Most modern dynamic economic models are too complex and ambiguous to support macro trading. A practical alternative is a simplified static model of the “New Keynesian” tradition that combines basic insights from dynamic equilibrium theory with an intuitive and memorable representation. Macro traders can analyse real life events in this framework my shifting curves in a simple diagram. In this way they can analyse the effect of fiscal policy shocks, monetary policy shocks, inflation expectation shocks, economic supply shocks and so forth. Part 1 of this post focuses on a model for a large closed economy (or the world as a whole).

Evan Tanner (2017), “The Algebraic Galaxy of Simple Macroeconomic Models: A Hitchhiker’s Guide”, IMF Working Paper, May 2017, WP/17/123.

The post ties in with this sites’ lecture on systematic value through macro trends, particularly the part on best practices for tracking macro trends.

The below are excerpts from the paper. Emphasis and cursive text have been added to explain key points from the angle of macro trading. The presentation of formulas and their explanations have been modified for easier reading.

The case for simplicity

Dynamic stochastic general equilibrium (DSGE) models do an important job: they help us understand the behavior of macroeconomic aggregates in a way that reflects rational, optimizing behavior by economic agents. The dominance of DSGE models in the economics profession is…evidence of their rigor and potential…However, this current in macroeconomic modeling is not without its critics. Several prominent authors…have raised doubts about the merits of these…models precisely because they are so complex and difficult to understand—even for seasoned professionals.”

“Simpler…traditional, static macroeconomic models…in macroeconomics can be superior to more complex ones… For applied professionals…[a simple model] often serves as the default ‘mental model’ to guide policy discussions. But, simple static models can do even more. A bare-bones version of the New Keynesian model—direct heir to the ‘IS/LM model’—can generate equilibrium values of standard ‘core’ macroeconomic variables: the output gap, the interest rate, the inflation rate, and the real exchange rate.”

“Applied macroeconomists should find the framework useful in their day-to-day assessments. The methods are especially useful for comparing alternative scenarios for one or several periods.”

A large (closed) economy model

“A New Keynesian model for a closed economy consists of three core equations: the IS curve [Investment-Savings curve, representing the relation between aggregate goods demand and the real interest rate], a monetary policy reaction function [in form of a Taylor rule for the short-term interest rate], and an aggregate supply or Phillips curve relationship [representing the relation between the economy’s output gap and inflation]. These three equations yield three equilibrium core values: the output gap, the interest rate (real and nominal), and the inflation rate.”

The case of a small open economy will be discussed in a subsequent post.

The IS curve

In this context, the IS curve describes the relation between an economy’s output gap and its real interest rate. It is based on the economy’s goods market equilibrium, which requires aggregate consumption, investment and real government spending to be equal to aggregate output.

Real consumption is described by a standard (Keynesian) function: of a constant and a variable component that depends of short-term fluctuations in after-tax income modified by propensity to consume from this temporary income:

C[t] = ac + (1 – sc) × ((1 – tr) × (Y[t] – YP) – TP[t])

C[t] is aggregate consumption in period t,
ac is a constant consumption component,
sc is the cyclical (short-run) savings rate, which is between 0 and 1,
Y[t] is aggregate output and demand of the economy,
tr is a proportionate tax rate,
TP[t] is a special lump-sum taxation (“tax policy shock”).

“The second term on the right-hand side tells us the short-run or cyclical component of consumption—including the effect of one-off tax measures…[depending on] the marginal propensity to consume out of transitory income.”

The constant consumption component equals potential output, diminished by the long-term savings and tax rates,

ac = (1 – tr) × (1 – sl) × YP

tr is a proportionate tax rate,
sl is the long-term savings rate (versus the short-term savings rate sc),
YP is the potential (full-employment) output of the economy.

“That constant explicitly informs us about household consumption and disposable income in the long run…[depending on] the long-run savings rate but now adjusted for taxes.”

Aggregate investment in the economy is mainly a function the real interest rate:

I[t] = ai – ar × RN – ar × (R[t] – RN)

I[t] is aggregate investment in period t,
R[t] is the real interest rate in period t,
RN is the natural real rate of interest,
ar is the sensitivity of investment with respect to the real interest.

Government real spending is divided into its normal level plus a ‘government policy’ shock, i.e. a temporary unanticipated deviation from its long-term state:

G[t] = ag + GP[t]

G[t] is aggregate real government spending in period t,
ag is a constant real government spending component,
GP[t] is a government spending ‘shock’ in period t

The goods market equilibrium requires real consumption, investment and government spending to be equal to the output of the economy:

C[t] + I[t] + G[t] = Y[t]

C[t] is aggregate consumption in period t,
I[t] is aggregate investment in period t,
G[t] is aggregate real government spending in period t,
Y[t] is aggregate output and demand of the economy.

“[By solving the goods market equilibrium from the demand side] we may…obtain the IS curve whose output and the interest rate are respectively measured as percentage gaps from potential output and the neutral interest rate, and the fiscal shocks are measured as a percent of potential… We then subtract and divide both sides of the equation by potential output…[we] obtain an expression for the output gap IS curve:”

R[t] = RN – (ssc × gap[t] + gp[t] + (1- sc) × tp[t]) / ar

R[t] is the real interest rate in period t,
RN is the natural rate of interest,
gap[t] is the output gap,
    i.e. the difference between actual and potential in % of potential,  (Y[t] – YP) /YP,
ssc is a tax-adjusted savings rate, 1- (1 – sc) × (1 – tr),
gp[t] is the government spending ‘shock’ in period t, in % of potential output,
tp[t] is a tax policy shock in % of potential output,
sc is the cyclical (short-run) savings rate, which is between 0 and 1,
ar is the sensitivity of investment with respect to the real interest (absolute value)

Note that the IS curve is downward sloping, i.e. as the output gap increases the real interest rate decreases, reflecting that this curve represents the demand side of the economy. Aggregate demand tends be higher when the real interest rate is lower. Note also that this curve is ‘pushed to the right’, i.e. same real rate corresponds to higher output gap, if the government spending shock is positive and the tax shock is negative (or if there was a positive shock to the natural rate of interest).

