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Wednesday, 4 April 2018

A Three Variable Macro Model : Simulation


This blogpost shall simulate a simple 3 variable macroeconomic model of the business cycle. The first variable is the annual increase in general prices or inflation. The second is the “output-gap” which measures the log-difference between actual aggregate production and potential/full capacity GDP. It can also be thought of as a measure of excess demand for goods and services at the aggregate level. The third variable is the interest rate, which is the price of loans and credit in the economy, for both households and firms. 
These three variables are contenders for the most important variables in macroeconomics and influence each other over the business cycle. By simulation, I mean that I will run a virtual experiment. The system will start at rest, and I will introduce different types of ‘shocks’ at various locations, and then observe the dynamic movements of these three variables due to these shocks.
Since this is a purely imaginary economy, I make very simplifying assumptions and am in complete control over what is going on here. STATA Codes to replicate are at the end.

The Model

The following 5 equations fully describe the economy. One unit time in this tiny model economy is a quarter. 

Here (1) is the Aggregate Supply (AS) curve which says that inflation is a function of inflation expectations, output gap and supply shocks. Firms observing excess aggregate demand for goods and services will raise prices in the next quarter. Workers’ expectations of inflation influences how they negotiate for wages; higher the inflation expectations, the higher the wage negotiated. This leads firms to raise product prices. Supply shocks cover unpredicted changes in inflation due to oil prices movements, monsoon & climatic conditions in agriculture, etc.   


(2) is the dynamic Investment-Savings (IS) schedule that says that ex-ante real interest rates effect output gap with 1 quarter lag. Ex-ante real interest rates is defined in (3) and are nothing but interest rates in the same period minus inflation expectations for the next quarter; and measure the real price of credit & borrowing. 
If State Bank of India offers a home loan today, for an interest of 5%, a 10 lakh home will cost half a lakh in interest above the instalments for the principal. But if prices rise 5% tomorrow (and consequently your wage rises 5% with it), you will effectively end up paying zero interest for that loan! Therefore the ex-ante cost of credit depends on the interest rate today and the expectation of inflation tomorrow.
When the expected cost of credit is low consumers will execute plans take a home or car loans, investors and industrialists will take loans to invest in businesses. Therefore spending on goods and services will exceed the optimal capacity of the economy. However, transmission from interest rate to demand takes a full quarter.  
(2) Also says that at some price of credit (r*) the excess demand is zero. This is the “natural” rate of real interest rate, consistent with the economy’s capacity to produce. Apart from ex-ante real interest rates, we also have 𝜂t which is the demand shock. This could consist of unpredictable changes in spending due to varying preferences for savings, changes in wealth caused by variations in asset prices, exchange rate fluctuations causing adjustments in exports & imports, abrupt movements in consumer and investor sentiment, etc.  


(4) is a Taylor rule, which describes how the central bank behaves. In our imaginary economy, the central bank’s sole mandate is to try and keep inflation at a target (𝜋*). It has control over the interest rate, because it is the banker to all commercial banks and at what rate it lends to them governs how they price credit. 
When inflation is on target (𝜋t=𝜋*), the central bank keeps interest rates at r* + 𝜋*, so that (given inflation expectations are also equal to target), the ex-ante real interest rate is at its natural levels - and thereby there is no excess demand in the economy. If inflation is below its target, then the central bank lowers the interest rate below r* + 𝜋*. This raises the real interest rate, and according to (2) will create excess demand for goods and services. Excess demand will then, according to (1), raise inflation. The central bank will keep interest rates low until inflation is back on target. 
Apart from this, interest rates also changes due to policy shocks (vt) which can arise due to frictions in transmission of monetary policy through commercial banks, or simply if the central bank decides to break from its rule of strict inflation targeting.  


(5) tells us that households, investors and workers always expect inflation to be what it was a quarter earlier. This is a very rudimentary rule of thumb, because obviously inflation will not be what it was a quarter earlier, because of changes in demand (whether caused by the central bank or by demand shocks) and supply shocks. Thus agents overlook the behaviour of firms and the central bank in determining inflation outcomes. But this is a simplifying assumption.
In summary: Monetary policy, is entirely concerned about inflation, effects aggregate demand through interest rates with one period lag and demand pressure generates inflation one period later. All three variables are interconnected, across time, to each other. And shocks intrinsic to each of them (demand, supply, policy) generate or power the dynamics in the economic system. 
Furthermore, we can describe the economic system in a more concise manner by solving (1) - (5) in terms of (𝜋t,Rt,yt)' the state vector, and writing in Vector notation: 


Stability of the system occurs when the state variables (𝜋t,Rt,yt) always tend to return to their steady-state values (𝜋*, r* + 𝜋*, 0) regardless of the magnitude of the shocks. This is assured only for certain ranges of parameter values (β, ϕ,θ), in all other cases the shocks whack the state variables permanently away from their steady state values. Stationarity occurs when the coefficient matrix of the lagged state vector is such - that it’s eigenvalues are within -1 and 1. We ensure this in the simulation. 

