How Does Being in a Monetary Union Affect
the Welfare of a Small Open Economy
A DSGE Model Approach, Calibrated on The Netherlands
Master Thesis
University of Amsterdam
Dylan AmericaSupervised by Dhr. Dr. M. Zouain Pedroni
10/08/2016
Student number: 10001682 Academic year: 2015/2016
Statement of Originality
This document is written by Dylan America who declares to take full responsibility for the contents of this document.
I declare that the text and the work presented in this document is 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
Abstract
In this paper, two DSGE models have been presented that consist of a continuum of small open economies. One for a small open economy outside a monetary union, and one for an economy within a monetary union. The models contain the price rigidity introduced by Calvo (1983), and show how the mechanisms differ between the systems. A country outside a monetary union does a better job in stabilizing the domestic inflation and output gap. A country within a monetary union on the other hand, stabilizes the CPI inflation better. For the Netherlands, the welfare outside a monetary union is higher compared to the scenario inside the union. The benefits of joining a monetary union increases as the correlation between the domestic country and the union increases. Countries with a high trade openness experience a welfare gain from joining a monetary union. The elasticity of labor supply does not influence this trade-off.
Content
1. Introduction ... 6
2. Related Literature ... 10
3. The Model: Outside a Monetary Union ... 13
3.1 Environment ... 13
3.1.1 Households ... 13
3.1.2 The Firms ... 16
3.1.3 International Risk Sharing ... 17
3.2 Equilibrium ... 18
3.3 Definitions ... 19
3.4 Log Linearizing and Modelling ... 20
3.4.1 Definitions ... 20
3.4.2 The Phillips Curve ... 21
3.4.3 The IS Curve ... 23
3.4.4 Formulas to Complete the Model ... 24
3.4.5 Monetary Policy Rule ... 25
3.5 Foreign Country ... 25
3.5.1 The Phillips Curve ... 25
3.5.2 The IS Curve ... 26
3.5.3 Monetary Policy Rule ... 27
4. The Model: Inside a Monetary Union ... 28
5. Welfare ... 30
6. Data and Results ... 33
6.1 Calibration ... 33
6.2 Mechanism after a shock... 35
6.2.1 Domestic productivity shock ... 35
6.2.2 Foreign productivity shock ... 38
6.3 Welfare Analysis ... 40
7. Conclusion ... 43
8. References ... 45
1. Introduction
In 1961, Robert Mundell published an article introducing the Theory of Optimal Currency Areas, which concerns areas that benefit from having a pegged exchange rate or a single currency. Mundell (1961) argued that the costs of being in a currency area were mainly on the macro-economic side of the economy, while the benefits were more on the micro-macro-economic side. The most important costs were assumed to be a loss of independent monetary policy, no flexibility in the nominal exchange rate, and restrictions on the use of fiscal policy. For an area with a pegged exchange rate, monetary policy is needed to sustain the peg. This can be deduced from the “impossible trinity”, also known as the “trilemma”, whereby countries can only preserve two out of the following three options; a pegged exchange rate, free capital mobility, and an independent monetary policy (Obstfeld, Shambaugh and Taylor, 2005). Countries involved in a monetary union have chosen to share the same currency, and therefore also a single monetary policy. The European Monetary Union is established in 1999 as a currency area with a single currency for the whole area. Therefore, the individual countries of this area do not have their own monetary policy, but a monetary policy is conducted on behalf of the whole union.
In this paper, the differences will be discussed between a country with an own monetary policy, and a country inside a monetary union. For the country within a monetary union, the costs of a loss of an independent monetary policy and a loss of a flexible exchange rate are taken into account. This analysis will be done with a DSGE model approach, whereby the assumption is made that the world consists of a continuum of small open economies. A rigidity ala Calvo (1983) for the price setting mechanism is used, and a subsidy on labor is applied to neutralize the effects of the firm’s mark-up. The model has been calibrated on the Netherlands, and for the sake of simplicity, the European Monetary Union will be seen as the rest of the world. The first model presented is the one for a small open economy outside a monetary union and is, apart from some changes, in line with the model used by Gali and Monacelli (2005). They created a model for a small open economy in a world that consists of a continuum of small open economies. The production of firms is only dependent on labor and technology, and all countries are assumed to have the same preferences, production function, and market structure. To adapt this model for a country inside a monetary union, some changes have been made to capture the fact that there is no difference in nominal exchange rates between countries in a monetary union. Furthermore, the fact that within a monetary
union individual countries have no independent monetary policy, is taken into account in the model. An important thing to note is that most benefits of joining a monetary union, as described by Mundell (1961), are not captured in the model. Benefits of joining a monetary union are for example, lower transaction costs, higher price transparency, deeper and wider growth of financial markets, and less influence of speculators on the exchange rate. Glick and Rose (2002) among others1, found that when countries merge into a monetary union, trade between these countries doubles. Their research has been done through a large panel data regression. The analysis made in this paper, on the other hand, just shows the effects of having a fixed exchange rate and a loss of independent monetary policy, on the choice of joining a monetary union. This is done by comparing the welfares of the two different situations, and by calculating the permanent consumption increase that is needed for the country with the lower welfare, to get to the same welfare level as the country with the higher welfare. To decide if a country is better off within a monetary union, articles like the one from Glick and Rose (2002), can be compared with this paper.
