies to invest in newer projects and thus they decide to reduce planned investments. Now, since there is a reduce in investments, the central bank reduces the rate of interest to encourage investments.

$$s\uparrow ⟹y\downarrow ⟹\overline{i}\downarrow ⟹r\downarrow$$

Rate of Interest Increases

When the rate of interest increases, then there is a reduce in the demand for loans. This leads to reduction in planned investments $\overline{i}\downarrow$ by the firms. Thus, production decreases, leading to a fall in the employment reducing the overall income $y\downarrow$. With lower income, the savings of the consumers come down $s\downarrow$.

$$r\uparrow ⟹\overline{i}\downarrow ⟹y\downarrow ⟹s\downarrow$$

The benefit of using the $IS$ curve is the cascading effect that it provides.

Effect of rise in $g$ (expansionary fiscal policy)

The $IS$ curve shifts right due to increase in the government expenditure

• Rise in $g$ from $g_0\to g_1$ would lead to outward shift of the $i+g$ curve in the NW Quad.
• Rise in $g$ would lead to a rise in $y$ through the multiplier process, assuming investment is unchanged. (the green curve shifts left because of $g$ alone.)
• Thus, rise in $g$ from $g_0\to g_1$ would rise the level of income from $y_0$ to $y_1$ at the initial rate of interest $r_0$.
• Since, $r$ is constant and $i$ depends on $r$, the investment is also constant.
• Thus, rise in $g$^[expansionary fiscal policy] shifts the $IS$-curve to the right.

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Doubts³

Footnotes

1. Four quadrant: SE, SW, NW, NE ↩

2. or, This leads to firms and companies reduce production and thus lay off employees, reducing the aggregate income. ↩

3. todo connect why $r$ is not crowded out (and how do we say that it could have been?). Why people don’t want to hold money as money? ↩