Monetary economics

Monetary Economics – Monetarist Model

Monetarist model overview

The monetarist model is composed of two equations:

AD: π_t = m_t - (y_t - y_L)

AS: y_t – y_L = θ(π_t - π_et)

π_t - π_et = γ(y_t – y_L)

AS is derived straightforwardly from the Phillips curve. We wrote both forms used by this model: one in output-gap form but it can also be expressed as a difference in effective and expected inflation, where θ = 1/γ. We will use the latter one. It is positively sloped, giving the standard idea that the inflation rate on the supply side depends on the parameter γ, which represents the market power of trade unions but can also reflect general imperfections in the labor market.

Aggregate demand: AD

In this form, assuming m is the money growth rate (with velocity of circulation constant as zero v_t = 0), the AD is written as a function of the output-gap (y - y_L) and tells us we have null inflation if both the money growth is null and the output-gap is null (output, GDP, or pil are equal to the long-run level, the equilibrium level). Conversely, inflation is positive if either there is positive growth in money supply or a positive output-gap.

Or the inflation rate is exactly equal to the money growth rate if the output-gap is null (and thus if GDP is at its level of long-period equilibrium). In the long run, the output-gap is by definition null, or by assumption null to be more precise (because in the long run we reach the equilibrium state ‘L’), the inflation rate is just like m_t. Moreover, this equation expresses a decreasing relation among π and Y (given m_t and Y_L).

Model equations and expectations

We have two equations and three unknowns (Y_t, π_t, π_et). Therefore, we need expectations information. This model is completed by the ‘extrapolative expectation’ hypothesis: π_et = π_t-1. The inflation expected for time t is the expectation experienced in time t-1. The model also admits a steady state, a situation in which:

• y_t = y_t-1 = y_L
• π_t = π_t-1 = π_L = m_t
• Money is neutral.

Steady state (SS) is the equilibrium situation of this model: expectations are fulfilled and do not change, inflation is constant, output is constant at its long-run level.

Key characteristics of the monetarist model

γ is a structural parameter of the economy, to change it requires structural reforms such as reforming labor, goods, services, or the financial market for a very long period. That’s why it is called a structural parameter: it depends on the structure of the economy, which changes only slowly over time even if you implement reforms, as these reforms change market behavior slowly. It describes features and habits of the economy for each country.

Keynesian models were useful for explaining fluctuations of output with fixed prices, whereas this model is useful for gaining insight into the economy's behavior in the medium period after a shock in monetary growth. To do that, we just lag one period on the AD (we need π_t-1 in order to have π_et).

Lagged aggregate demand and inflation variation

AD (lagged once): π_t-1 = m_t-1 – (y_t-1 – y_L)

We get past inflation rate, past money growth rate, and the past output-gap. Subtracting this equation from the original AD and rearranging:

π_t – π_t-1 = (m_t – m_t-1) – (y_t – y_t-1)

We get an equation that expresses the variation in the inflation rate as the difference in variation in m and y over the period. Through the term in the box, we can now substitute the equation previously obtained into AS since, with extrapolative expectations, π_et = π_t-1:

y_t = y_L + θ[(m_t – m_t-1) – (y_t – y_t-1)]

y_t = y_L + θ(Δm - Δy)

y_t = y_L + θΔm – θy_t-1 + θy

y_t(1 + θ) = y_L + θΔm + θy_t-1 divide by (1+θ)

y_t = (1-a)y_L + ay_t-1 + aΔm, where a = θ/(1+θ) and (1-a) = 1/(1+θ)

Solution of the model

Output at time t depends on, or is equal to, a weighted average of the natural level of output, output at t-1, and a differential between money growth rates. So, money is not neutral in the medium run; money growth is in that formula. Monetary policy may affect output in the medium run. Moreover, the medium run equilibrium, if it exists, is not a steady state because in SS we have by definition π_t = m_t = π_t-1; in this situation, we don’t have that.

Let’s look at the medium-run dynamics in the case of a shock, as if the monetary policy decides to influence the output, starting from a SS. Suppose the Central Bank (CB) at t=1 decides to increase the money growth rate from m₀ to m₁ = Δ permanently, to try to boost y over y_L.

Periods: 0 and 1

• 0: SS, with (y₀ = y_L, π₀ = π_t-1 = π_L, m₀).
• 1: Find the point of equilibrium y₁ and π₁

y₁ = (1-a)y_L + ay_L + aΔm → y₁ = y_L + aΔm

Using the AD: m₁ + Δ(1-a)π₁ = m₀ – (0) = m₀ + Δ(1-a) – (0) →

π₁ = π₀ + (1-a)Δm

First, by definition, looking at the aggregate supply, as the shock Δm is not zero, the y will not be equal to y_L. We get out of the steady state: a part of the Δm increases y (in a measure equal to ‘a’) and a part increases π in a measure of (1-a).

