Dynamics of Inflation, Unemployment and Money Supply –Explore

“Workhour Haircut”, a new instrument in times of economic crisis

Ramanujam Madhavan

March 20, 2016

1 Introduction

1.1 Objective

The objective of this paper is to theoretically study the effects of governments capping the total workhours put in by employees as a means to combat high unemployment during recession. To explainthis a bit more,consider this scenario: The government enacts a policy where the employers are expectedto reduce the working hours by a small percentage, consequently bringing down pay cheques by thesame proportion. This should bring down the unemployment rate, resulting in better income spread andincreased demand . I call this instrument workhour haircut. We will study the dynamics of workhourhaircut using the mathematical tools in Dynamic System Analysis

1.2 Elaboration of the core idea

The idea is that if the working hours of individual employees are reduced by a certain percentage (hair-cut to work hours and income), the employers would be encouraged to increase the employee headcount(permanent or temporary) to compensate for the loss of productive hours, which will in turn result inreduction in unemployment, better income distribution and increased demand for goods.

Perhaps, the small reduction in income in exchange for time would not hurt households badly, butit could have profound effects on the economy. For example, a 10% decrease in household income neednot necessarily result in reduced consumption, but may only impact savings, as savings is the generaltendency during economic crisis. Moreover, the price fall due to deflation could even alleviate the fallin income to an extent. On the other hand, the increased employment rate would certainly increaseconsumption.

In the early 2000s, Nobel Laureate economist Paul Krugman repeatedly criticized the Bush tax cuts,both before and after they were enacted. Krugman argued that the tax cuts enlarged the budget deficitwithout improving the economy, and that they enriched the wealthy, worsening income distributionin the US. Krugman advocated that lower interest rates, and increased government spending would havea larger stimulus effect. These views of Krugman support the core idea of this paper.

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1.3 Refresher on Economics

Low inflation, low unemployment and steady expansion of GDP are indicators of healthy economy. Re-cession is a business cycle contraction which results in general slowdown in economic activities. Duringrecessions, macroeconomic indicators such as GDP, Investments, Capacity Utilization, Household Income,Profits and Inflation fall, while Unemployment Rate and Bankruptcy rise.

Depression is a sustained long-term downturn in economic activity in one or more economies whichis more severe downturn than the recession.

Governments usually respond to recessions by adapting expansionary macroeconomic policies throughfiscal and monetary policies and tend to increase money supply in the economy. Fiscal policy is aboutgovernments increasing the expenditure and reducing the taxation. Monetary policy is about the Cen-tral Banks cutting down interest rates to encourage investment by borrowing and discourage savings.

When the central banks cut down interest rate to increase the money supply, it may arise a liquiditytrap situation, where the interest rate has gotten close to zero that it is no longer an instrument toencourage spending and the people hold cash since they expect deflation. Pauls Krugman advocatesaggressive fiscal policy and unconventional monetary policy to avoid liquidity trap.

1.4 References

The background economic concepts and equations are borrowed from the book titled Economic Dynam-ics by Roland Shone. The core idea has been built on top by incorporating the necessary parametersand analyzing the dynamics arising out if it.

2 Dynamics of Inflation and Unemployment

2.1 Phillips Curve

Phillips Curve explains an inverse relationship between Unemployment and Inflation backed by ob-servations in many economies. Simply put, decreased unemployment in an economy will correlate withincrease in inflation rates. However, there is a short term inverse correlation between unemployment andinflation, in the long run this correlation does not hold good. The result was that there is natural rateof unemployment which does not impact the inflation.

Phillips curve stipulates a relationship between price inflation, , and unemployment, u, augmentedfor inflationary expectations, πe.

π = f(u) + ξπe 0 < ξ ≤ 1 (1)

This relationship is the expectations-augmented Phillips curve.