The contribution of this paper to the existing literature focuses on two points. Firstly, the difference in dynamics after a domestic and foreign productivity shock between a country outside a monetary union and a country inside a monetary union are shown. For a country outside a monetary union with domestic inflation targeting and a high trade openness, domestic inflation and the output gap are stabilized better than if the country was inside a monetary union. CPI inflation on the other hand, is better stabilized within a monetary union, due to the fact that there is no imported inflation. Secondly, a comparison between the welfare of a country outside a monetary union and a country inside a monetary union is made. This shows that welfare for the country within a monetary union is lower than for the country that is not involved. The calibration has been done for the Netherlands, and the results are therefore only valid for a country with the same economical structure as the Netherlands. Analyzing the permanent increase of consumption that is needed to equalize the welfare of the two different systems leads to a value of 0.0003 percent. As this value is relatively low, it is hard to make any statements about it. It is however, interesting to see in what direction the welfare gains go if certain specifications of the model change. An increase in the correlation of a country with the rest of the union results in lower welfare losses from joining a monetary union. The reason for this can be found in the way a central bank acts. The central bank wants to stabilize
1 Rose and Van Wincoop (2001) and Micco, Stein and Ordoñez (2003) have the same findings although the latter
labor to reduce the variance of the output gap. If the correlation between a country and the union is high, it is more likely that the country and the union are hit by symmetric shocks. This makes the central bank of a monetary union more likely to act in the same way the domestic central bank would, and therefore lowers the variance of labor. Changing the trade openness of a country results in an interesting finding. Countries with a high trade openness, experience benefits of joining a monetary union. This can be traced back to the rigidities introduced in this economy. The price rigidity is introduced in the domestic and the foreign country, and a country which is open to trade, experiences inefficiency’s due to both these rigidities. Increasing the trade openness makes a country more vulnerable to the foreign rigidity, and therefore increases the benefits from a stable exchange rate. The elasticity of labor supply on the other hand, has barely any effect on the welfare gains of joining a monetary union. A surprising result is that the costs or benefits of joining a monetary union are relatively low. The reason for this result can be traced back to the magnitude of the shocks. As the Netherlands and the European Monetary Union as a whole have a relatively stable growth of GDP, this implies that productivity has been fairly stable. Increasing the variance of these shocks, results in higher costs or benefits of joining a monetary union. Countries like China, India and Brazil could have much more variation in their shocks and therefore are likely to have higher costs or benefits in case they join a monetary union.
The main difference of this paper with respect to the existing literature is the fact that most existing literature focuses on a single state of the economy. The focus is either on a country outside a monetary union or within a monetary union, like Benigno (2004) and Gali and Monacelli (2008), who analyze the optimal monetary policy within a monetary union. The papers that actually make the comparison between the two different systems, for example Kollmann (2004) and Ca’ Zorzi, de Santis and Zampolli (2005), do this with a two country model approach, whereby the countries affect the state of the other country. In a model with a continuum of small open economies, as used in this paper, the state of a single country does not affect the world variables. Since the Dutch economy is relatively small compared to the euro zone as a whole, using a continuum of small open economies is a more representable model for the Netherlands.
The rest of this paper is organized as follows. In section two, the related literature is analyzed and compared with this paper. Section three sets the layout for the model for the country outside a monetary union. This includes not only the layout for the domestic country, but also for the rest of
the world. In the part that follows, the changes that had to be made to make the model a fit for a country within a monetary union are explained. The fifth section explains the calculations of the welfare, and the way to measure the difference in welfare. Section six explains the data and results, starting with the calibration, and ending with the explanation of the impulse response functions and the welfare. The last section consists of a conclusion and some possibilities for further research.
2. Related Literature
To give an introduction to what has been studied already, the most relevant related articles will be discussed. The main goal and methodology will be shown, but also the similarities and differences in relation with this paper.
The first important article is written by Gali and Monacelli (2005). The fundamentals of this paper come from their article, since they created a DSGE model for a small open economy. The main goal of their paper is to analyze three different monetary policy rules, namely a domestic inflation targeting, CPI inflation targeting, and a currency peg. The model they use is a DSGE model, created with a continuum of small open economies that represent the world. The main difference between their model and the model in this paper, used for the country outside a monetary union, are the specifics for the world. Gali and Monacelli use a simplification where the world inflation is set equal to zero, and they make foreign output an AR(1) function with a foreign output shock. In this paper however, a separate model for the world has been added to make the mechanisms of the world more realistic and observable. Gali and Monacelli find that there is a trade-off between the domestic inflation and output gap on the one hand, and the nominal exchange rate and terms of trade on the other hand. The welfare of a specific domestic country is maximized when the volatility of domestic inflation and output gap are minimized. To determine this formula, they made the assumption that the elasticity of consumption and the substitutability between domestic and foreign goods, and between different foreign goods, equals one. They find that the domestic inflation targeting best stabilizes the domestic inflation and output gap, and therefore appoint it as the optimal monetary policy for a small open economy. This is only valid under the specified conditions. The conclusion of their paper will be used in this paper, as the domestic inflation targeting will be taken as the monetary policy rule for the country outside a monetary union. This article was a perfect start to create the model for the country inside the monetary union.
The second article is written by Benigno (2004), who tries to find the optimal monetary policy within a currency area suffering from asymmetric shocks. The model he uses in his research, is a general equilibrium model based on two countries that include monopolistic competition and price stickiness. The main difference between his model and the one exploited in this paper, is the fact that instead of two countries, a world consisting of a continuum of small open economies is used
relatively small economy, since they hardly affect the state of the union as a whole. Another difference is that money enters the utility function in the paper of Benigno. Benigno finds that the optimal monetary policy is dependent on the nominal rigidities. When these rigidities are equal between the two countries, monetary policy should be targeting a weighted average of the inflation in the countries. If the nominal rigidities in a country are higher than in the other country, a higher weight should be given to the inflation of that country. Benigno finds a relation between the rigidities and the optimal policy, where Gali and Monacelli (2008), as in this paper, assume that these rigidities are equal between the different countries. By assuming rigidities to be equal among countries, our monetary policy rule for the union as a whole meets the findings of Benigno.
