University of Amsterdam
MSc in Economics
Master thesis
OPTIMAL MONETARY SIMPLE RULE
IN THE PRE-VOLCKER PERIOD IN THE UNITED STATES
Author: Supervisor:
Yuxi Jin (11374632)
Christian A. Stoltenberg
Statement of Originality
This document is written by Student Yuxi Jin who declares to take full responsibility for the contents of this document.
I declare that the text and the work presented in this document are original and that no sources other than those mentioned in the text and its references have been used in creating it.
The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.
Abstract
In this thesis, I find the Optimal Monetary Simple Rule in the pre-Volcker period in the U.S. to be active in accordance with the Taylor principle. My baseline model comes from Kriwoluzky and Stoltenberg (2015), who equip standard New Keynesian model with explicit transaction role of money. I take welfare loss of Ramsey policy as the benchmark and compare it with the losses of various simple rules. I discover that inflation, output gap and past real money balances are necessary in formulating Optimal Simple Rule. The effect of past interest rate, real money balances and past inflation is rather limited.
Keywords: Optimal Simple Rules, Ramsey Policy, Transaction Role of Money, Welfare
Contents
1 Introduction ... 1 2 Literature Review ... 3 3 Economic Environment ... 5 4 Welfare Analysis ... 8 4.1 Loss Function ... 9 4.2 Ramsey Policy ... 94.3 Optimal Simple Rule... 10
4.4 Comparison between Ramsey Policy and Optimal Simple Rule ... 10
4.5 Calibration... 12
5 Quantitative Results... 12
5.1 Analyzing Procedure and the Optimal Simple Rule ... 13
5.2 Effectiveness of the Instruments ... 16
5.3 Influence of Money Velocity... 17
6 Conclusion ... 20
References ... 21
Appendix ... 23
A. Source Code of Ramsey Policy... 23
1 Introduction
The main objective of monetary policy is to maintain price stability. Price stability is important for the macroeconomic environment because it promotes the efficiency of resource allocation and avoids inflation costs, like risk premia and distortions in taxes. To achieve that, central banks have been taking nominal interest rate as the instrument of monetary policy. Nevertheless, the underlying mechanism of interest rate policy can be rather complicated. The efficiency of monetary policy varies based on different environment and agents. Therefore, a lot of economic models have been created to capture the mechanism, so that more efficient policies can be formulated. Then, how does interest rate influence price stability? Which model best describes the reality? How do we assess the efficiency of a monetary policy? I will focus on a representative case in the United States and find out the answers.
After Paul Volcker was appointed chairman of the Federal Reserve in 1979, the high inflation periodof the United States during 1970s came to an end. The monetary policies of the Federal Reserve board, led by Paul Volcker, were widely credited with curbing the inflation rate and the expectations that inflation would continue. It is generally believed that the change in carrying out the interest rate policy in 1979 could be a milestone in the US monetary policy history, (Goodfriend, 1993; Song, 2004; Ireland, 2009). Meanwhile, conventional wisdom has it that monetary policy before 1979 is badly managed. These ideas arouse my interest in the years around 1979. How does the change in monetary policy realize economic stability? Is it the Federal Reserve to blame for the instability before 1979? What is the monetary policy that should have been conducted?
Clarida et al. (2000) captured the change in 1979 by splitting the US economic environment into pre-Volcker period and Volcker-Greenspan period. They found interest rate policy in the later period to be much more sensitive to changes in expected inflation. The tolerance of inflation in the pre-Volcker period permits greater macroeconomic instability through the self-fulfilling changes in expectations on inflation and output. Favero and Monacelli (2005) described such transition as the onset of the Taylor principle for monetary policy, which
required the monetary authority to adjust the nominal interest rate actively (i.e., more than one-for-one) to fight changes in inflation. Lubik and Schorfheide (2004) arrived at a similar conclusion that there was a switch in anti-inflationary stance from passive to active in 1979. Against the traditional view, Kriwoluzky and Stoltenberg (2015) found that, rather than monetary policy, it was the decreasing role of money in transactions that could rationalize the transition in the interest rate policy. They pointed out that the passive interest rate policy before Volcker was difficult to capture with a standard cashless New Keynesian model (Woodford. 2003), as it would cause indeterminacy. Therefore, the possibility of sunspot fluctuation is incurred, leading to macroeconomic instability. Such problem is solved by adding explicit transaction role of money to a standard cashless New Keynesian model. In this way, determinacy could be achieved in a passive interest rate environment. Their explanation was supported by empirical estimation that there was a decline in the relevance of money in facilitating transactions before 1979. But they did not dig further into the optimal monetary policy and social welfare analysis. Based on their latest model, I will further explore the Optimal Simple Rule in the pre-Volcker era in the United States.
More specifically, I will take the model with explicit transaction role of money from Kriwoluzky and Stoltenberg (2015) as the baseline model. In order to reach the optimal monetary policy in the pre-Volcker era, I use Dynare to compute the welfare loss for both Ramsey policy and Optimal Simple Rule (OSR). Although Ramsey policy offers a straightforward way to carry out welfare analysis and leads to a lower loss than Optimal Simple Rule, the instruments employed to implement the policy remain unknown. Hence, I intend to find out the closest Optimal Simple Rule to the corresponding Ramsey policy under the same loss function. To achieve that, I will try various combinations of policy instruments for the Optimal Simple Rule, and compute the welfare loss. The closer the welfare loss is, the closer the policies will be. Besides, I will calculate the coefficient of interest rate in the loss function with different money velocity in the pre-Volcker period to attain better consequence.
I find that the Optimal Monetary Simple Rule in the pre-Volcker era is active interest rate policy rule, contrary to what Kriwoluzky and Stoltenberg (2015) suggest. The policy stance is
switched by adding past real money balances into the simple rules. It is the explicit transaction role of money that affects price stability. Besides, inflation 𝜋𝑡, output 𝑦𝑡,and past real money balances 𝑚𝑡−1 , are essential in constructing Optimal Simple Rules. Inflation is the most effective policy instrument. Past policy rate, past inflation and current real money balances will lead to indeterminacy and even interact negatively with other instruments. The adjustment in coefficient of the loss function matters for the choice of instruments.
The remainder of the paper is organized as follows. Section 2 is the literature review. I will describe economic environment in Section 3. In Section 4, I will elaborate the welfare analysis and specify the loss function. I will demonstrate the quantitative results in Section 5. The Last section concludes.
2 Literature Review
In this section, I will go through the models capturing the policy change in 1979, and display the corresponding explanation of such change provided by the model. Then, I will review the methods to conduct welfare analysis. In the end, I will point out my inspiration based on these literatures.
When describing and explaining for the transition in monetary policies, Ireland (2004) conducted maximum likelihood estimates of a small, structural model of the business cycle with the U.S. quarterly time-series data in the Volcker-Greenspan era. It turned out that, even after correcting for money demand shocks, money had a minor role in the monetary business cycle. Similarly, Castelnuovo (2012) found declining role of money in explaining business cycle fluctuation.
Sims and Zha (2006) casted doubt on Clarida et al. (2000) in that their analysis of policy rule change were potentially subject to bias. Sims and Zha concluded that the ‘Great Moderation’ in output and inflation volatility was largely due to good luck rather than a desirable change in policy.
indeterminacy region to determinacy region, but was due to the Volcker disinflation effect on the level of trend inflation. They provided an alternative explanation for the determinacy in the Volcker-Greenspan era. Aside from inflation targeting monetary policy, output growth targeting (rather than output gap) and price-level targeting also played pivotal role in restoring determinacy.
Similarly, Orphanides (2004) interpreted the pre-Volcker period as too activist policy in reacting to overambitious perceived output gaps. Nevertheless, the Taylor principle was satisfied before 1979 in his forward-looking monetary policy reaction function.