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Next we make a simple assumption about expected inflation, namely

π = β(π − πe) β > 0 (2)

When the actual rate of inflation exceeds the expected rate, expectations are revised upwards andwhen the actual rate is below the expected rate, then expectations are revised downwards.

Suppose the government attempts to maintain unemployment at some constant level, u∗. We furthersuppose that they are successful and so f(u) is a constant and known. To establish the implications ofsuch a policy, differentiate the Phillips curve relationship (1) with respect to time under the assumptionthat u = u and substitute in equation (2). Then

π = ξπe = ξβ(π − πe) = β(ξπ − ξπe)

But ξπe = π − f(u∗) and so

π = βf(u∗) − β(1 − ξ)π (3)

which is linear with intercept βf(u∗) and slope −β(1 − ξ). The relationship is illustrated in figure 1

Figure 1: Graph between π versus π

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Let us establish if the fixed point exists π = 0

i.e βf(u∗) − β(1 − ξ)π∗ = 0

π∗ =f(u∗)

(1 − ξ)(4)

Now to study the stability of the fixed point π∗ let us compute the following:

dπ

dπ= −β(1 − ξ)

From above, we infer that if 0 < ξ < 1, then π∗ is asymptotically globally stable. If ξ = 1 the infla-tion is always correctely anticipates and π = πe. In this instance the rate of unemployment is constantregardless of the rate of inflation. This unemployment rate is called the natural rate of unemploymentand demoted by un

The situation is illustrated in the more conventional diagram, in figure 2. However, we can go further.Since π = βf(u) and f ′(u) < 0, if u = u∗ < un then it follows that π > 0; while if u = u∗ > un, thenπ∗ < 0. This implies that if the government maintains the level of unemployment below the natural level(here we ignore u∗ > un) then permanent inflation will be the result. This is because expected inflationalways lags behind actual inflation and the economy is forever trying to catch up with what it observes.

Figure 2: Graph between π versus u

It is common in a number of studies to assume a relationship between inflation, π, and real income,y, and expected inflation, πe. In particular it is common to express this relationship in the form

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π = α(y − yn) + πe α > 0 (5)

where y and yn are in natural logarithms and yn is the natural level of income (the income level as-sociated with un) , and the relationship is referred to as ’the Phillips Curve’. This is not the relationshipbetween inflation and umemployment but embodies two reaction functions.

Following the original formulation of the Phillips curve, we postulate a relationship of the form

π = −γ1(u− un) + πe γ1 > 0 (6)

This is the first reaction function indicating the response of price inflation to the unemployment gap.It implies a specific functional form for f(u).

The second reaction function is a formulation of Okuns law3 and is given by

u− un = −γ2(y − yn) γ2 > 0 (7)

Note: the above equation assumes the standard working hoursSubstituting this into the previous equation we obtain

π = γ1γ2(y − yn) + πe

or

π = α(y − yn) + πe (8)

The above is nothing but equation 5.

2.2 Incorporate the Core Idea

Let us plug in our core idea of rugulating the work hours into the equations. Let ωn be the normalworking hours of employees. And ω is the regulated working hours.

The regulated working hours ω would be in such a way that ω < ωn and (ωn−ω)ωn

in the range of say8-10%.

Now, let us study the impact of the ω on equations 6 and 7. If we make a key assumption that thesmall reduction in income due to ω < ωn will only make a negligible impact on comsumption,since the

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comsumption is inelastic in the short run. Moreover, the saving is the one that takes the hit in times ofsalary cuts. Hence the equation 6 will remain unchanged.

On the other hand, the income takes the direct beating and hence the equation 6 is modified as blow:

u− un = −γ2(ψy − yn) where ψ =

(ω

ωn

)and γ2 > 0 (9)

Hence the modified equation 8 will be as follows after incorporating for ω and ωn

π = α(ψy − yn) + πe (10)

The above equation will be the basis of our analyze in the next section

3 Deflationary death spirals

Let us first understand the deflationary death spiral discussed in the book. We will then experiment tosee if workhour haircut has any positive impact on the dynamics.