Thirdly, Gali and Monacelli (2008) wrote an extension to their own paper. The main goal of their paper is to find the optimal fiscal and monetary policy to maximize welfare for a monetary union as a whole. The model is an extension of the earlier model of Gali and Monacelli (2005) and also consist of a continuum of small open economies. The main difference in the model for the domestic country in their earlier paper compared to this paper, is the government component. Fiscal policy is conducted on a country specific level, and the government spending is fully on domestic products. Gali and Monacelli (2008) implement this government spending in the utility function of the households, and also the market clearing condition includes this government spending. Monetary policy will be conducted on a union wide level. The model in this paper differs from their model and focusses on a different aspect. The main difference is that government spending is not included in the model. This is due to the fact that the main point of this paper is to analyze the difference in welfare coming from the change of monetary power. Gali and Monacelli (2008) find that, to maximize union wide welfare, the central bank needs to stabilize the union’s inflation. On the other hand, the national governments have to provide goods to extend households utility, but also spend in a way to stabilize domestic inflation and output gap. The unions central bank’s policy is only optimal if the national governments provide their share to stabilize the domestic variables. Instead of analyzing the best way to organize the system within a monetary union, in this paper, the difference between being in a monetary union and staying outside of it will be analyzed.
The fourth article is by Kollmann (2004). He analyzes the same question as in this paper, comparing the welfare of a country inside and a country outside a monetary union. Just as Benigno (2004), he makes use of a two country model instead of the continuum of economies represented in this paper.
The main contribution of his paper is the fact that he not only uses a productivity shock, but he also implements a shock to the UIP condition. Kollmann (2004) also added capital to his model, which is omitted in this paper. This UIP shock causes consumption to be more volatile and therefore decreases the welfare of the country involved. In his paper, he makes a distinction between two different relationships between countries, namely countries with a weak trade link, like US and Europe, and countries with a strong trade link, like countries within Europe. The high trading countries were much more affected by the UIP shock and their welfare decreased more. For countries within a monetary union, this UIP shock does not appear, as there is only one interest rate. Therefore, according to Kollmann (2004), high trading countries benefit from forming a monetary union. An interesting point is that he used a trade openness of 0.2 for the high trading countries, which is much lower than the openness of the Netherlands, which accounts for 70 percent imports of GDP. As most of the European countries have a higher openness than 0.2, he could improve his research by analyzing higher levels of trade openness.
The last reference is a working paper for the ECB written by Ca’ Zorzi, de Santis and Zampolli (2005). They analyzed the effects of joining a monetary union. They conduct their research with a general equilibrium model including two countries and two different sectors, namely tradable and non-tradable goods. This model brings an important role for trade. To have a positive effect on the economy, the increased output has to offset the negative effects of being in a monetary union. The output has to be higher if the country was smaller, has a higher difference in the standard deviations of the shocks, a lower correlation between the domestic shocks, or a higher variance of the real exchange rate. The most important factor for the costs of joining a monetary union according to Ca’ Zorzi, de Santis and Zampolli (2005) was the variance-covariance matrix of the supply shocks. The same conclusion can be found in this paper.
3. The Model: Outside a Monetary Union
The model consists of a continuum of small open economies who represent the whole world. Each of the economies is insignificantly small and therefore does not affect the world, which can be seen as a closed economy. All economies are assumed to have the same preferences, production functions and market structure. A rigidity for the price setting mechanism is used ala Calvo (1983). Firstly, the model will be discussed by analyzing the environment. This will be done by setting the assumptions and preferences for households and firms. These will be maximized to create optimal conditions for both parties. Secondly, the equilibrium conditions in the economy will be discussed, and thirdly, some important definitions are shown and combined to include the assumptions into the model. After that, the obtained formulas will be log linearized and combined, to get to the required functions for the domestic part of the model. The last part discusses the foreign country, which represents the whole world. All formulas needed to run the model for the country outside a monetary union can be found in appendix B and D.
3.1 Environment
3.1.1 Households
Households try to maximize their lifetime utility. This utility for households includes consumption and leisure, and since time can only be spent as either leisure or labor, labor is represented by a negative utility. 𝑀𝑎𝑥: 𝐸0∑ 𝛽𝑡 𝑈(𝐶𝑡, 𝑁𝑡) ∞ 𝑡=0 𝑈(𝐶𝑡, 𝑁𝑡) = 𝐶𝑡1−𝜎 1 − 𝜎− 𝑁𝑡1+𝜑 1 + 𝜑 ( 1 )
Consumption consists of the consumption of domestic and foreign goods, in which η stands for the substitutability between domestic and foreign goods. α represents the amount of imports of a country as a percentage of GDP, and is therefore a good measure of the openness of a country.
𝐶𝑡= [(1 − 𝛼) 1 𝜂 (𝐶 𝐻,𝑡) 𝜂−1 𝜂 + 𝛼1𝜂 (𝐶 𝐹,𝑡) 𝜂−1 𝜂 ]𝜂−1𝜂 ( 2 )
The consumption of domestic goods is the sum of all consumption goods, taken into account the elasticity of substitution between the different goods within a country which is denoted by ε. j is the notation for the different goods and could be any value between zero and one.