From the viewpoint of asset market participation, Bilbiie and Straub (2013) developed a standard sticky-price dynamic stochastic general equilibrium model, which displayed a positive IS curve at low asset market participation rate. When such non-Keynesian feature emerged, passive monetary policy actually ensured the equilibrium determinacy.
Schmitt-Grohe (2004) investigated optimal monetary policy under sticky prices time with invariant Ramsey approach. The model was an infinite-horizon production economy with imperfectly competitive product markets and sticky prices. Money demand came from its role of transactions. They computed dynamics by solving first- and second- order logarithmic approximations to the Ramsey planner’s policy functions around a non-stochastic Ramsey steady state.
Paustian and Stoltenberg (2008) concentrated on optimal nominal interest rate stabilization rather than inflation stabilization. The investigation was done through the linear–quadratic framework and an MIU model with transaction friction (Walsh, 2003a). They approximated their MIU model around the optimal steady state and derived a second-order approximation to households’ utility as what Woodford (2003a, Chapter 6) did.
From the perspective of welfare cost, Ireland (2009) conducted dynamic OLS estimates on the measure provided by Lucas (2000). He proved that the shift in 1979 reduced welfare cost of inflation, which was brought by money demand distortions.
Contrary to conventional view, Bilbiie and Straub (2013) found that when asset market participation rate was low, passive monetary policy maximized welfare. They used Bayesian estimation techniques in providing the empirical evidence that pre-Volcker passive policy was closer to optimal.
Rather than simple rule, Chen, et al (2013) described optimal policy with various forms – commitment, discretion and quasi-commitment. They found that there had been significant welfare gains to the conservatism in policy making which was adopted following the Volcker disinflation. Much of the high inflation in pre-Volcker era could have been avoided had policy makers been able to commit instead of conducting discretion.
To sum up, the policy change in 1979 was noticed and admitted by most economists. Lots of effort has been devoted to describe and explain for such transition from various aspects. Monopolistic competition and nominal rigidity was introduced in most models. Welfare analysis was conducted with log-linearized approximation. Nevertheless, Optimal Simple Rule in this period has not been investigated. Thus, I will explore the feasible Optimal Simple Rue with the framework of Kriwoluzky and Stoltenberg (2015). Because their model is one of the most convincing and the most recent models.
3 Economic Environment
In this section, I will present the basics of the model I employ. The model is a New Keynesian model equipped with transaction friction. That is to say, purchasing consumption goods takes additional transaction cost, while holding money reduces such cost. However, holding money will let go the interest and utility brought by consumption. Therefore, households will choose to reach the balance between the opposite effects.
The economy comprises of infinitely lived households indexed by j ∈ [0,1]. They have the same asset endowments and preferences, and they face the following budget constraint in each period t
𝑀𝑗𝑡+ 𝐵𝑗𝑡+ 𝔼𝑡[𝑞𝑡,𝑡+1𝑋𝑗𝑡+1] + 𝑃𝑡𝑐𝑗𝑡+ 𝑃𝑡𝜙 (𝑐𝑗𝑡, 𝑧𝑡𝑀𝑗𝑡−1 𝑃𝑡 ) ≤ 𝑀𝑗𝑡−1+ 𝑅𝑡−1𝐵𝑗𝑡−1+ 𝑋𝑗𝑡+ 𝑃𝑡𝑤𝑗𝑡𝑙𝑗𝑡+ ∫ 𝐷𝑗𝑖𝑡𝑑𝑖 1 0 − 𝑃𝑡𝜏𝑡, (1)
where 𝑀𝑗𝑡 denotes end-of-period nominal balances. Household j carries government bonds
𝐵𝑗𝑡−1 from the previous period with gross nominal interest rate 𝑅𝑡−1. A portfolio of state- contingent claims on other households with random payoff 𝑋𝑗𝑡 is also available. In period t,
such claims materializing in t+1 is priced as 𝔼𝑡[𝑞𝑡,𝑡+1𝑋𝑗𝑡+1] . 𝑃𝑡 represents the aggregate price level, 𝑐𝑗𝑡 a consumption, 𝜔𝑗𝑡 the real wage rate for labor services 𝑙𝑗𝑡 , 𝜏𝑡 a lump sum tax. Households act as monopolistic suppliers and firms are monopolistically competitive. 𝐷𝑗𝑖𝑡 is dividends from firms indexed by i. Purchasing consumption good is assumed to be costly, and such real resource costs of transaction are captured by 𝜙(𝑐𝑗𝑡, ℎ𝑗𝑡), where ℎ𝑗𝑡 = 𝑧𝑡𝑀𝑗𝑡−1/𝑃𝑡 denotes effective real money balances. 𝑧𝑡 is a transaction-cost technology shock with mean of 1. Moreover, transaction costs are assumed to increase in consumption (𝜙𝑐 ≥ 0), and to decrease strictly in real money balances (𝜙ℎ < 0 ). Marginal transaction costs of consumption are non-increasing in real money balances (𝜙𝑐ℎ ≤ 0 ) and non-decreasing in consumption (𝜙𝑐𝑐 ≥ 0 ). Marginal resource gains of holding money are strictly decreasing (𝜙ℎℎ < 0).
Household j maximizes the following objective
𝔼𝑡0∑ 𝛽
𝑡[𝑢(𝑐
𝑗𝑡, 𝜈𝑡) − 𝑣(𝑙𝑗𝑡)], ∞
𝑡=𝑡0 (2)
where 𝛽 ∈ (0,1) represents the discount factor. We assume the utility function to be non-decreasing in consumption, non-decreasing in labor supply, and strictly concave. 𝜈𝑡 is the taste shock, with mean 1. Besides, consumption and real money balances are normal goods. Households set wage by supplying differentiated types of labor 𝑙𝑗𝑡.
Nominal rigidity in the form of staggered price setting is introduced through the Calvo Pricing model. The final consumption good 𝑦𝑡 is an aggregate of differentiated goods produced by monopolistically competitive firms indexed 𝑖 ∈ [0,1] , and defined as 𝑦𝑡𝜍−1𝜍 =
∫ 𝑦𝑖𝑡
𝜍−1 𝜍
1
0 𝑑𝑖, with 𝜍 > 1 . Production function is 𝑦𝑖𝑡 = 𝑎𝑡𝑙𝑖𝑡, where 𝑎𝑡 denotes a productivity shock with mean 1. In each period, a fraction 𝛼 ∈ [0,1) of firms may reset their prices according to the simple rule 𝑃𝑖𝑡 = 𝜋̅𝑃𝑖𝑡−1, where 𝜋̅ denotes the average inflation rate. While 1 − 𝛼 of randomly selected firms could set new price 𝑃̃𝑖𝑡.
The central bank conducts monetary policy rule to control short-term interest rate 𝑅𝑡. The simple feedback rule is
𝑅̂𝑡 =𝜌𝜋𝜋̂𝑡+𝜌𝜋1𝜋̂𝑡−1+𝜌𝑦𝑦̂𝑡+𝜌𝑅𝑅̂𝑡−1+𝜌𝑚𝑚̂𝑡+𝜌𝑚1𝑚̂𝑡−1+ 𝜖𝑡, (3)
with 𝑓𝜋 > 0, 𝑟𝑡𝑚 a monetary policy shock with mean 1. Government budget is constrained by 𝑀𝑡−1+ 𝑅𝑡−1𝐵𝑡−1+ 𝑃𝑡𝑔𝑡 = 𝑀𝑡+ 𝐵𝑡+ 𝑃𝑡𝜏𝑡, where 𝑔𝑡 denotes exogenous government expenditures. 𝑔𝑡 has a mean of 𝑔̅, which is restricted to a constant fraction of steady-state output, 𝑔̅ = 𝑦̅(1 − 𝑆𝑐). 𝑆𝑐 is the output share of private consumption.