By mid-2001, Japan was in a recession and the USA began to experience a serious downturn enoughfor some economists to wonder whether a major deflation worldwide was likely. In explaining such apossibility, interest has returned to the concept of the liquidity trap. The floor of zero on the nominalinterest rate leads to what Groth (1993) has called a dynamic liquidity trap. Here we shall present asimplified version of the model outlined in Groth (1993) and similar to the one utilised by Krugman early2000

The following model is used in the book:

Goods Market

1. c = a+ b(1 − t)y

2. i = i0 − h(r − πe)

3. y = c+ i+ g

Money Market

4. md = ky − ur

5. ms = m− p

6. md = ms

Inflation

7. π = α(yψ − yn) + πe Note: ψ is incorporated

8. ˙πe = β(π − πe)

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Where

c = consumptiony = incomei = investmentr = nominal interest rateπe = expected inflationg = real givernment spendingmd = demand for real money balancesms = supply of real money balancesm = nominal money supplyp = price levelyn = natural level of incomeπ = inflationπe = dπ/dtψ = ω/ωn

ω = Regulated working hourωn = Normal working hour

g, yn and m are assumed constant, as are all autonomous expenditures (a and i0) and all parameters(b,t,h,k,u, α and β). The first six equations are the familiar IS-LM model, equation (7) is the expecta-tions augmented Phillips curve and equation (8) specifies adaptive expectations.

The dynamics of the model is analysed in terms of (ms,πe)-phase space. Hence we need to derive two

equations of the form

ms = f(ms, πe)

πe = g(ms, πe)

From equation (5), and noting m is constant, we have ms = −π and substituting equation (7) intothis we have

ms = −[α(yψ − yn) + πe] (11)

From equation (7) we immediately have π − πe = α(yψ − yn), which on substitution into equation(8), gives

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πe = αβ(yψ − yn) (12)

In order to eliminate income, y, in each dynamic equation, we require to solve the IS-LM componentof the model embedded in equations (1) to (6). Combining (1), (2) and (3), we derive the IS curve

r =

(a+ io + g

h

)+ πe − [1 − b(1 − t)]y

h(13)

From equations (4), (5) and (6) we obtain the LM-curve

r =−ms

u+ky

u(14)

Substituting equation (14) into equation (13), we derive an expression for equilibrium income

y∗ =(a+ i0 + g) + hπe + (h/u)ms

1 − b(1 − t) + (kh/u)(15)

Substituting equation (15) into equation (11), we obtain

ms =

{−α(a+ i0 + g)ψ

1 − b(1 − t) + (kh/u)+αyn

}−{

α(h/u)msψ

1 − b(1 − t) + (kh/u)

}−{

αhψ

1 − b(1 − t) + (kh/u)+1

}πe (16)

Substituting equation (15) into equation (12), we obtain

πe =

{αβ(a+ i0 + g)ψ

1 − b(1 − t) + (kh/u)− αβyn

}+

{αβ(h/u)msψ

1 − b(1 − t) + (kh/u)

}+

{αβhπeψ

1 − b(1 − t) + (kh/u)

}(17)

which is also linear in ms and πe.We shall simplify these linear equation by writing them in the form

ms = A+Bms + Cπe

πe = D + Ems + Fπe

Using these equations we can define the (ms,πe)-phase plane with nullclines ms = 0

and πe = 0. We shall now pursue this model by means of a numerical example.

Consider the model :

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a = 60b = 0.75 t = 0.2i0 = 430 h = 4g = 330k = 0.25 u = 10m = 450 p = 0yn = 2000α = 0.1β= 0.08

Case 1:(The Book version)for ω = ωn and ψ = 1, hence no reduction in workhour

ms = 36 − 0.08ms − 1.8πe

πe = −2.88 + 0.0064ms + 0.064πe

Setting ms = 0 and πe= 0 we derive the two isoclines

ms = 0 πe = 20 − 0.0444ms

πe = 0 πe = 45 − 0.1ms (18)

with fixed point (ms, πe) = (450, 0).