The RR curve

The RR curve represents the monetary policy and inflation side of the economy and is complementary to the IS curve.

“The monetary policy reaction function is typically phrased along the lines of John Taylor’s interest rate rule for central banks…The interpretation of this equation is an appealing one: the central bank has a dual mandate—to stabilize both prices and output.”

NR[t] = RN + PE + bp × (P[t] – PT) + bg × gap[t] + RD[t]

NR[t] is the nominal short-term interest rate,
PE is expected long-term inflation,
P[t] is the current rate of inflation,
PT is the central bank’s inflation target,
bp is the sensitivity of the short-term interest rate to excess inflation,
gap[t] is the current output gap,
bg is the sensitivity of the short-term interest rate to the output gap,
RD[t] is a discretionary deviation of the policy rate from the rule (“monetary tightening shock”).

“The inflation rate is determined by the following simple Phillips-curve relationship [i.e. a relationship between actual inflation, expected inflation and a supply shock-adjusted output gap].”

P[t] = PE + cg × (gap[t] – ss[t])

PE is expected long-term inflation,
ss[t] is a supply shock
gap[t] is the current output gap,
cg is the sensitivity of inflation to excess demand.

“We then obtain the equilibrium real interest rate from monetary policy and the supply side…The first step is to substitute in Phillips curve equation into Taylor rule curve equation to obtain a reduced-form expression for the nominal interest rate. This substitution has an appealing interpretation: the Phillips Curve poses a constraint for the central bank whose goal is to stabilize prices and output. Then… obtain an expression for the real interest rate consistent with both the central bank reaction function and the Phillips curvethe ‘RR’ schedule:”

R[t] = RN + bp × (PE – PT) + (bg + cg × (bp -1)) ×gap[t] – cg × (bp – 1) × ss[t] + RD[t]

R[t] is the real interest rate in period t,
RN is the natural rate of interest,
bp is the sensitivity of the interest rate to excess inflation (> 1, for hawkish central banks),
cg is the sensitivity of inflation to excess demand,
bg is the sensitivity of the short-term interest rate to the output gap,
gap[t] is the output gap,
ss[t] is a supply shock,
RD[t] is a discretionary deviation of the policy rate from the rule.

Like the IS curve, the RR curve describes a relation between the real interest rate and the output gap. However, it is upward sloping for inflation-averse or ‘hawkish’ central banks, meaning that a higher output gap is typically associated with a higher real interest rate. Note also that a monetary tightening shock pushes the RR curve to the right, i.e. the real interest rate will be higher for the same output gap. A positive supply shock will push it to the left, meaning that the real interest rate will be lower for the same output gap. Also, positive shocks to the natural rate of interest and inflation expectations will push the curve to the right, while positive shocks to the inflation target will push it to the left.

Overall macroeconomic equilibrium for this closed economy

The macroeconomic equilibrium of this closed economy is the combination of real interest rate and output gap for which both the IS and RR curve conditions are met. Formally, we would have to set the solutions for the real interest rate equal and subsequently solve for the equilibrium output gap, real interest rate, inflation and nominal interest rates. Since the resulting terms are large and take some time to interpret, a more practical analysis for macro traders can make use of two simple graphs.

We first draw into the real rate/ output gap diagram the downward sloping IS curve and the upward sloping RR curve. Their intersection marks the equilibrium in terms of real rate and output. Second, we draw into the inflation/ output gap diagram the upward sloping Phillips curve. The inflation rate that corresponds to the equilibrium output gap is the equilibrium inflation rate. Note also that the equilibrium nominal interest rate is the sum of the real interest rate and the inflation rate.

The practical use of the model and this representation is that we can simulate various types of shocks that can be observed in the economy and the market, by shifting the IS/RR curves accordingly and inferring the stylized consequences for the real and nominal interest rates from the changed equilibrium values.

The above first example simulates a fiscal expansion (government spending expansion or tax cut), by shifting the downward sloping IS curve to the right. The consequences will be a higher real interest rate and a higher output gap (intersection of red dashed with blue solid line). This will also lead to a higher inflation rate and – accordingly – a yet higher nominal interest rate. The above second example simulates a combination of fiscal expansion and monetary expansion. The consequence will be an uncertain effect on the real interest rate, but a positive impact on the output gap (intersection of red dashed and blue dashed line) and an increase in the inflation rate.

“Figure 2 shows the graphical representation of these scenarios. The upper diagram shows the IS and RR schedules in output gap/real interest rate space. Under alt (i) the downward sloping red line shifts to the right, from the solid to the dotted/dashed line, along the RR curve, which remains in its original position. Thus, we see the joint increase of output and the real interest rate under this scenario. Under alt(ii), we see no further shift of the IS curve but the RR curve now shifts to the right. Accordingly, output under alt(ii) is higher but the interest rate is lower than under alt(i). Directly below, we see the Phillips Curve outcome: inflation rises under alt(i) and higher still under alt(ii)—as reflected in movements along the green Phillips Curve schedule.”