Aggregate resource constraint is given by
𝑦𝑡= 𝑎𝑡𝑙𝑡/Δ𝑡, (4)
where Δ𝑡= ∫ (𝑃𝑖𝑡/𝑃𝑡)−𝜍 1
0 𝑑𝑖 ≥ 1 captures price dispersion. Goods market clearing requires
𝑐𝑡+ 𝑔𝑡+ 𝜙(𝑐𝑡, ℎ𝑡) = 𝑦𝑡. (5) There are six kinds of shocks existing in this system, namely productivity technology shock 𝑎𝑡, government expenditure shock 𝑔𝑡, wage mark-up shock 𝜇𝑡 , taste shock 𝜐𝑡 , transaction cost technology shock 𝑧𝑡 and monetary policy shock 𝑟𝑡𝑚. They follow an AR (1) process with autocorrelation coefficients 𝜌𝑎, 𝜌𝑔, 𝜌𝜇, 𝜌𝜐, 𝜌𝑧 ∈ [0,1).
The recursive equilibrium requires that households maximize their utility (2) subject to the budget constraints (1), and firms chosen to set the new price maximize their profit. Also, the aggregate resource constraint and market clearing condition need to be satisfied. At last, the transversality condition should be fulfilled. As analyzed in Kriwoluzky and Stoltenberg (2015),
the equilibrium must satisfy the following log-linearized first order conditions. 𝜎̃𝔼𝑡𝑐̂𝑡+1− 𝔼𝑡𝜐̂𝑡+1− 𝜂𝑐ℎ(𝑚̂𝑡− 𝔼𝑡𝜋̂𝑡+1+ 𝔼𝑡𝑧̂𝑡+1) = 𝜎̃𝑐̂𝑡− 𝜐̂𝑡− 𝜂𝑐ℎ(𝑚̂𝑡−1− 𝜋̂𝑡+ 𝑧̂𝑡) + 𝑅̂𝑡− 𝔼𝑡𝜋̂𝑡+1, (6) ω(𝑦̂𝑡− 𝑎̂𝑡) + 𝜇̂𝑡− 𝑤̂𝑡 = −𝜎̃𝑐̂𝑡+ 𝜂𝑐ℎ(𝑚̂𝑡−1− 𝜋̂𝑡+ 𝑧̂𝑡) + 𝜐̂𝑡, (7) 𝜋̂𝑡 = 𝛽𝔼𝑡𝜋̂𝑡+1+ κ(𝑤̂𝑡− 𝑎̂𝑡), (8) 𝑦̂𝑡 = 𝑠𝐶𝑐̂𝑡+ 𝑔̂𝑡, (9) 𝑚̂𝑡= −𝜂𝑅𝑅̂𝑡+𝜂hc 𝜎ℎ 𝔼𝑡𝑐̂𝑡+1+ 𝔼𝑡𝜋̂𝑡+1+ 1−𝜎ℎ 𝜎ℎ 𝔼𝑡𝑧̂𝑡+1, (10) where 𝜎𝑐 = −𝑢𝑐𝑐𝑐/𝑢𝑐 , ω = 𝑣𝑙𝑙𝑙/𝑣𝑙 , 𝜅 = (1 − 𝛼)(1 − 𝛼𝛽)/𝛼 , 𝜂𝑐ℎ = −𝜙𝑐ℎℎ/(1 + 𝜙𝑐) , 𝜂ℎ𝑐 = 𝜙ℎ𝑐𝑐/𝜙ℎ , 𝜂𝑅 = 𝑅/𝜎ℎ(𝑅 − 1) , 𝜎ℎ = −𝜙ℎℎℎ/𝜙ℎ , and 𝜎̃ = 𝜎𝑐 + 𝜂𝑐𝑐 , 𝜂𝑐𝑐 = 𝜙𝑐𝑐𝑐/ (1 + 𝜙𝑐). Here 𝑥̂𝑡 denotes the percentage deviation of a variable 𝑥𝑡 from its steady state value 𝑥 . 𝜂𝑐ℎ captures transaction frictions and the importance of money. Different from the simple Taylor rule provided by Kriwoluzky and Stoltenberg (2015), I add other instruments to form the Optimal Simple Rule 𝑅̂𝑡= 𝜌𝜋𝜋̂𝑡+𝜌𝜋1𝜋̂𝑡−1+𝜌𝑦𝑦̂𝑡+𝜌𝑅𝑅̂𝑡−1+𝜌𝑚𝑚̂𝑡+𝜌𝑚1𝑚̂𝑡−1+ 𝜖𝑡, (11)
The upcoming thesis investigate optimal monetary policy under such framework through welfare analysis.
4 Welfare Analysis
In this section, I will first point out the loss function applied. Then, I will explain the procedure of the two ways to calculate loss, i.e. Ramsey policy and the Optimal Simple Rule. At last, I will show how to convert them into a comparable level.
Kriwoluzky and Stoltenberg (2015) mainly focused on the Taylor rule that revises the policy rate in response to changes in inflation and output. However, optimal monetary policy is not limited by these two variables. Other instruments could also be considered in minimizing welfare loss. Dynare provides two methods to get the optimal monetary policy. One is Ramsey Policy, the other is Optimal Simple Rule. Before going further into the two policies, I will show the exact loss function that I use for both policies.
4.1 Loss Function
I adopt the loss function from Paustian and Stoltenberg (2008), as their model also includes transaction frictions. The loss function takes the form of:
𝑈𝑡0 = −𝛺𝔼𝑡0∑∞ 𝛽𝑡−𝑡0 𝑡=𝑡0 [𝜆𝑥𝑦̂ 2+ 𝜋̂ 𝑡2+ 𝜆𝑅𝑅̂𝑡 2 ] + 𝑡. 𝑖. 𝑠. 𝑝 + 𝒪(∥∙∥3), (12)
where 𝑡. 𝑖. 𝑠. 𝑝 represents the policy independent terms, 𝒪(∥∙∥3) the terms of order higher than 2, 𝛺 = 𝑢𝑐𝑦𝜃(𝜔 + 𝜎)/(2𝜅) and 𝜅 = (1 − 𝛼)(1 − 𝛼𝛽)/𝛼. In the objective function (12), the weights on output gap, inflation and nominal interest rate are 𝜆𝑥, 1, 𝜆𝑅
respectively. 𝜆𝑥 and 𝜆𝑅 are defined as the following:
𝜆𝑥= 𝜅
𝜃 , (13)
𝜆𝑅 = 𝜂𝑅𝜆𝑥
𝑣𝑛(𝜔+𝜎) , (14)
where 𝑣𝑛 = 𝑦/𝑚 indicates the money velocity and 𝜂𝑅 demonstrates the interest elasticity of money demand in the steady state. I take 𝑣1 = 5.419 as the velocity of M1 money stock in the United States from Q1 1959 to Q4 1979, and 𝑣2 = 1.730 as the velocity of M2 money stock. After manual computation, 𝜆𝑥 ≈ 0.2273, 𝜆𝑅1 ≈ 0.1851 and 𝜆𝑅2 ≈ 0.5797.
4.2 Ramsey Policy
choose the interest rate minimizing the inter-temporal welfare loss and commit not to change the policy any more in the future. Such policy requires the central bank to make every effort exploring and using all information with perfect foresight. Hence, the loss of Ramsey policy could be taken as a standard of welfare analysis.
The command ‘ramsey_policy’ in Dynare computes the first-order approximation of the policy that maximizes the policy maker’s objective function subject to the constraints provided by the equilibrium path of the private economy and under commitment to this optimal policy. The planner’s objective must be declared with the command ‘planner_objective’, which only creates the expanded model and does not perform any computations. In the output, I can find the planner’s objective value that reflects social welfare loss.