𝐶𝐻,𝑡 = (∫ 𝐶𝐻,𝑡(𝑗)𝜀−1𝜀 𝑑𝑗 1 0 ) 𝜀 𝜀−1 ( 3 )
The same holds for the consumption of foreign goods, where γ is the elasticity of substitution between the different countries.
𝐶𝐹,𝑡 = (∫ 𝐶𝑖,𝑡 𝛾−1 𝛾 𝑑𝑖 1 0 ) 𝛾 𝛾−1 ( 4 ) where, 𝐶𝑖,𝑡 = (∫ 𝐶𝑖,𝑡(𝑗)𝜀−1𝜀 𝑑𝑗 1 0 ) 𝜀 𝜀−1 ( 5 )
As households maximize their utility, they have a budget constraint, given by
∫ 𝑃𝐻,𝑡(𝑗)𝐶𝐻,𝑡(𝑗)𝑑𝑗 1 0 + ∫ ∫ 𝑃𝑖,𝑡(𝑗)𝐶𝑖,𝑡(𝑗) 1 0 𝑑𝑗𝑑𝑖 1 0 + 𝐸𝑡{𝑄𝑡,𝑡+1𝐷𝑡+1} ≤ 𝐷𝑡+ 𝑊𝑡𝑁𝑡+ 𝑇𝑡 ( 6 )
The first part of the left hand side of this equation consists of expenditures on domestic goods, and the second part consists of expenditures on foreign goods. The expected proceedings from the portfolio in period t+1 are given by 𝐸𝑡{𝑄𝑡,𝑡+1𝐷𝑡+1}. 𝐷𝑡 is the amount that has been invested in the
portfolio in period t, and 𝑇𝑡 are the lump-sum taxes or transfers.
The optimal conditions that can be obtained from these conditions are:
𝐶𝐻,𝑡(𝑗) = ( 𝑃𝐻,𝑡(𝑗) 𝑃𝐻,𝑡 ) −𝜀 𝐶𝐻,𝑡 ( 7 )
𝐶𝑖,𝑡(𝑗) = (𝑃𝑖,𝑡(𝑗) 𝑃𝑖,𝑡 ) −𝜀 𝐶𝑖,𝑡 ( 8 ) 𝐶𝑖,𝑡 = (𝑃𝑖,𝑡 𝑃𝐹,𝑡) −𝛾 𝐶𝐹,𝑡 ( 9 ) 𝐶𝐹,𝑡 = 𝛼 (𝑃𝐹,𝑡 𝑃𝑡 ) −𝜂 𝐶𝑡 ( 10 )
where 𝑃𝐻,𝑡 𝑎𝑛𝑑 𝑃𝑖,𝑡 are the indexes for respectively the domestic prices and the prices in country i, denoted in the currency of the home country. 𝑃𝐹,𝑡 is the index for the prices of imported goods and 𝑃𝑡 is the CPI price in the domestic country, which is a weighted average of domestic and foreign
prices, 𝑃𝑡 ≡ [(1 − 𝛼)𝑃𝐻,𝑡1−𝜂 + 𝛼𝑃𝐹,𝑡1−𝜂]
1
1−𝜂. Taking all this into account, the household budget
constraint could be written as,
𝑃𝑡𝐶𝑡+ 𝐸𝑡{𝑄𝑡,𝑡+1𝐷𝑡+1} ≤ 𝐷𝑡+ 𝑊𝑡𝑁𝑡+ 𝑇𝑡
( 11 )
Where, 𝑃𝑡𝐶𝑡 = 𝑃𝐻,𝑡𝐶𝐻,𝑡+ 𝑃𝐹,𝑡𝐶𝐹,𝑡.
After taken derivatives, the optimal conditions for the households turn out to be
𝐶𝑡𝜎𝑁𝑡𝜑 =𝑊𝑡 𝑃𝑡 ( 12 ) 𝛽(𝐶𝑡+1 𝐶𝑡 ) −𝜎 𝑃𝑡 𝑃𝑡+1 = 𝑄𝑡,𝑡+1 ( 13 )
If we take expectations on both sides, the last equation can be written as
𝛽𝑅𝑡𝐸𝑡[(𝐶𝑡+1 𝐶𝑡 ) −𝜎 𝑃 𝑡 𝑃𝑡+1] = 1 ( 14 ) in which 𝑅𝑡 = 1
3.1.2 The Firms
There are two types of firms in this model, intermediate good firms and final good firms. Final good firms act in an environment with perfect competition and flexible prices. Intermediate good firms on the other hand, deal with monopolistic competition and sticky prices. Intermediate good firms in the domestic country, and due to symmetry also in foreign countries, produce with a simple production function, which is only dependent on labor supply and technology. The production of a differentiated product j is given by
𝑌𝑡(𝑗) = 𝐴𝑡 𝑁𝑡(𝑗)
( 15 ) Technology is given by an AR(1) function and includes a shock variable which indicates a domestic productivity shock. j ϵ [0,1] is the index for the different products within the economy. For the economy as a whole, the production function will be 𝑌𝑡 = 𝐴𝑡 𝑁𝑡. The production function for the
final good firms, consists of a combination of intermediate goods, 𝑌𝑡= [∫ 𝑌01 𝑗,𝑡𝑞𝑑𝑗]
1 𝑞
. As the final good firms try to maximize their profit, they will do this subject to their production function.
max 𝑌𝑗,𝑡 𝑃𝑡𝑌𝑡− ∫ 𝑃𝑗,𝑡𝑌𝑗,𝑡𝑑𝑗 1 0 𝑠. 𝑡. 𝑌𝑡 = [∫ 𝑌𝑗,𝑡 𝑞 𝑑𝑗 1 0 ] 1 𝑞 This leads to 𝑌𝑗,𝑡= [ 𝑃𝐻,𝑗,𝑡 𝑃𝐻,𝑡] −𝜀 𝑌𝑡 in which 𝜀 = 1
1−𝑞 and q is related to the elasticity of substitution
between any two input goods in the production of the final good.
