🔢 The modeling of Keynesian macroeconomics

Economic formalization of the Keynesian model

Some neoclassical inheritance

With the Keynesian model, there are fewer graphical representations, but equations remain important.

As we have seen, Keynes was not a huge modeler, so most of the mathematical formalizations came after his death. John Hicks led this effort.

Keynes was still influenced by neoclassicals. We still have $Y=C+I$ and $Y=C+S$ (where $S$ = Saving, which empowers $I$ = Investment). But, there is an inverse causality.

• For neoclassicals: $Y\to C\text{ and }I$.
• For Keynes: $C\text{ and }I\to Y$.

So, there is a choice between Consumption and Savings. Also, for Keynes, $I\ne S$, as we will see in the investment section.

The propensity to consume

Keynes shows that households have a different average propensity to consume (propension moyenne à consommer). This can be computed with $c=\frac{C}{Y}$.

The richer you are, the smaller the part of your revenue consumed. So, you save more. The marginal propensity to consume is computed as $c_m=\frac{\partial C}{\partial Y}$.

The consumption function is:

$$C=c\cdot Y+\bar{C}$$

with $C_0$ being the autonomous consumption (minimal consumption needed to survive).

The 45-degree diagram

The 45-degree diagram (also called the "Keynesian Cross") is a graphical formulation by Paul Samuelson, one of the most important Neo-Keynesians. With John Hicks, they built the Synthesis, a work linking neoclassical and Keynesian research.

This graph represents when $Y=Z$, i.e., when aggregated production is equal to global demand.

• Before the equilibrium point $Y^{\ast }$, there is under-production (Demand > Supply).
• After $Y^{\ast }$, there is over-production (Supply > Demand).

$Y^{\ast }$ should lead to full employment ($Y_{FE}$), but it often does not. As we saw with Walras's Law, if one market is in equilibrium, others might not be. Also, some unemployment is inevitable, such as due to people moving or geographical mismatches, which is called Structural Unemployment.

These reasons are not the same as for Neoclassicals.

Keynesian multiplier’s computation

Example case: No State

Without the State, the equilibrium can be computed with $Y=C+I$, which gives:

$$Y=\frac{1}{1-c}(\bar{C}+I)$$

Here, the multiplier $k=\frac{1}{1-c}$ is always higher than one (since $0<c<1$). So, any increase in investment will trigger a beneficial circle of increased production, limited by the Fundamental Psychological Law.

Fundamental Psychological Law

Keynes’ Fundamental Psychological Law states that as income rises, people increase their consumption, but by less than the increase in income (i.e., the marginal propensity to consume is positive but $<1$).

This behavior gives the Keynesian multiplier its force: because each round of spending is only partially re-spent, the initial increase in demand creates a multiplied—yet finite—expansion of total income.

Open Economy

In an open economy (with the State), the model can be enriched:

$$Y=\frac{1}{1-c+m+t}(\bar{C}+I+G+X)$$

The value of the multiplier $\frac{1}{1-c+m+t}$ is not always superior to 1 (due to imports $m$ and taxes $t$). Thus, we need to look closely at any state intervention to be sure it has a beneficial impact overall.

In these two systems, Investment is a central part that empowers the rest.

Application: Document 4

1. Given $C=0.8Y+580$ (so $c=0.8$ & $\bar{C}=580$) and $I=600,G=700$.

   $$Y=C+I+G$$$$Y=0.8Y+580+600+700$$$$Y-0.8Y=1880$$$$0.2Y=1880$$$$Y=\frac{1880}{0.2}=9400$$$$C=0.8\times 9400+580=7520+580=8100$$

2. If $Y=8900$ and $C=7700$: $\mathrm{\Delta }G=-100⟹\mathrm{\Delta }Y=k\cdot \mathrm{\Delta }G=5\cdot (-100)=-500$. New $Y=9400-500=8900$. $\mathrm{\Delta }C=c\cdot \mathrm{\Delta }Y=0.8\cdot (-500)=-400$. New $C=8100-400=7700$.

Investment drivers

The importance of profitability

We need to acknowledge the difference bewteen Profits vs profitability. For classics and neoclassic, Profit ($\pi$) drive production. A firm is sick of profit, and will do anything they can to maximize their profits. To achieve this goal, you can :

• Minimize the Cost Function $C(q)=q^2$
• Maximize the Output $p\times q$ So, the profit function can be computed as followed:

$$\pi =p\times q-C(q)$$

One small example with p = 10 and q = 1000: $\pi =10,000-1,000,000=-90,000$ Or, C(q) is driver by Savings (= Investments thanks to Say's Law).

He consider that capitalists will decided the level of employment that will maximize their profit according to the goods and services demand. In the short run equilibrium, their observations are right. Once their done their computations, all productions on any market do depends on demand. But this demand is from revenues that are from production. Demand determine production which determine demand. When you hire new workers to produce, you also create a source of revenue that increase the global demand. Capitalists, on goods & services markets, will se their profits rise as long as their paid the workers. Investment is not coming from savings, but more that an anticipated profit (noted ${\pi }^e$, powers serves only as notations)

We are going to a concept of profitaiblity

The Tobin's Q is a law to see if they can anticipate if an investment will be interesting or not. Its computed as follow :

$$Q=\frac{\text{Equity market value for the firm}}{\text{Replacement value of the fixed capital}}$$

Equity market value for the firm is basically the shares of the firm (actions), so the number of shares times their values. The Replacement value of the fixed capital is the interest rate. If Q is higher than one, then the firm is profitable. If not, then it's not.

Tobin's Q is deeply related to the Keynesian Anticipation. Profitability is what Keynes used to call the "business climate".

BUT, If you have a good profitability, then the market thinks it's fine for you and so create a sort self-fulfilling prophecy.

Eventually, unlike neoclassicals that explain investment by profit, Keynesians explain investment by profitability lead by the anticipation of the market.

The accelerator effect

This other effect didn't grabbed as fame as the Keynesian Multiplicator. The accelerator effect, unlike the multiplicator, will take a look at the supply. We will see how do investment impact supply and then demand.

Note

The idea of the accelerator is previous than the Multiplicator, and also previous from Keynes as it was formalized by Clark in 1917.

We are in a similar reflection to Kalecki. The idea is that the stock of capital K is proportional to the level of production ($\gamma =a\times K$). There is no production output that is not used (every machine is used).

This model is the first to take into account investment, and to compute the investment's accumulation.

$$K_t=v\times {\gamma }_t\text{ with }v=\frac{K}{\gamma }⟺K_{t+1}-K_t=v({\gamma }_{t+1}-{\gamma }_t)=I_t$$

$v$ is supposed to be constant (independent from the period, factor of the economy's structure).

Example

Generally, $0<v<1$, but here, to simplify computations, we will take $v=2$. When $\mathrm{\Delta }\gamma =+10,{\gamma }_{t+1}-{\gamma }_t=10$ so $10\times v=10\times 2=20$

In the simple accelerator model, if production decrease, then investment will decrease too. This model implies an endogenous explanation of the economy.

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Koyck also worked on a similar model, where the demand is not forcibly lead by the demand but more by the past demand.

If the demand falls, inverters will invest less. They will under-use they production capacity, and so have a margin if the economy recovers. So, an increase of demand can lead... to no investment.

-> Past Capital Cost

Investment adapt with a delay. We call that viscosity.

It's computed with $K_t=v\times Y_t^e$, Where $Y^e$ is the expected production.

Summary

Investment is based on past production, past socks and a anticipated demand. How do we analyse this anticipated demand?