4.3 Optimal Simple Rule
Optimal Simple Rule specifies its targeting variables. I will use Dynare to figure out the optimal value of the coefficients in front of each variable. The command ‘osr’ computes a subset of model parameters to minimize the weighted (co)-variance of a specified subset of endogenous variables, subject to a linear law of motion implied by the first order conditions of the model. To fully exploit the potential policy instruments, I employ the simple rule as:
𝑅̂𝑡= 𝜌𝜋𝜋̂𝑡+𝜌𝜋1𝜋̂𝑡−1+𝜌𝑦𝑦̂𝑡+𝜌𝑅𝑅̂𝑡−1+𝜌𝑚𝑚̂𝑡+𝜌𝑚1𝑚̂𝑡−1, (15)
where the optimal set of parameters 𝜌𝜋, 𝜌𝜋1, 𝜌𝑦, 𝜌𝑅, 𝜌𝑚, 𝜌𝑚1 is to be discovered. Other than
inflation, I try to target at past inflation, output gap, past interest rate, current real money balances and past real money balances.
4.4 Comparison between Ramsey Policy and Optimal Simple Rule
The most important thing to find the closest Optimal Simple Rule is to compare the welfare loss between Optimal Simple Rule and Ramsey Policy. Nevertheless, the comparison can not be done straightforwardly. The key difference lies in that Ramsey minimizes a discounted sum
of conditional variances, while Optimal Simple Rule minimizes a sum of discounted unconditional variances. More precisely, the Ramsey policy is computed by approximating the equilibrium system around the steady state of the Lagrange multipliers. In this case, the initial Lagrange multiplier is set to 0. The Ramsey planner exploits its ability to surprise households in the first period of implementing Ramsey policy. This is the value of implementing optimal policy for the first time and committing not to re-optimize in the future, which is in line with the explanation in section 4.2.
Thus, the ‘planner objective value’ displayed by Ramsey policy and the ‘objective function’ in the output of Optimal Simple Rule are of distinct foresight. Therefore, I manually compute the loss by inserting variances from the oo_.var file in both programs into the following equation:
𝐿 = 𝜆𝑥𝑣𝑎𝑟(𝑦̂𝑡) + 𝑣𝑎𝑟(𝜋̂𝑡) + 𝜆𝑅𝑣𝑎𝑟(𝑅̂𝑡). (16)
Since these variances are of identical meaning, the welfare loss becomes comparable. It is worth mentioning here that the loss is measured in utils. For better readability and
4.5 Calibration
Table 1. Parameters and Corresponding Values
Notation Value Interpretation
𝛼 0.3304 Calvo pricing coefficient
𝛽 0.99 Discount factor
𝜔 3.5 Frisch elasticity
𝜃 6 Loss function coefficient
𝑠𝑐 0.8 Average output share of private consumption
𝑅𝑠𝑠 1.0146 Steady state nominal interest rate
𝑣1 5.419 Velocity of 𝑀1 money stock
𝑣2 1.730 Velocity of 𝑀2 money stock
𝜎̃ 1 Relative risk aversion coefficient 𝜂𝑐ℎ 0.4587 Transaction friction proxy
𝜂ℎ𝑐 3.0792 Cross derivative of real money balances and consumption
𝜎ℎ 4.1656 Output elasticity of real money balances
𝜎𝑐 1 Relative risk aversion
𝜎𝑣 0.0112 Standard deviation taste shock
𝜎𝑧 0.0513 Standard deviation transaction cost technology shock
𝜎𝑎 0.0068 Standard deviation productivity shock
𝜎𝑔 0.0084 Standard deviation government expenditure shock
𝜎𝜇 0.0417 Standard deviation wage mark-up shock
𝜎𝑚 0.0051 Standard deviation monetary policy shock
𝜌𝑣 0.6637 AR1 coefficient taste shock
𝜌𝑧 0.8943 AR1 coefficient transaction cost technology shock
𝜌𝑎 0.9221 AR1 coefficient productivity shock
𝜌𝑔 0.8535 AR1 coefficient government expenditure shock
𝜌𝜇 0.8501 AR1 coefficient wage mark-up shock
𝜌𝜋 0.9692 Monetary policy coefficient inflation
5 Quantitative Results
In this section, I will illustrate the results of welfare analysis. First, I will go through the analyzing procedure. Second, I point out the optimal monetary policy which is the core of the thesis. Third, I will explain the effectiveness of each targeting instrument. Fourth, I will shed light on the influence of changing money velocity.
5.1 Analyzing Procedure and the Optimal Simple Rule
My investigation of Optimal Simple Rule consists of two groups that employ velocity of M1 and M2 in the loss function accordingly. I conduct six subgroups of welfare analysis in each group. Above all, I compute the loss of Ramsey Policy and set it as the benchmark of the group. Then, I categorize all 63 policy instrument combinations into six subgroups, based on the number of targeting variables. Next, I calculate the loss of each simple rule and record the corresponding optimal value of the coefficients. To get reasonable results, I restrict the feedback coefficients to be non-negative and the one on past interest rates to be in the interval [0,1]. The losses of the simple rules are converted into relative losses, which are the original values divided by the benchmark values of the Ramsey policy. The results are shown in Table 2.