For the intermediate good firms in this model, a pricing friction has been added to make the model more realistic. This friction is called the Calvo (1983) pricing model, whereby firms cannot alter their prices every period. The chance a firm is not allowed to change their prices, is captured by the parameter θ. Therefore, firms set their prices with the knowledge that there is a chance of θ that the price has to be the same in the next period and they take into account the optimality condition from the final good firms.
max 𝑃𝐻,𝑗,𝑡 𝐸𝑡∑ 𝜃𝑖 ∞ 𝑖=0 ∆𝑖,𝑡+𝑖[𝑃𝐻,𝑗,𝑡 𝑃𝐻,𝑡+𝑖 𝑌𝑗,𝑡+𝑖− 𝑀𝐶𝑡+𝑖𝑌𝑗,𝑡+𝑖] ( 16 ) 𝑠. 𝑡. 𝑌𝑗,𝑡 = [𝑃𝐻,𝑗,𝑡 𝑃𝐻,𝑡 ] −𝜀 𝑌𝑡
where ∆𝑖,𝑡+𝑖 is the discount factor between period t and t+i, and can be noted as ∆𝑖,𝑡+𝑖 = 𝛽𝑖[𝐶𝑡+𝑖
𝐶𝑡 ]
−𝜎
and real marginal cost is noted as 𝑀𝐶𝑡. When the producers of a certain good maximize this function with respect to their prices they arrive at,
𝑃𝐻,𝑗,𝑡 𝑃𝐻,𝑡 = 𝜇 𝐸𝑡∑ 𝜃𝑖𝛽𝑖𝑀𝐶 𝑡+𝑖[ 𝑃𝐻,𝑡+𝑖 𝑃𝐻,𝑡 ] 𝜀 𝑌𝑡+𝑖1−𝜎 ∞ 𝑖=0 𝐸𝑡∑ 𝜃𝑖𝛽𝑖[𝑃𝐻,𝑡+𝑖 𝑃𝐻,𝑡 ] 𝜀−1 𝑌𝑡+𝑖1−𝜎 ∞ 𝑖=0 ( 17 ) where 𝜇 ≡ 𝜀
𝜀−1 and corresponds to the mark up for a firm. This is the optimal condition for the
intermediate firms in this model.
3.1.3 International Risk Sharing
Assumed that security markets are completely competitive, it can be taken for granted that function (13) also holds for the different countries outside the domestic country. The following formula is equal due to the fact that we assume that all the small open economies are perfectly symmetric.
𝛽 [𝐶𝑡+1 𝑖 𝐶𝑡𝑖 ] −𝜎 𝑃𝑡𝑖 𝑃𝑡+1𝑖 ℰ𝑡𝑖 ℰ𝑡+1𝑖 = 𝑄𝑡,𝑡+1 ( 18 )
If the two equations for risk sharing are being equalized, a formula arises that links consumptions in different countries.
𝐶𝑡 = 𝐶𝑡𝑖 𝒬𝑖,𝑡 1 𝜎
( 19 )
In this step, the law of one price is used. It is written as 𝑃𝑖,𝑡(𝑗) = ℰ𝑖,𝑡𝑃𝑖,𝑡𝑖 (𝑗) , and will be explained
further in part 3.3. We can assume that in the symmetric perfect foresight equilibrium, 𝐶𝑡= 𝐶𝑡𝑖 =
𝐶𝑡∗ and also 𝒬
𝐶𝑡= 𝐶𝑡∗ 𝒬 𝑡 1 𝜎 ( 20 )
3.2 Equilibrium
Assuming goods market clearing for the domestic country, every good that has been produced in the country has to be consumed by either the domestic or one of the foreign countries.