Simulations: Multiple Scenarios

We set the following parameters to be the following for the baseline case. The initial values for state variables are given to be their steady state values.


  1. Demand, Supply & Policy Shocks 

First I introduce a positive supply shock of unit magnitude at quarter 5, which falls by 0.25 units every consecutive quarter till it reaches zero. What happens to the economy? The economy begins at steady state. Then in quarter 5, the supply impulse hits inflation, which rises above target. This has two effects in quarter 6. First, inflation expectations for quarter 6 rise. So anticipated real interest rate in quarter 6 falls. Thus demand rises in quarter 6, albeit temporarily. Second, the central bank responds to inflation overshooting target in quarter 5, by raising interest rates in quarter 6. This reverses the temporary increase in demand and causes a contraction in demand. This contraction in demand reduces inflation and brings it back to target levels. A negative supply shock, would reduce inflation first, then interest rates would fall after which demand would rise - restoring inflation back up to target levels. 

  
Next I introduce a positive demand shock at quarter 5, which also dies out like the  supply impulse. In this situation, it is demand which rises first in quarter 5 (a boom). This is followed by rising inflation in quarter 6, which has two effects - one, it further bolsters demand by raising inflation expectations and lowering real interest rates and two, it forces the central bank to swing into action in quarter 7. Interest rates thus respond last in this schemata, but are pivotal in bringing the economy back to target. A negative demand would do the reverse: lower demand, then lower inflation, which would have the central bank lower rates, which would reverse the recession and deflation.  



A similar positive policy shock is generated at quarter 5. So the central bank raises rates from in quarter 5, which it only gradually reduces. This reduces demand in quarter 6, which in turn, reduces inflation at quarter 7. Now the central bank must quickly reduce interest rates to bring inflation back up to target.

B. A Dovish Central Bank

The parameter ϕ tells us how hard the central bank reacts to inflation deviations from target, therefore it is a measure of how dovish or hawkish the central bank is. We reduce the value of ϕ to 1.2 and re-simulate the impulses. The key differences (a) the state variables, once disturbed, take much longer to get back to long run equilibrium. (b) the dovish central bank, by responding weakly, allows inflation to climb well above target.
   


C. Disinflation: Gradualist or Cold-Turkey

Here we simulate two different strategies to bring inflation to a lower target level. The goal of disinflation is to bring inflation down from 4% to 2%. The first approach is mild (call it “Gradualist”) while the other is aggressive (call it “Cold-Turkey”). 
 The Gradualist approach seeks to first reduce the target to 3% from quarter 5 onwards and then further reduce the target to 2% from quarter 25. I also keep the the central bank dovish by setting ϕ = 1.2. Cold-Turkey is when a hawkish central banker goes for immediate dis-inflation. In our simulation I re-set ϕ = 1.5 and have the central bank immediately begins to target inflation at 2% from quarter 5 onwards.   

What are the main differences? Gradualism takes a lot of time. Interest rates are not raised very high. It also does not push aggregate demand far below potential. In contrast, Cold-Turkey goes for heavy rate hikes that cause recession. This recession is used to disinflation the economy. Due to adaptive expectations, Cold-Turkey ends up creating serious output loss but achieves the task quicker than Gradualism. 

D. What a Dynamic Stochastic Economic System can look like

I re-simulate, bombarding the economy with random disturbances. The baseline parameters have been taken. The supply, demand and policy shocks that I have given are independent, identical and normally distributed with zero mean and certain variance. 



The resulting dynamics are complex and fascinating - shocks push the system away from its equilibrium but causal forces keep the system stable. 



Disclaimer - This is a purely imaginary economy. It borrows plot elements from the real world and plays out like a Bollywood movie. Any reference to any person or phenomenon, living or dead, is purely accidental and solely expositional!  

Why Simulate? 
  1. it is expositional, gives clarity and visualisation
  2. tells you what effect each parameter or shock has on the systems dynamics
  3. since we built it, we can do whatever we like - we are the omnipresent and omniscient gods - we can explore ideas! 
  4. is fun
  5. helps test the power of estimation techniques - if estimation techniques are unable to figure out the parameters that were used to generate simulated data then they are probably not very useful against real world data

The next blogpost will cover (e) where we will using only this very simulated data, and see if empirical techniques called the Structural Vector Auto-Regression (SVAR) can get us back to square one i.e where we started from!