Table 2. Losses and Feedback Coefficients of the Policies M1 Instruments Relative Losses Feedback Coefficients 𝜌𝜋 𝜌𝜋1 𝜌𝑦 𝜌𝑅 𝜌𝑚 𝜌𝑚1 𝜋𝑡 1.438 0.426 𝜋𝑡−1 N/A 𝑦𝑡 111.707 0.236 𝑅𝑡−1 N/A 𝑚𝑡 N/A 𝑚𝑡−1 25.272 0.036 𝜋𝑡 𝜋𝑡−1 N/A 𝜋𝑡 𝑦𝑡 1.404 0.451 0.056 𝜋𝑡 𝑅𝑡−1 N/A 𝜋𝑡 𝑚𝑡 N/A 𝜋𝑡 𝑚𝑡−1 1.229 9.582 99.263 𝑦𝑡 𝑅𝑡−1 110.661 0.404 0.712 𝑦𝑡 𝑚𝑡−1 25.488 0.001 0.036 𝑦𝑡 𝜋𝑡−1 N/A 𝑦𝑡 𝑚𝑡 N/A 𝜋𝑡−1 𝑅𝑡−1 N/A 𝜋𝑡−1 𝑚𝑡 N/A 𝜋𝑡−1 𝑚𝑡−1 1.979 0.078 0.056 𝑅𝑡−1 𝑚𝑡 N/A 𝑅𝑡−1 𝑚𝑡−1 25.594 0.645 0.059 𝑚𝑡 𝑚𝑡−1 N/A 𝜋𝑡 𝜋𝑡−1 𝑦𝑡 1.404 0.451 0.000 0.056
Instruments Relative Losses Feedback Coefficients 𝜌𝜋 𝜌𝜋1 𝜌𝑦 𝜌𝑅 𝜌𝑚 𝜌𝑚1 𝜋𝑡 𝜋𝑡−1 𝑅𝑡−1 N/A 𝜋𝑡 𝜋𝑡−1 𝑚𝑡 N/A 𝜋𝑡 𝜋𝑡−1 𝑚𝑡−1 1.293 54.753 60.177 62.022 𝜋𝑡 𝑦𝑡 𝑅𝑡−1 1.404 0.451 0.056 0.000 𝜋𝑡 𝑦𝑡 𝑚𝑡 N/A 𝜋𝑡 𝑦𝑡 𝑚𝑡−1 1.229 9.693 0.003 99.982 𝜋𝑡 𝑅𝑡−1 𝑚𝑡 N/A 𝜋𝑡 𝑅𝑡−1 𝑚𝑡−1 1.229 9.680 0.082 99.562 𝜋𝑡 𝑚𝑡 𝑚𝑡−1 1.229 9.851 0.025 99.485 𝜋𝑡−1 𝑦𝑡 𝑅𝑡−1 N/A 𝜋𝑡−1 𝑦𝑡 𝑚𝑡 N/A 𝜋𝑡−1 𝑦𝑡 𝑚𝑡−1 2.599 76.739 0.002 28.343 𝜋𝑡−1 𝑅𝑡−1 𝑚𝑡 N/A 𝜋𝑡−1 𝑅𝑡−1 𝑚𝑡−1 2.593 74.465 0.422 27.505 𝜋𝑡−1 𝑚𝑡 𝑚𝑡−1 N/A 𝑦𝑡 𝑅𝑡−1 𝑚𝑡 N/A 𝑦𝑡 𝑅𝑡−1 𝑚𝑡−1 25.657 0.001 0.683 0.060 𝑦𝑡 𝑚𝑡 𝑚𝑡−1 1053.413 0.722 0.934 0.860 𝑅𝑡−1 𝑚𝑡 𝑚𝑡−1 N/A 𝜋𝑡 𝜋𝑡−1 𝑦𝑡 𝑅𝑡−1 N/A 𝜋𝑡 𝜋𝑡−1 𝑦𝑡 𝑚𝑡 N/A 𝜋𝑡 𝜋𝑡−1 𝑦𝑡 𝑚𝑡−1 1.229 9.573 0.082 0.009 99.608 𝜋𝑡 𝜋𝑡−1 𝑅𝑡−1 𝑚𝑡 N/A 𝜋𝑡 𝜋𝑡−1 𝑅𝑡−1 𝑚𝑡−1 1.229 9.582 16.647 0.214 98.550 𝜋𝑡 𝜋𝑡−1 𝑚𝑡 𝑚𝑡−1 N/A 𝜋𝑡 𝑦𝑡 𝑅𝑡−1 𝑚𝑡 N/A 𝜋𝑡 𝑦𝑡 𝑅𝑡−1 𝑚𝑡−1 1.229 9.721 0.014 0.101 99.483 𝜋𝑡 𝑦𝑡 𝑚𝑡 𝑚𝑡−1 1.229 9.701 0.001 0.001 99.978 𝜋𝑡 𝑅𝑡−1 𝑚𝑡 𝑚𝑡−1 1.229 9.663 0.090 0.024 99.506 𝜋𝑡−1 𝑦𝑡 𝑅𝑡−1 𝑚𝑡 N/A 𝜋𝑡−1 𝑦𝑡 𝑅𝑡−1 𝑚𝑡−1 1.994 0.081 0.003 0.008 0.057 𝜋𝑡−1 𝑦𝑡 𝑚𝑡 𝑚𝑡−1 N/A 𝜋𝑡−1 𝑅𝑡−1 𝑚𝑡 𝑚𝑡−1 N/A 𝑦𝑡 𝑅𝑡−1 𝑚𝑡 𝑚𝑡−1 360.948 0.825 0.826 0.908 0.848 𝑤𝑖𝑡ℎ𝑜𝑢𝑡 𝜋𝑡 N/A 𝑤𝑖𝑡ℎ𝑜𝑢𝑡 𝜋𝑡−1 1.229 9.829 0.008 0.069 0.028 99.516 𝑤𝑖𝑡ℎ𝑜𝑢𝑡 𝑦𝑡 N/A 𝑤𝑖𝑡ℎ𝑜𝑢𝑡 𝑅𝑡−1 1.246 18.972 0.473 0.244 23.555 90.536 𝑤𝑖𝑡ℎ𝑜𝑢𝑡 𝑚𝑡 1.229 9.717 0.061 0.010 0.053 99.598 𝑤𝑖𝑡ℎ𝑜𝑢𝑡 𝑚𝑡−1 N/A 𝑎𝑙𝑙 𝑖𝑛𝑠𝑡𝑟𝑢𝑚𝑒𝑛𝑡𝑠 1.276 80.337 1.066 6.446 0.297 2.438 58.560
After computation, the loss of Ramsey policy is indeed the smallest compared to all other sets of Optimal Simple Rules. It can be proved by the fact that all relative losses are bigger than one. At the same time, the relative losses of the simple rules cover a wide range from 1.229 to 1053.413, but mainly concentrate in several scales. For clearer comparison, I sequence the results according to the value of the welfare losses and drop out the ones arousing indeterminacy or no stable equilibrium. Then, I get Table 3.
Table 3. Sorted Losses and Feedback Coefficients of the Policies M1 Instruments Set Relative
Losses Feedback Coefficients 𝜌𝜋 𝜌𝜋1 𝜌𝑦 𝜌𝑅 𝜌𝑚 𝜌𝑚1 𝜋𝑡 𝑦𝑡 𝑚𝑡 𝑚𝑡−1 1 1.229 9.701 0.001 0.001 99.978 𝜋𝑡 𝑦𝑡 𝑚𝑡−1 2 1.229 9.693 0.003 99.982 𝜋𝑡 𝑚𝑡−1 3 1.229 9.582 99.263 𝜋𝑡 𝑅𝑡−1 𝑚𝑡−1 4 1.229 9.680 0.082 99.562 𝜋𝑡 𝑚𝑡 𝑚𝑡−1 5 1.229 9.851 0.025 99.485 𝜋𝑡 𝜋𝑡−1 𝑅𝑡−1 𝑚𝑡−1 6 1.229 9.582 16.647 0.214 98.550 𝜋𝑡 𝜋𝑡−1 𝑦𝑡 𝑚𝑡−1 7 1.229 9.573 0.082 0.009 99.608 𝑤𝑖𝑡ℎ𝑜𝑢𝑡 𝑚𝑡 8 1.229 9.717 0.061 0.010 0.053 99.598 𝜋𝑡 𝑅𝑡−1 𝑚𝑡 𝑚𝑡−1 9 1.229 9.663 0.090 0.024 99.506 𝜋𝑡 𝑦𝑡 𝑅𝑡−1 𝑚𝑡−1 10 1.229 9.721 0.014 0.101 99.483 𝑤𝑖𝑡ℎ𝑜𝑢𝑡 𝜋𝑡−1 11 1.229 9.829 0.008 0.069 0.028 99.516 𝑤𝑖𝑡ℎ𝑜𝑢𝑡 𝑅𝑡−1 12 1.246 18.972 0.473 0.244 23.555 90.536 𝑎𝑙𝑙 𝑖𝑛𝑠𝑡𝑟𝑢𝑚𝑒𝑛𝑡𝑠 13 1.276 80.337 1.066 6.446 0.297 2.438 58.560 𝜋𝑡 𝜋𝑡−1 𝑚𝑡−1 14 1.293 54.753 60.177 62.022 𝜋𝑡 𝑦𝑡 15 1.404 0.451 0.056 𝜋𝑡 𝜋𝑡−1 𝑦𝑡 16 1.404 0.451 0.000 0.056 𝜋𝑡 𝑦𝑡 𝑅𝑡−1 17 1.404 0.451 0.056 0.000 𝜋𝑡 18 1.438 0.426 𝜋𝑡−1 𝑚𝑡−1 19 1.979 0.078 0.056 𝜋𝑡−1 𝑦𝑡 𝑅𝑡−1 𝑚𝑡−1 20 1.994 0.081 0.003 0.008 0.057 𝜋𝑡−1 𝑅𝑡−1 𝑚𝑡−1 21 2.593 74.465 0.422 27.505 𝜋𝑡−1 𝑦𝑡 𝑚𝑡−1 22 2.599 76.739 0.002 28.343 𝑚𝑡−1 23 25.272 0.036 𝑦𝑡 𝑚𝑡−1 24 25.488 0.001 0.036 𝑅𝑡−1 𝑚𝑡−1 25 25.594 0.645 0.059 𝑦𝑡 𝑅𝑡−1 𝑚𝑡−1 26 25.657 0.001 0.683 0.060 𝑦𝑡 𝑅𝑡−1 27 110.661 0.404 0.712 𝑦𝑡 28 111.707 0.236 𝑦𝑡 𝑅𝑡−1 𝑚𝑡 𝑚𝑡−1 29 360.948 0.825 0.826 0.908 0.848 𝑦𝑡 𝑚𝑡 𝑚𝑡−1 30 1053.413 0.722 0.934 0.860
I notice that the outcomes can be divided into three scales. The first scale is the closest to Ramsey policy, and the feedback coefficients of inflation are larger than one. That is to say, these simple rules are active interest rate policies. The second scale is close to Ramsey policy and 𝜌𝜋 is smaller than one. Hence, the simple rules in this scale are passive interest rate policies.