𝑌𝑡(𝑗) = 𝐶𝐻,𝑡(𝑗) + ∫ 𝐶𝐻,𝑡𝑖 (𝑗)𝑑𝑖 1
0 ( 21 )
where 𝐶𝐻,𝑡𝑖 (𝑗) represents the domestic produced goods consumed in country i. Taking into account some of the formulas obtained by the optimization of the households conditions, (3), (4) and (5), formula (19), and assuming symmetric preferences between countries, 𝐶𝐻,𝑡𝑖 (𝑗) =
𝛼 (𝑃𝐻,𝑡(𝑗) 𝑃𝐻,𝑡 ) −𝜀 ( 𝑃𝐻,𝑡 ℰ𝑖,𝑡𝑃𝐹,𝑡𝑖 ) −𝛾 (𝑃𝐹,𝑡 𝑖 𝑃𝑡𝑖) −𝜂
𝐶𝑡𝑖, formula (21) can be written as
𝑌𝑡(𝑗) = (𝑃𝐻,𝑡(𝑗) 𝑃𝐻,𝑡 ) −𝜀 [(1 − 𝛼) (𝑃𝐻,𝑡 𝑃𝑡 ) −𝜂 𝐶𝑡+ 𝛼 ∫ ( 𝑃𝐻,𝑡 ℰ𝑖,𝑡𝑃𝐹,𝑡𝑖 ) −𝛾 (𝑃𝐹,𝑡 𝑖 𝑃𝑡𝑖 ) −𝜂 𝐶𝑡𝑖𝑑𝑖 1 0 ] ( 22 )
The definition of total domestic output is given by 𝑌𝑡 = [∫ 𝑌𝑡(𝑗)1−
1 𝜀𝑑𝑗 1 0 ] 𝜀 𝜀−1 . When implementing formula (22) into this definition, eventually a formula will follow which includes 𝑆𝑡𝑖 and 𝑆
𝑖,𝑡. These
are the terms of trade between respectively country i and the rest of the world, and the domestic country and country i. So 𝑆𝑡𝑖 = 𝑃𝐹,𝑡
𝑖
𝑃𝑡𝑖 and 𝑆𝑖,𝑡 =
𝑃𝑖,𝑡
𝑃𝐻,𝑡. This results in:
𝑌𝑡 = ( 𝑃𝐻,𝑡 𝑃𝑡 ) −𝜂 𝐶𝑡[(1 − 𝛼) + 𝛼 ∫ (𝑆𝑡𝑖𝑆𝑖,𝑡)𝛾−𝜂𝒬𝑖,𝑡 𝜂−1𝜎 𝑑𝑖 1 0 ] ( 23 )
The goods market clearing condition for the whole world implies that
𝑌𝑡∗ = 𝐶𝑡∗ 𝑖𝑛 𝑤ℎ𝑖𝑐ℎ 𝑌𝑡∗ = ∫ 𝑌𝑡𝑖𝑑𝑖 1 0 𝑎𝑛𝑑 𝐶𝑡∗ = ∫ 𝐶𝑡𝑖𝑑𝑖 1 0 ( 24 )
3.3 Definitions
CPI inflation is, as noted before, defined as
𝑃𝑡 ≡ [(1 − 𝛼)𝑃𝐻,𝑡1−𝜂+ 𝛼𝑃𝐹,𝑡1−𝜂] 1 1−𝜂
( 25 )
The terms of trade are defined as the foreign price of a good, divided by the domestic price of a good, both given in domestic currency. This property holds for bilateral terms of trade 𝑆𝑖,𝑡 ≡ 𝑃𝑖,𝑡
𝑃𝐻,𝑡 ,
but also if we combine all countries together,
𝑆𝑡 ≡ 𝑃𝐹,𝑡
𝑃𝐻,𝑡 ( 26 )
Assuming that the law of one price applies,
𝑃𝑖,𝑡(𝑗) = ℰ𝑖,𝑡𝑃𝑖,𝑡𝑖 (𝑗)
( 27 ) 𝑃𝑖,𝑡(𝑗) is the price of good j produced in country i, expressed in the domestic currency. 𝑃𝑖,𝑡𝑖 (𝑗) on
the other hand, is the price of that same good only expressed in the currency of country i. Therefore, ℰ𝑖,𝑡 is the nominal exchange rate between the home country and country i. An increase in ℰ𝑖,𝑡
corresponds with a depreciation of the domestic currency, compared to the currency of country i.
The real exchange rate is defined as
𝒬𝑖,𝑡 ≡ ℰ𝑖,𝑡𝑃𝑡
𝑖
𝑃𝑡 ( 28 )
Net export as a percentage of steady state GDP, is the amount of output minus the consumption of a single country, divided by the natural level of output. This corresponds with the function
𝑁𝑋𝑡≡ 1 𝑌(𝑌𝑡− 𝑃𝑡 𝑃𝐻,𝑡 𝐶𝑡) ( 29 )
3.4 Log Linearizing and Modelling
3.4.1 Definitions
Some of the definitions will be log linearized and expanded, to become useful for the rest of the calculations. The log linearized function for the CPI prices will be
𝑝𝑡= (1 − 𝛼)𝑝𝐻,𝑡+ 𝛼𝑝𝐹,𝑡
( 30 ) and for the terms of trade,
𝑠𝑡 = 𝑝𝐹,𝑡− 𝑝𝐻,𝑡
( 31 ) Combining the two equations above, and taking first differences will give the function of the CPI inflation.
𝜋𝑡 = 𝜋𝐻,𝑡+ 𝛼 Δ𝑠𝑡
( 32 ) Log linearizing the law of one price and combining all the foreign countries together results in
𝑃𝐹,𝑡 = ∫ ℰ𝑖,𝑡 𝑃𝑖,𝑡𝑖 1 0 ( 33 ) 𝑝𝐹,𝑡= ∫ (𝑒𝑖,𝑡+ 𝑝𝑖,𝑡𝑖 ) 1 0 𝑑𝑖 𝑒𝑡 ≡ ∫ 𝑒𝑖,𝑡𝑑𝑖 1
0 being the nominal effective exchange rate and 𝑝𝑡
∗ ≡ ∫ 𝑝 𝑖,𝑡𝑖 1
0 being the world prices,
the following formula is constructed,
𝑝𝐹,𝑡 = 𝑒𝑡+ 𝑝𝑡∗
( 34 ) Mixing this outcome with the definition of the terms of trade to get 𝑠𝑡 = 𝑒𝑡+ 𝑝𝑡∗− 𝑝𝐻,𝑡 , and by taking first differences, a relationship follows, including foreign and domestic inflation.
