New Keynesian Adjustments for Inflation

One view of what to do with too much money!
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[NOTE: Edited on 11/13/2015 to ensure proper interpretation of the Taylor Rule.]

The New Keynesian views of inflation, interest rates, and aggregate demand (a synonym for GDP), and the links between them are captured in two relations. As discussed in our last post, the first is some kind of monetary rule for finding the interest rate the central bank wants to impose on the financial markets. This can be a formal monetary rule or some kind of ad hoc discretionary rule made up on the fly by the central bank. Whatever that rule is, Keynesians call it the Policy Reaction Function, or the PRF. In our discussion we will take the PRF as being the Taylor rule, which has the form

r_t = r_t* + α_π( π_t – π^T ) + α_y( y_t – y* )/y*                      (1)

where r_t is the real interest rate defined as the difference between the nominal interest rate, i_t in the time period t and the inflation rate π_t during that period.

r_t ≡ i_t – π_t                                                              (2)

The other variables in the Taylor rule are r_t*, which is a long-term average of the interest rate being adjusted minus inflation; α_π and α_y are constant parameters that are often set to 0.5; π^T is the target inflation rate; y_t is the economic output of the nation during the period t (i.e. the GDP); and y* is a Keynesian construction called the potential GDP. I will have more to say about the potential GDP later.The quantity ( y_t – y* )/y* is called the “output gap”, and it is clearly the fractional difference from potential GDP of the GDP during the period. Note that because of the output gap term, the interest rates in the other terms must be fractional differences between years. To convert them into percent differences, all terms must be multiplied by 100%. The New Keynesian priority is always to minimize the output gap while at the same time taking account of inflation.

The second relation is a New Keynesian assumption about the link between the inflation rate in the current time period to the inflation rate in the previous time period and the output gap.

π_t = π_t-1 + γ ( y_t – y* )/y* , γ > 0 .                         (3)

In this equation the parameter γ is a positive constant that can be changed by some event outside the theory, and can be different for different economies. Note that if the output gap is positive, i.e. expansionary, the inflation rate will increase over the previous period’s inflation rate; if the output gap is negative, i.e. contractionary, the inflation rate will decrease over the previous period’s inflation rate; and if the output gap is zero, the inflation rate will be the same as the last period’s.

We will now look at how Keynesians construct an analogue to Alfred Marshall’s demand curve, called the Aggregate Demand (AD) curve, which is a line in the πy-plane. First, consider what would happen if, for whatever reason, the inflation increases from a value of π₀ to a value of π₁. This increase in inflation motivates the central bank using the PRF to increase the interest rate from r₀ to r₁. If interest rates increase, it will be harder for companies to borrow and increase investments, and for households to borrow to make purchases, causing a fall in total demand and therefore of output y from y₀ to y₁. In this way an increase in inflation generates a decline in demand from y₀ to y₁ as shown in the figure below.

Repeating this exercise many, many times for other initial values of inflation causes interest changes required by the PRF  that  generate other declines in output. Connecting all these generated points results in the aggregate demand or AD curve.

Our next task is to generate the macroeconomic analogue of the microeconomic supply curve, called the Aggregate Supply (AS) curve. The concept of AS is motivated by considering the idea of inflation adjustment (IA), which is summarized by equation (3).

π_t = π_t-1 + γ ( y_t – y* )/y*

This shows that inflation adjusts to the output gap, with an expansionary output gap causing positive inflation, and a negative output gap causing deflation. Plotting inflation as a function of aggregate output gives us a positive sloped curve, which is the AS line for which we are looking. Where the AS and AD curves intersect is a point of both short-run and long-run equilibrium where the aggregate amount of goods and services supplied and demanded are both equal at the same inflation rate. Before we can get to this conclusion however, we must  define what “short run equilibrium” and “long run equilibrium” mean and why we must distinguish between them.

This is where the “sticky” prices and wages enter into New Keynesian theory. In long run equilibrium there is no output gap; output will equal potential GDP, no matter what the inflation rate is. (This last statement should already be ringing alarm bells in your head!) Also, inflation will become a constant value, since as long as there is no output gap, equation (3) tells us that in any year, the inflation rate will be the same as the last year. π_t = π_t-1. Therefore the Long Run Aggregate Supply line (LRAS) is a vertical line at y = y*. However, in the short run, sticky prices throw sand into the machinery because they will not immediately adjust to the market price of the law of supply and demand. Sticky prices (and wages) are prices suppliers are reluctant to change because of fear of the changing environment. With prices sticky, suppliers will supply any demand that consumers present since a larger volume of sales will always give them bigger profits. This is represented by a horizontal line at the current inflation rate, π₀, which represents the Short Run Aggregate Supply line (SRAS), which is also known as the Inflation Adjustment line (IA).

No matter what the AS line happens to be at the moment, whether it is short run or long run AS or something in between, the point the economy will occupy is where the AS line of the moment intersects the AD curve; then the amount of goods produced will equal the amount supplied at a common inflation rate. This defines a point of short run equilibrium. A point of long run equilibrium is defined by the intersection of the AD and LRAS lines. Therefore an economic condition of simultaneous short run and long run equilibria (a state we will describe as a general equilibrium) is represented by the figure below.

Although this figure shows the AD and the IA lines intersecting the LRAS at the same point, this is not generally the case. But if the aggregate demand line is somehow displaced so that it does not pass through the intersection of the long run aggregate supply and inflation adjustment lines, market forces will bring the economic system back into general equilibrium, albeit at a higher level of inflation. In what follows, we will assume that the system starts off in general equilibrium, but is forced off it by some shock. All the parameters defining the shape and position of the AD curve are outside the inflation-output model we are using and are explained by theory from different models. These variables are said to be exogenous to our current model. To be pushed from its current position passing through a point of general equilibrium, some exogenous economic shock must change some of the exogenous parameters defining the AD curve.

As a particular example of a displacement from general equilibrium, we will consider a positive demand shock. Something external to our model has caused aggregate demand to increase with no change in inflation. This will force the AD line to displace from its original position at AD₀ to AD₁ as shown in the figure below.

The economy’s system point has now been driven from (y*, π₀) to (y₁, π₀) with no increase in inflation since the system point was moved on IA₀. Recall that an important assumption of the model is that at constant inflation, companies will supply whatever is demanded, and therefore the economy will move along IA₀ until it reaches the new aggregate demand curve where it forms a new short run equilibrium. As a result the demand shock has opened up an expansionary output gap ( y₁ – y* )/y*. Because we have assumed that a positive, expansionary output gap will create an increase in inflation from equation (3), a new inflation adjustment line IA₁ will be formed at the inflation rate

π₁ = π₀ + γ (y₁ – y*)/y* .

The system point will then move along the AD₁ until it reaches IA₁ at (y₂, π₁), where the system again forms another short run equilibrium. What explains this last move is that the central bank has seen all the inflationary activity going on with factories open at all hours and people working overtime, and uses the PRF to calculate an increase in interest rates that reduces aggregate demand along the AD₁ line. However, once the system forms another short run equilibrium, there is still an expansionary output gap of ( y₂ – y* )/y*. This procedure is then repeated for as many steps as necessary until the final inflation adjustment line (in the case of our figure IA₂) where the AD₁ line intersects IA₂ at (y*, π₂), and the output gap totally disappears. The system is again at both short run and long run equilibrium, but now with a higher inflation rate π₂ than its original interest rate of π₀.

In the next post we will show how the New Keynesians explain the stagflation of the late 1970s using this theoretical structure.

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