The third scale is far from the benchmark. The Optimal Simple Rule boils down to set 1, the combination of 𝜋𝑡 𝑦𝑡 𝑚𝑡 𝑚𝑡−1. More intuitively, it takes the form of:
𝑅̂𝑡 =9.701𝜋̂𝑡+0.001𝑦̂𝑡+0.001𝑚̂𝑡−1+99.978𝑚̂𝑡−1. (16) Since 𝜌𝜋 > 1, the Optimal Simple Rule is an active interest rate policy that obeys the Taylor
Principle. In other words, for each percent increase in inflation, the central bank should raise the nominal interest rate by 9.701 percent to fight instability.
5.2 Effectiveness of the Instruments
It is interesting to see the results of the second last subgroup in Table 2. Optimal Simple Rule without inflation, output gap and past real money balances does not exist. Hence, I conclude that these three instruments are indispensable when constructing Optimal Simple Rule with my six candidate instruments. It can be verified by the Optimal Simple Rule highlighted above. I observe that the top 18 sets in Table 3 all incorporate inflation as a policy instrument. Further, only the simple rules targeting on inflation cover the first two scales. Therefore, I discover that inflation is the most important instrument a policy maker has. Comparing the first scale and the second scale, I find that it is the inclusion of past real money balances that switches the interest rate policy stance. By adding past real money balancesto the simple rule (set 18) from Kriwoluzky and Stoltenberg (2015), the Optimal policy turns from passive (𝜌𝜋=0.426) to active.
It is because that the transaction role of money provides another way to affect interest rate and price stability. Besides, the weights put on past real money balances are very large, which reach the upper bound I set. According to set 26-27, output gap does not work well without inflation. Although it is necessary in formulating Optimal Simple Rule, the weights put on output gap are relatively small.
It is also interesting to notice set 8,11,12 in Table 3. When each of real money balances, past inflation and past interest rate is taken out of all the instruments, the loss is fairly low. Moreover, all these three variables arouse no stable equilibrium. Hence, I believe these instruments are trivial in minimizing welfare loss.
Without the essential inflation, past inflation only works with past real money balances according to set 19-22. Comprising past inflation will increase welfare loss, which can be proved by set 4 and 6, set 2 and 7. Past interest rate also raises loss (see set 3 and set 4). Thus, interest rate smoothing does not work in this case. Current real money balance only emerges when past real money balance shows up. The influence of real money balance is ambiguous. Comparing set 1 with set 2, I find that real money balance reduce loss. But set 3 and set 5 show the opposite effect.
To sum up, each instrument has various effectiveness in Optimal Simple Rule. The interactions among the variables also influence the final consequence. It is not the case that the more variables I incorporate, the lower the welfare loss become.
5.3 Influence of Money Velocity
In the following part, I adjust the velocity of money stock from M1 to M2 following the same procedure stated above. In the first place, I focus on the social welfare loss. To compare the results in an obvious way, I put two groups of results together in Table 4.
I see similar loss ranking and similar relative loss values with the one under M1. Also, the benchmark provided by Ramsey policy are similar. The scales remain the same, with most of the switches caused by past interest rate. None of the changes results from output gap or inflation seeing the second scale. It can be explained by the definition of 𝜆𝑅, as money velocity only shows up there. The decrease in velocity raises the weight put on interest rate in the objective function, which leads to the changes on interest rate and real money balances related simple rules. Moreover, relative losses are reduced because of the change in money velocity, except for the second scale. It is due to the increase in 𝜆𝑅 as well. The Optimal Simple Rule
combines 𝜋𝑡 𝑦𝑡 𝑚𝑡−1, with a bit change in the coefficients. Now, the policy reaction function follows:
𝑅̂𝑡 =8.937𝜋̂𝑡+0.001𝑦̂𝑡+99.981𝑚̂𝑡−1, (17) with less attention paid to inflation and more paid to real money balances.