𝑠𝑡= 𝑠𝑡−1+ 𝑒𝑡− 𝑒𝑡−1+ 𝜋𝑡∗− 𝜋𝐻,𝑡
The real exchange rate between the domestic country and country i is given by 𝒬𝑖,𝑡 ≡ ℰ𝑖,𝑡𝑃𝑡𝑖
𝑃𝑡 . The
log linearized version of this formula is stated
𝑞𝑡 = ∫ (𝑒𝑖,𝑡+ 𝑝𝑡𝑖 − 𝑝𝑡)𝑑𝑖 1 0 ( 36 ) 𝑞𝑡 = 𝑒𝑡+ 𝑝𝑡∗− 𝑝𝑡 𝑞𝑡= 𝑠𝑡+ 𝑝𝐻,𝑡− 𝑝𝑡 𝑞𝑡= (1 − 𝛼)𝑠𝑡 ( 37 ) where the formulas of the CPI price level and the terms of trade are used to derive the last two steps. This results in a relation between the real exchange rate and the terms of trade.
3.4.2 The Phillips Curve
To get to the Philips curve, the first assumption is that all the domestic firms have the same technology. Also, the real marginal cost is assumed to be,
𝑚𝑐𝑡 = −𝑣 + 𝑤𝑡− 𝑝𝐻,𝑡− 𝑎𝑡 ( 38 )
In this formula, 𝑣 is a function of the employment subsidy (𝜏) and is noted as 𝑣 ≡ −log (1 − 𝜏).
Log linearizing formulas (15), (12) and (20), which are the production functions, the optimal level of labor, and the formula received by the international risk sharing condition, results in respectively
𝑦𝑡 = 𝑎𝑡+ 𝑛𝑡 ( 39 ) 𝑤𝑡− 𝑝𝑡 = 𝜎𝑐𝑡+ 𝜑𝑛𝑡 ( 40 ) 𝑐𝑡 = 𝑐𝑡∗+1 𝜎𝑞𝑡 ( 41 ) 𝑐𝑡 = 𝑐𝑡∗+1 − 𝛼 𝜎 𝑠𝑡 ( 42 )
where this last formula is obtained by combining the former formula with formula (37). By using the formulas above and inserting them into formula (38), the following formula will be created,
𝑚𝑐𝑡= −𝑣 + 𝜎𝑦𝑡∗+ 𝜑𝑦𝑡+ 𝑠𝑡− (1 + 𝜑)𝑎𝑡
( 43 ) Log linearizing the domestic goods market clearing condition (23), expanded with the international risk sharing condition (19), results in
𝑦𝑡 = 𝑐𝑡+ 𝛼𝛾𝑠𝑡+ 𝛼(𝜂 −1
𝜎)𝑞𝑡 ( 44 )
𝑦𝑡= 𝑐𝑡+𝛼𝜔 𝜎 𝑠𝑡
where 𝜔 ≡ 𝜎𝛾 + (1 − 𝛼)(𝜎𝜂 − 1). The last step is acquired by using the relation between the real exchange rate and the terms of trade (37).
Combining formula (44) with (24) and (42), results in
𝑦𝑡= 𝑐𝑡∗+ 1
𝜎𝛼𝑠𝑡 ( 45 )
where 𝜎𝛼 = 𝜎
(1−𝛼)+𝜔𝛼. Using this in the previous function for the marginal cost will result in
𝑚𝑐𝑡 = −𝑣 + (𝜎𝛼+ 𝜑)𝑦𝑡+ (𝜎 − 𝜎𝛼)𝑦𝑡∗− (1 + 𝜑)𝑎 𝑡
( 46 ) To get the natural level of output, marginal cost plus the mark up for firms should equal to zero, so 𝑚𝑐𝑡 = −𝜇. −𝜇 = −𝑣 + (𝜎 − 𝜎𝛼)𝑦𝑡∗+ (𝜎 𝛼+ 𝜑)𝑦̅𝑡− (1 + 𝜑)𝑎𝑡 ( 47 ) 𝑦̅𝑡= Ω + Γ𝑎𝑡+ 𝛼Ψ𝑦𝑡∗ ( 48 ) where, Ω ≡ 𝜐−𝜇 𝜎𝛼+𝜑 , Γ ≡ 1+𝜑 𝜎𝛼+𝜑 , Ψ ≡ − 𝜎𝛼Θ 𝜎𝛼+𝜑 and Θ = ω − 1.
The output gap is divined as 𝑥𝑡≡ 𝑦𝑡− 𝑦̅𝑡. To get a relation between the marginal cost and the output gap, the natural level of output will be added and subtracted to formula (46). This will result in,
𝑚𝑐𝑡 = −𝑣 + (𝜎𝛼+ 𝜑)𝑥𝑡+ [𝛼Ψ(𝜎𝛼+ 𝜑) + (𝜎 − 𝜎𝛼)]𝑦𝑡∗+ [−1 − 𝜑 + (𝜎 𝛼+ 𝜑)Γ]𝑎𝑡 + (𝜎𝛼+ 𝜑)Ω 𝑚𝑐𝑡= −𝜐 + (𝜎𝛼+ 𝜑)𝑥𝑡+ (𝜎𝛼+ 𝜑)Ω 𝑚𝑐̂𝑡= (𝜎𝛼+ 𝜑)𝑥̂𝑡 ( 49 ) This last function is only valid under the condition that the employment subsidy is used to neutralize the rigidity, caused by the mark up of firms, so 𝜐 = 𝜇.