The eigen values for the above system are : λ1 = −0.008 + i0.079699 and λ2 = −0.008 − i0.079699

Figure 3: Phase plane diagram for Case 1

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Initially in Figure 3, we see a stable spiral that moves towards a fixed point where there is near zeroinflation and certain money supply

Here fixed point infer the equilibrium state and the spiral of trajectory refers to the way the economymove toward the fixed point, in other words come out of recession. The more circular the spiral,the moretime it takes to recover. As we clearly notice in the phase plane diagram, it takes several spirals beforethe economy settles to the equilibrium.

In the next few cases, we will vary the workhour in small percentages and analyse it’s impact on thedynamics.

Case 2:In this case we change the value of ψ from 1 to 0.975 or in other words there is 2.5% cut in hour and sothe income of an individual. Hence, we get the equation of (ms,π

e) as below:

ms = 48.875 − 0.09706ms − 1.73πe

πe = −3.91 + 0.007805ms + 0.058976πe

The eigen values for the above system are : λ1 = −0.0019442 + i0.0861153 and λ2 = −0.0019442 −i0.0861153.

Figure 4: Graph represent Case 2

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The figure 4 shows that by small reduction in working hour, trajectory has less number of spiralwhich means that the trajectory move at a more faster rate towards the fixed point. In other words, theeconomy comes out of the recession faster, much less time than in case 1.

Case 3:In this case, we change the value of ψ from 1 to 0.95 and we get the equation of (ms,π

e) as below:

ms = 66.10938 − 0.125ms − 1.65313πe

πe = −5.28875 + 0.01ms + 0.05225πe

The eigen values for the above system are : λ1 = −0.036375 + i0.0931499 and λ2 = −0.036375 −i0.0931499.

Figure 5: Graph represent Case 3

The figure 5 shows that by further small reduction in working hours, the trajectory is moving evenfatser.

Case 4:In this case, we change the value of ψ from 1 to 0.925 ( about 7.5% reduction in workhour) and we getthe equation of (ms,π

e) is given below:

ms = 147.2857 − 0.28571ms − 1.25714πe

πe = −11.7829 + 0.022857ms + 0.020571πe

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The eigen values for the above system are : λ1 = −0.13257 + i0.0726804 and λ2 = −0.0.13257 −i0.0726804.

Figure 6: Graph represent Case 4

Case 4 shows that trajectory almost loses it’s spiral property and convert into stable node. This ishuge for an economy that it recovers in a very short span of time.

We didn’t experiment with further reduction in workhour since the unemployment goes into negativefor the given initial condition of 12%.

4 Loose Ends and Further research

The idea is oversimplified in its implementation in this analysis. Perhaps, in practice the workhourregulation has to be sectoral and category specific based on the study of available workforce, skillset,trainability, and so on. Moreover, the employers need the means to accommodate the increased head-count through shift systems, etc.More research is required to know how people will utilize the extra free time and how does it impactthe economy. For example, worker may take on some part time jobs or spend time caring for children’seducation and so on.The analysis is based on only one set of initial conditions drawn from the book. We need to try withvarious initial conditions based on factual data from economies around the world and repeat the experi-ment.The system modelled here assumes that workhour reduction will better unemployment unemployment

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immediately. In practice, it would take some time and this time lag should be addressed in the model

5 Conclusion

From the above brief analysis, there are strong indications that the work hour regulation as a policy willhave significant impact on the dynamics of economics during periods of sustained recession or depressionswith high unemployment. Given the limited time and effort spent on this experiment (relative to theimplications), we cannot claim any breakthroughs. However, we can definitely lay the need for dedicatedand serious research towards this topic, perhaps the topic of my PhD thesis!

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