Table 4. Comparison of Welfare Loss between M1 and M2 Set Instruments M1 Relative
Losses M1 Instruments M2 Relative Losses M2 1 𝜋𝑡 𝑦𝑡 𝑚𝑡 𝑚𝑡−1 1.229 𝜋𝑡 𝑦𝑡 𝑚𝑡−1 1.228 2 𝜋𝑡 𝑦𝑡 𝑚𝑡−1 1.229 𝜋𝑡 𝑦𝑡 𝑅𝑡−1 𝑚𝑡−1 1.228 3 𝜋𝑡 𝑚𝑡−1 1.229 𝜋𝑡 𝜋𝑡−1 𝑚𝑡−1 1.228 4 𝜋𝑡 𝑅𝑡−1 𝑚𝑡−1 1.229 𝜋𝑡 𝜋𝑡−1 𝑦𝑡 𝑚𝑡−1 1.228 5 𝜋𝑡 𝑚𝑡 𝑚𝑡−1 1.229 𝜋𝑡 𝑅𝑡−1 𝑚𝑡−1 1.228 6 𝜋𝑡 𝜋𝑡−1 𝑅𝑡−1 𝑚𝑡−1 1.229 𝜋𝑡 𝑦𝑡 𝑚𝑡 𝑚𝑡−1 1.228 7 𝜋𝑡 𝜋𝑡−1 𝑦𝑡 𝑚𝑡−1 1.229 𝜋𝑡 𝑚𝑡 𝑚𝑡−1 1.228 8 𝑤𝑖𝑡ℎ𝑜𝑢𝑡 𝑚𝑡 1.229 𝑤𝑖𝑡ℎ𝑜𝑢𝑡 𝜋𝑡−1 1.228 9 𝜋𝑡 𝑅𝑡−1 𝑚𝑡 𝑚𝑡−1 1.229 𝜋𝑡 𝑅𝑡−1 𝑚𝑡 𝑚𝑡−1 1.228 10 𝜋𝑡 𝑦𝑡 𝑅𝑡−1 𝑚𝑡−1 1.229 𝜋𝑡 𝜋𝑡−1 𝑅𝑡−1 𝑚𝑡−1 1.228 11 𝑤𝑖𝑡ℎ𝑜𝑢𝑡 𝜋𝑡−1 1.229 𝑤𝑖𝑡ℎ𝑜𝑢𝑡 𝑚𝑡 1.228 12 𝑤𝑖𝑡ℎ𝑜𝑢𝑡 𝑅𝑡−1 1.246 𝜋𝑡 𝑚𝑡−1 1.259 13 𝑎𝑙𝑙 𝑖𝑛𝑠𝑡𝑟𝑢𝑚𝑒𝑛𝑡𝑠 1.276 𝑎𝑙𝑙 𝑖𝑛𝑠𝑡𝑟𝑢𝑚𝑒𝑛𝑡𝑠 1.275 14 𝜋𝑡 𝜋𝑡−1 𝑚𝑡−1 1.293 𝑤𝑖𝑡ℎ𝑜𝑢𝑡 𝑅𝑡−1 1.275 15 𝜋𝑡 𝑦𝑡 1.404 𝜋𝑡 𝑦𝑡 1.405 16 𝜋𝑡 𝜋𝑡−1 𝑦𝑡 1.404 𝜋𝑡 𝜋𝑡−1 𝑦𝑡 1.405 17 𝜋𝑡 𝑦𝑡 𝑅𝑡−1 1.404 𝜋𝑡 𝑦𝑡 𝑅𝑡−1 1.405 18 𝜋𝑡 1.438 𝜋𝑡 1.439 19 𝜋𝑡−1 𝑚𝑡−1 1.979 𝜋𝑡−1 𝑚𝑡−1 1.978 20 𝜋𝑡−1 𝑦𝑡 𝑅𝑡−1 𝑚𝑡−1 1.994 𝜋𝑡−1 𝑦𝑡 𝑚𝑡−1 1.985 21 𝜋𝑡−1 𝑅𝑡−1 𝑚𝑡−1 2.593 𝜋𝑡−1 𝑦𝑡 𝑅𝑡−1 𝑚𝑡−1 2.008 22 𝜋𝑡−1 𝑦𝑡 𝑚𝑡−1 2.599 𝜋𝑡−1 𝑅𝑡−1 𝑚𝑡−1 2.594 23 𝑚𝑡−1 25.272 𝑚𝑡−1 25.268 24 𝑦𝑡 𝑚𝑡−1 25.488 𝑦𝑡 𝑚𝑡−1 25.485 25 𝑅𝑡−1 𝑚𝑡−1 25.594 𝑅𝑡−1 𝑚𝑡−1 25.590 26 𝑦𝑡 𝑅𝑡−1 𝑚𝑡−1 25.657 𝑦𝑡 𝑅𝑡−1 𝑚𝑡−1 25.653 27 𝑦𝑡 𝑅𝑡−1 110.661 𝑦𝑡 𝑅𝑡−1 110.646 28 𝑦𝑡 111.707 𝑦𝑡 111.692 29 𝑦𝑡 𝑅𝑡−1 𝑚𝑡 𝑚𝑡−1 360.948 𝑦𝑡 𝑅𝑡−1 𝑚𝑡 𝑚𝑡−1 360.895 30 𝑦𝑡 𝑚𝑡 𝑚𝑡−1 1053.413 𝑦𝑡 𝑚𝑡 𝑚𝑡−1 1053.294
The complete sequenced Optimal Policy Rules are displayed in Table 5. I discover that the effectiveness of the instruments discussed above does not change. Inflation works the best, then is real money balances. Output gap remains trivial and the other three instruments still brings negative effect in minimizing welfare loss.
Table 5. Sorted Losses and Optimal Coefficients of the Policies (M2) Instruments Set Relative
Losses Feedback Coefficients 𝜌𝜋 𝜌𝜋1 𝜌𝑦 𝜌𝑅 𝜌𝑚 𝜌𝑚1 𝜋𝑡 𝑦𝑡 𝑚𝑡−1 1 1.228 8.937 0.001 99.981 𝜋𝑡 𝑦𝑡 𝑅𝑡−1 𝑚𝑡−1 2 1.228 8.944 0.001 0.003 99.981 𝜋𝑡 𝜋𝑡−1 𝑚𝑡−1 3 1.228 8.945 0.014 99.344 𝜋𝑡 𝜋𝑡−1 𝑦𝑡 𝑚𝑡−1 4 1.228 8.970 0.047 0.008 99.630 𝜋𝑡 𝑅𝑡−1 𝑚𝑡−1 5 1.228 8.948 0.087 99.546 𝜋𝑡 𝑦𝑡 𝑚𝑡 𝑚𝑡−1 6 1.228 8.976 0.008 0.022 99.629 𝜋𝑡 𝑚𝑡 𝑚𝑡−1 7 1.228 8.881 0.033 99.583 𝑤𝑖𝑡ℎ𝑜𝑢𝑡 𝜋𝑡−1 8 1.228 9.448 0.011 0.060 0.025 99.514 𝜋𝑡 𝑅𝑡−1 𝑚𝑡 𝑚𝑡−1 9 1.228 8.839 0.051 0.024 99.518 𝜋𝑡 𝜋𝑡−1 𝑅𝑡−1 𝑚𝑡−1 10 1.228 9.058 0.048 0.115 99.399 𝑤𝑖𝑡ℎ𝑜𝑢𝑡 𝑚𝑡 11 1.228 8.900 0.080 0.012 0.098 99.585 𝜋𝑡 𝑚𝑡−1 12 1.259 55.230 62.083 𝑎𝑙𝑙 𝑖𝑛𝑠𝑡𝑟𝑢𝑚𝑒𝑛𝑡𝑠 13 1.275 79.420 1.187 6.069 0.276 3.596 59.047 𝑤𝑖𝑡ℎ𝑜𝑢𝑡 𝑅𝑡−1 14 1.275 77.561 1.111 6.139 4.085 57.067 𝜋𝑡 𝑦𝑡 15 1.405 0.428 0.049 𝜋𝑡 𝜋𝑡−1 𝑦𝑡 16 1.405 0.428 0.000 0.049 𝜋𝑡 𝑦𝑡 𝑅𝑡−1 17 1.405 0.428 0.049 0.000 𝜋𝑡 18 1.439 0.408 𝜋𝑡−1 𝑚𝑡−1 19 1.978 0.076 0.056 𝜋𝑡−1 𝑦𝑡 𝑚𝑡−1 20 1.985 0.077 0.001 0.056 𝜋𝑡−1 𝑦𝑡 𝑅𝑡−1 𝑚𝑡−1 21 2.008 0.082 0.002 0.023 0.058 𝜋𝑡−1 𝑅𝑡−1 𝑚𝑡−1 22 2.594 73.099 0.646 27.014 𝑚𝑡−1 23 25.268 0.036 𝑦𝑡 𝑚𝑡−1 24 25.485 0.001 0.036 𝑅𝑡−1 𝑚𝑡−1 25 25.590 0.645 0.059 𝑦𝑡 𝑅𝑡−1 𝑚𝑡−1 26 25.653 0.001 0.683 0.060 𝑦𝑡 𝑅𝑡−1 27 110.646 0.404 0.712 𝑦𝑡 28 111.692 0.236 𝑦𝑡 𝑅𝑡−1 𝑚𝑡 𝑚𝑡−1 29 360.895 0.825 0.826 0.908 0.848 𝑦𝑡 𝑚𝑡 𝑚𝑡−1 30 1053.294 0.722 0.934 0.860
6 Conclusion
I have illustrated that the Optimal Monetary Simple Rule in the pre-Volcker period in the U.S. differs under different money velocity. In the case of M1, it is the combination of targeting inflation 𝜋𝑡, output 𝑦𝑡, current real money balances 𝑚𝑡 and past real money balances 𝑚𝑡−1. Under M2, it is optimal to take out current real money balances 𝑚𝑡. Both policies are active interest rate policies contrary to what Kriwoluzky and Stoltenberg (2015) suggest. The change in policy stance is achieved by incorporating real money balances into the Optimal Simple Rule. Because the explicit transaction role enables real money balances to work. I also discover that inflation, output gap and past real money balances are indispensable instruments in maintaining stability. Inflation is the most effective instrument among the six. The adjustment in money velocity matters for the interest rate and real money balances related simple rules.
In this thesis, the loss function is not derived but given by Paustian and Stoltenberg (2008). It would be more accurate if the loss function can be derived theoretically from the utility function, or calculated by Dynare. Also, the losses would be more precise if they could be measured in certainty equivalent consumption rather than utils.