Part of the shock is transferred to prices (π) and part as output (y). The effectiveness of the monetary policy (the ability to change output) depends on ‘a,’ so depends on θ/(1+θ), so on the slope of the curve. If you have a flat AS, the impact of a monetary shock will be more on output rather than inflation.

In the graph we see: the steady state with AS that is the vertical line AS describing the long-run SS (y = y_L), because π = π_L, and the black AD down-sloped, function of m₀.

The shock at period T1

• CB increases m₁, the blue AD shifts upward (m₁ > m₀) by the measure m₁ - m₀.
• AS doesn’t get shocked (its shifting parameter is π that in T is π_t-1, still equal to π₀) and its slope-changing parameter is y.

New temporary equilibrium whose coordinates (y₁; π₁) are given by the equations previously derived.

Is this a steady state? NO, π₁ ≠ π₀, that is π_et; people’s expectations are not fulfilled, people expected an inflation lower than actual, moreover y₁ is greater than y_L, there is a positive output-gap, THE ECONOMY IS EXPERIENCING A BOOM, SO THIS IS NOT A STEADY STATE.

Why this is not an equilibrium

Suppose all economic agents expect π to be 3%, then it turns out to be 5%. If inflation is higher than expected, real wages will be lower than expected. Firms, thinking they are paying a lower real wage, may be willing to give money wage increases to their workers because it would cost nothing more. Firms are pricing according to the expected inflation of 3, but if it is 5, they know their competitors are selling at higher prices, their relative prices decrease, and raw materials, energy, and all other inputs in the production process will cost more, so they will mark-up their prices.

In the financial market, people will not be happy with their financial contracts. If a debt contract was signed expecting to pay a rate equal to 4, the real rate, according to expectations, would be 4 - π (3) = 0, they will not be happy if the real rate then is -1. Creditor will not be happy, financial contracts will be revised, or new contracts will have a different interest rate.

Outside the steady state, no contract will be constant: wages, prices, and interest rates will be revised. Outside SS, there is no constancy. Moreover, people will revise their expectations. If actual inflation is 5, why should they expect inflation next period to be 3 according to the extrapolative rule of expectations?

If people revise inflation expectations upward, AS will not stay constant because it is parametrized by expected inflation; it will shift.

Period T2

As anticipated, inflation expectations are revised upward, and AS shifts up. Substitute π₂ with π₁ that we found. Now there is no more monetary shock because m₂ once increased is constant on the new higher level.

New equilibrium:

• y₂ = y_L + aΔm

y₂ < y₁ because a is lower than 1, so squaring it reduces its effect on output, is weaker, the positive output-gap begins to shrink.

π₂ = π₀ + (1-a)Δm

π₂ > π₁

The impact of the initial money growth shock, his distribution, will now reduce on output and grow in inflation. AD doesn’t shift anymore because m is now constant at the new level.

Will this be a steady state? NO: y₂ > y_L; π₂ > π₁ (that is π_et). People will change their choices and expectations: AS will shift.

Periods T3 and beyond

If we keep going with the process, we will find many equilibrium points so that y₁ > y₂ > y₃ > y₄, and π₁ < π₂ < π₃, etc. Inflation keeps increasing and absorbing a larger part of the shock until y_n = y_L.

Equilibrium π_n and y_n will be an equilibrium with higher inflation, but not a steady state.

The dynamic process will be longer or shorter according to the value of ‘a’ (or θ, or γ), which defines how deep the shock is at the beginning on the AD, and how quickly expectations are revised, that is how long it will take for the AS to shift to the right equilibrium of SS.

After all the monetary shock caused by the CB increasing m will finally be absorbed, and the economy will return to its SS after a period of uncertain length. Money will be neutral once again; money will not have an output effect, or the effect will only be temporary, ineffective in the long run. Monetary policy is effective in the short run.

So why should the CB shock the economy if it is only temporary? Only to accelerate processes of shock generated by other parameters or exogenous shocks. The monetarist model even in case of recession, so with y < y_L, the economy tends to move to a SS, but if the CB thinks the process will take too long, they may simply accelerate the process.

Disinflation and monetary policy

Suppose y < y_L, in that situation, prices are lower than expected, a process of downgrade of expectations will move down the AS until it will cross the AS at y_L (output-gap shrinks, π becomes lower and lower). Disinflation is (temporarily) costly (costly means in terms of output-gap: you have to reduce output for a period of time here). Monetary policy may be a shortcut to accelerate the process to get the economy back to SS (because in this case, if CB set a higher AD by increasing m, the convergence to SS is accelerated). But convergence itself will take place anyway in this monetary model.