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Appendix
A. Source Code of Ramsey Policy //Endogenous variables
var c, y, m, w, R, r, pi, v, z, a, g, mu;
//Exogenous variables
varexo e_v, e_z, e_a, e_g, e_mu;
//Parameters
parameters beta alpha omega theta kappa s_c v_1 v_2 R_ss sigma_tilt eta_ch eta_hc eta_R sigma_h sigma_c
rho_pi rho_v rho_z rho_a rho_g rho_mu sigma_v sigma_z sigma_a sigma_g sigma_mu sigma_m
lambda_x lambda_R;
//Initialisation
alpha = 0.3304; // Calvo pricing probability beta = 0.99; // discount factor
omega = 3.5; // Frisch elasticity theta = 6; // parameter output gap
s_c = 0.8; // average output share of private consumption v_1 = 5.419; // M1 velocity
v_2 = 1.730; // M2 velocity
R_ss = 1.0146; // steady state nominal interest rate sigma_tilt = 1; // coefficient relative risk aversion eta_ch = 0.4587; // transaction friction proxy
eta_hc = 3.0792; // cross derivative of real money balances and consumption(S1)
sigma_h = 4.1656; // output elasticity of real money balances sigma_c = 1; //relative risk aversion
kappa = ((1 - alpha)*(1 - alpha * beta))/alpha; eta_R = R_ss/(sigma_h *(R_ss - 1));
rho_pi = 0.9692; //
rho_v = 0.6337; // AR1 coefficient taste shock
rho_z = 0.8943; // AR1 coefficient transaction cost technology shock rho_a = 0.9221; // AR1 coefficient productivity shock
rho_g = 0.8535; // AR1 coefficient government expenditure shock rho_mu = 0.8501; // AR1 coefficient wage mark-up shock
sigma_v = 0.0112; // standard deviation taste shock
sigma_z = 0.0513; // standard deviation transaction cost technology shock
sigma_a = 0.0068; // standard deviation productivity shock
sigma_g = 0.0084; // standard deviation government expenditure shock sigma_mu = 0.0417; // standard deviation wage mark-up shock
sigma_m = 0.0051; // standard deviation monetary policy shock lambda_x = kappa/theta; lambda_R = (eta_R*lambda_x/(sigma_tilt+omega))*(1/ v_2); //My model model(linear);
c(+1) = (v(+1)+ eta_ch *(m - pi(+1)+ z(+1))+ sigma_tilt * c - v - eta_ch *(m(-1) - pi + z) + R - pi(+1))*sigma_tilt;//Euler Equation omega *(y - a)+ mu - w = - sigma_tilt * c + eta_ch*(m(-1) - pi + z) + v;
pi = beta * pi(+1) + kappa*(w - a); y = s_c * c + g;//aggregate resource
m = - eta_R * R +(eta_hc/sigma_h)* c(+1)+ pi(+1)+ ((1 - sigma_h)/ sigma_h)* z(+1);//money demand
r = R - pi(+1); // fisher equation
v = rho_v * v(-1) + e_v; //taste shock
z = rho_z * z(-1) + e_z; //transaction cost technology shock a = rho_a * a(-1) + e_a; //productivity shock
g = rho_g * g(-1) + e_g; //government expenditure shock mu = rho_mu * mu(-1) + e_mu; //wage mark-up shock
end;
//shocks shocks;
var e_v = sigma_v^2; var e_z = sigma_z^2; var e_a = sigma_a^2; var e_g = sigma_g^2; var e_mu = sigma_mu^2; end;
initval; v = 0.5; z = 0.5; a = 0.5; g = 0.5; mu = 0.5; end; //Ramsey policy
planner_objective pi^2 + lambda_x *y^2 +lambda_R * R^2; ramsey_policy(planner_discount=0.99);
stoch_simul c y pi m R r; check;
B. Source Code of Optimal Simple Rule //Endogenous variables
var c, y, m, w, R, r, pi, v, z, a, g, mu;
//Exogenous variables
varexo e_v, e_z, e_a, e_g, e_mu;
//Parameters
parameters beta alpha omega theta kappa s_c v_1 v_2 R_ss sigma_tilt eta_ch eta_hc eta_R sigma_h sigma_c
rho_pi rho_v rho_z rho_a rho_g rho_mu sigma_v sigma_z sigma_a sigma_g sigma_mu sigma_m
lambda_x lambda_R rho_pi1 rho_y rho_m rho_m1 rho_R;
//Initialisation
alpha = 0.3304; // Calvo pricing probability beta = 0.99; // discount factor
omega = 3.5; // Frisch elasticity theta = 6; // parameter outputgap
s_c = 0.8; // average output share of private consumption v_1 = 5.419; // M1 velocity
v_2 = 1.730; // M2 velocity
R_ss = 1.0146; // steady state nominal interest rate sigma_tilt = 1; // coefficient relative risk aversion
eta_ch = 0.4587; // transaction friction proxy
eta_hc = 3.0792; // cross derivative of real money balances and consumption
sigma_h = 4.1656; // output elasticity of real money balances sigma_c = 1; //relative risk aversion
kappa = ((1 - alpha)*(1 - alpha * beta))/alpha; eta_R = R_ss/(sigma_h *(R_ss - 1));
rho_pi = 0.9692; //
rho_v = 0.6337; // AR1 coefficient taste shock
rho_z = 0.8943; // AR1 coefficient transaction cost technology shock rho_a = 0.9221; // AR1 coefficient productivity shock
rho_g = 0.8535; // AR1 coefficient government expenditure shock rho_mu = 0.8501; // AR1 coefficient wage mark-up shock
sigma_v = 0.0112; // standard deviation taste shock
sigma_z = 0.0513; // standard deviation transaction cost technology shock
sigma_a = 0.0068; // standard deviation productivity shock
sigma_g = 0.0084; // standard deviation government expenditure shock sigma_mu = 0.0417; // standard deviation wage mark-up shock
sigma_m = 0.0051; // standard deviation monetary policy shock rho_pi1=0.85; rho_y=0.85; rho_m=0.85; rho_m1=0.85; rho_R=0.85; lambda_x = kappa/theta; lambda_R = (eta_R*lambda_x/(sigma_tilt+omega))*(1/ v_2); //My model model(linear);
c(+1) = (v(+1)+ eta_ch *(m - pi(+1)+ z(+1))+ sigma_tilt * c - v - eta_ch *(m(-1) - pi + z) + R - pi(+1))*sigma_tilt;//Euler Equation omega *(y - a)+ mu - w = - sigma_tilt * c + eta_ch*(m(-1) - pi + z) + v;
pi = beta * pi(+1) + kappa*(w - a); y = s_c * c + g ;//aggregate resource
m = - eta_R * R +(eta_hc/sigma_h)* c(+1)+ pi(+1)+ ((1 - sigma_h)/ sigma_h)* z(+1);//money demand
r = R - pi(+1); // fisher equation
R = rho_pi*pi + rho_pi1*pi(-1) + rho_y*y + rho_R*R(-1) + rho_m*m + rho_m1*m(-1);//Taylor Rule
v = rho_v * v(-1) + e_v; //taste shock
z = rho_z * z(-1) + e_z; //transaction cost technology shock a = rho_a * a(-1) + e_a; //productivity shock
g = rho_g * g(-1) + e_g; //government expenditure shock mu = rho_mu * mu(-1) + e_mu; //wage mark-up shock
end;
//shocks shocks;
var e_v = sigma_v^2; var e_z = sigma_z^2; var e_a = sigma_a^2; var e_g = sigma_g^2; var e_mu = sigma_mu^2; end; initval; v = 0.5; z = 0.5; a = 0.5; g = 0.5; mu = 0.5; end;
//Optimal simple rule optim_weights; pi 1; y lambda_x; R lambda_R; end;
osr_params rho_pi rho_pi1 rho_y rho_R rho_m rho_m1; osr_params_bounds; rho_pi,0,100; rho_pi1,0,100; rho_y,0,100; rho_R,0,1; rho_m,0,100; rho_m1,0,100; end; osr(opt_algo=1) ;