The instrument mainly used by the CB today is interest rate, not growth rate, so the monetarist debates that the CB should not intervene by lowering interest rates or by opening money flows. They thought the economy would converge back to SS somehow and in some time. The process could be long, but it doesn’t matter; the adjustment process will be safer and healthier than if the CB intervenes. We will analyze the rule of Friedman suggesting: follow a constant m policy, CB should announce a m and stick to that, then don’t intervene anymore. The economy has the instrument to reach the steady state, and CB will have π equal to that m. So if you want π = 2%, then set m = 2% and you will have that, sooner or later. Monetary policy according to monetarists is this, and it is very simple. Keep a constant m.

Adaptive expectation

Let’s go deeper on the issue of expectations; we assumed a static no learning rule of extrapolative expectations, stating: π_t = π_t-1. To get closer to reality, we can now add that economic agents revise their expectations each time they observe a difference between what was expected and what turned out to be. This formula says: “the difference between the expectation for this year and the expectation for the previous year (the exp adjustment) is a proportion of differences between actual inflation and expected inflation, lagged once.” Or: ‘I adjust my expectation rather than keeping the previous year’s expectation, according to the difference that there has been in the actual inflation and my expectation last year, if any. And weighted by λ’

π_et - π_et-1 = λ(π_t-1 - π_et-1)

Or, to remember it better, it is the same as for the extrapolative, but with both sides subtracted by π_et-1. And then the right-hand side multiplied by coefficient λ that stands for the entity of the adjustment:

• λ = 0, no learning, agents never adjust their expectation, static
• λ = 1, total learning, agents are extrapolative
• 0 < λ < 1 (with equals too) partial learning

NOTICE: extrapolative expectations are just a particular case of adaptive expectations. In other words, people form expectations based on past experiences, gradually "correcting" over time any past forecast error.

Understanding λ

To understand better λ, we adjust the formula into:

π_et = λπ_t-1 + (1-λ)π_et-1

Stating that π_et is a weighted average of the inflation at t-1 and what people expected at t-1 (when they were in t-2): people are supposed to stick to the previous expectation, and the degree of stickiness depends on λ. ‘My expectations this period t are in part due to past inflation, in part due to what I expected last period’

If I expected inflation last year to be 3, and I experienced 2, somehow, if lambda is less than 1, I think that inflation will still go to 3, I revise but not entirely, in the back of my mind there is an attachment to past expectations, this kind of inertia tells us how, λ is the speed of this expectations revision.

Excursus – first problem

We can clearly see how the expectation of inflation for period t, which is a non-observable variable, depends on past expectations, which are again unobservable variables because they were expected. It looks like these adaptive expectations are not useful to solve any model: we cannot have data. But even if λ is different than 1, you may solve the model by lagging it once, and getting:

π_et-1 = λπ_et-2 + (1-λ)π_et-2

And then substituting it inside the previous formula:

π_et = λπ_t-1 + (1-λ)[λπ_et-2 + (1-λ)π_et-2] = λπ_t-1 + λ(1-λ)π_et-2 + (1-λ)²π_et-2

The term with expected inflation is sort of pushed back into the past and multiplied for 1-λ squared, which is lower than just 1-λ, which is better because it reduces the weight attached to expected inflation. And then, lagging it again... and again... we will observe inflation expectations to be a weighted average of effective values of past inflation, with weight reduced the more we go far in the past. And the lightest on the value referring to the expectation in the past, that remains unobservable, but until negligible.

Now it is possible to appreciate the meaning of λ. The greater, the faster expectation adjustment to the precedent.

Systematic errors – second problem

Still, a problem is addressed to this kind of expectation formulation system. Suppose inflation expectations today is 2% and actual turns out to be 3. So, according to extrapolative (which is the fastest adjustment), I expect it to be 3 next year. Then it turns out to be 0.5 now I expect it to be 0.5 and it turns out to be 0.3 then 4... Regardless of λ, a great problem of adaptive expectations is that agents continuously commit systematic errors because they look excessively at past values of the variable they have to formulate expectation, without using other information they have, like for example if OPEC countries announce an increase in oil price. It is probable that they will have difficulties in calculating which weight this can have but is improbable that everyone ignores that this has an impact and awaits an effective increase in prices to revise upward expectations. Moreover, their expectations will keep on being wrong during the whole process of adjustment seen before until the return to SS.

This has a cost as we saw before (choices made on wrong expectations are sub-optimal, firms may decide to hire more or fewer employees than would let them maximize their profits, or trade unions may fight for lower minimum wages than would maximize the utility of their enrolled workers).

Numerical example

Look at the systemic error with a Phillips curve:

π_t = π_et + γ(u + u_L)

With γ = 2, u_L = 8%, π = 3%, the economy in long-run equilibrium SS, π_t = π_L = π_t-1 = π_L.

CB considers the natural (and actual) unemployment rate too high (at 8), even if in equilibrium, and decides to make an expansive aggregate demand policy to reduce it to the target 5. We assume also λ = 1 to make it faster to adjust.

Plugging values, π_t+1 = 9, we will have an expectation error equal to ε = 6

Then plug in the same target unemployment rate.