Formula (17), which is the optimal condition for firms, shows to be log linearized as
𝜋𝐻,𝑡 = 𝛽𝐸𝑡{𝜋𝐻,𝑡+1} + 𝜆𝑚𝑐̂𝑡
( 50 )
In which 𝜆 =(1−𝛽𝜃)(1−𝜃)
𝜃 . These derivations are shown in appendix A. Combining the dynamics of
inflation (50) and the relation between the real marginal cost and the output gap (49), result in the well-known Phillips curve,
𝜋𝐻,𝑡 = 𝛽𝐸𝑡{𝜋𝐻,𝑡+1} + 𝜅𝛼𝑥𝑡
( 51 ) where 𝜅𝛼 = 𝜆(𝜎𝛼+ 𝜑)
3.4.3 The IS Curve
To derive the IS curve, formula (44) has to be combined with the log linearized version of the Euler equation, 𝑐𝑡 = 𝐸𝑡{𝑐𝑡+1} − 1 𝜎(𝑟𝑡− 𝐸𝑡{𝜋𝑡+1} − 𝜌), and results in 𝑦𝑡 = 𝐸𝑡{𝑐𝑡+1} − 1 𝜎(𝑟𝑡− 𝐸𝑡{𝜋𝑡+1} − 𝜌) + 𝛼𝜔 𝜎 𝑠𝑡 ( 52 ) 𝑦𝑡 = 𝐸𝑡{𝑦𝑡+1} − 1 𝜎(𝑟𝑡− 𝐸𝑡{𝜋𝐻,𝑡+1} − 𝜌) − 𝛼Θ 𝜎 𝐸𝑡{Δ𝑠𝑡+1}
in which Θ ≡ (𝜎𝛾 − 1) + (1 − 𝛼)(𝜎𝜂 − 1) = 𝜔 − 1. The last step made use of formula (32), and it also made use of the fact that if formula (44) is taken one period forward, it results in 𝐸𝑡(𝑐𝑡+1) =
𝐸𝑡(𝑦𝑡+1) − 𝛼𝜔
𝜎 𝐸(𝑠𝑡+1). In the next step, taking first differences of formula (45) gives 𝐸𝑡{Δ𝑠𝑡+1} =
𝜎𝛼[𝐸𝑡{𝑦𝑡+1} − 𝑦𝑡− 𝐸𝑡{Δ𝑦𝑡+1∗ }], and is used to substitute for 𝐸
𝑡{Δ𝑠𝑡+1}. 𝑦𝑡 = 𝐸𝑡{𝑦𝑡+1} − 1 𝜎𝛼 (𝑟𝑡− 𝐸𝑡{𝜋𝐻,𝑡+1} − 𝜌) + 𝛼Θ 𝐸𝑡{Δ𝑦𝑡+1∗ } ( 53 ) 𝑥𝑡= 𝐸𝑡{𝑥𝑡+1} − 1 𝜎𝛼(𝑟𝑡− 𝐸𝑡{𝜋𝐻,𝑡+1} − 𝜌) + 𝛼Θ 𝐸𝑡{Δ𝑦𝑡+1 ∗ } − Ω − Γ𝑎 𝑡− 𝛼Ψ𝑦𝑡∗+ Ω + Γ𝑎𝑡+1 + 𝛼Ψ𝑦𝑡+1∗ 𝑥𝑡 = 𝐸𝑡{𝑥𝑡+1} − 1 𝜎𝛼(𝑟𝑡− 𝐸𝑡{𝜋𝐻,𝑡+1} − 𝑟𝑟̅̅̅𝑡) 𝑟𝑟 ̅̅̅𝑡 = 𝜌 − 𝜎𝛼Γ(1 − 𝜌𝑎)𝑎𝑡+ 𝛼𝜎𝛼(Θ + Ψ)𝐸𝑡{Δ𝑦𝑡+1∗ } ( 54 )
From the first to the second equation above, the natural level of output has been subtracted and added on both sides to get to the output gap.
3.4.4 Formulas to Complete the Model
In this part, the missing formulas are being log linearized to complete the model for the domestic country. As the goods market clearing condition is stated, 𝑦𝑡∗ = 𝑐
𝑡∗, function (45) can be written as
𝑦𝑡 = 𝑦𝑡∗+ 1
𝜎𝛼𝑠𝑡 ( 55 )
The definition of the net exports, formula (29), can be log linearized as
𝑛𝑥𝑡 = 𝑦𝑡− 𝑐𝑡− 𝛼𝑠𝑡 ( 56 )
𝑛𝑥𝑡= 𝛼 (
𝜔
𝜎− 1) 𝑠𝑡
3.4.5 Monetary Policy Rule
To complete the domestic part of the first model, the monetary policy rule will be analyzed. The result of the research by Gali and Monacelli (2005) was that the domestic inflation target gave the lowest welfare loss for the domestic country. Therefore, a Taylor rule will be used, which is dependent on the domestic inflation instead of the CPI inflation of a country.
𝑟𝑡 = 𝜑𝜋𝜋𝐻,𝑡+ 𝑥𝑡
( 57 )
3.5 Foreign Country
The foreign country can be seen as a closed economy, and is therefore not affected by the domestic variables in this model. The domestic country on the other hand will be affected by changes in the world output and inflation. The assumptions for this foreign country are therefore slightly different, and calculations show that the Phillips curve and the IS curve have different properties as well.
3.5.1 The Phillips Curve
To derive the Phillips curve for the whole world, the same assumption for the real marginal cost is used as for the domestic country, meaning that,
𝑚𝑐𝑡∗ = −𝑣 + 𝑤
𝑡∗− 𝑝𝑡∗− 𝑎𝑡∗
( 58 )
Also due to the assumption that all countries have the same preferences, formulas (39) and (40) also apply for the world as a whole. The goods market clearing on the other hand, will be 𝑦𝑡∗ = 𝑐𝑡∗ for the world. Applying these formulas in the function of the real marginal cost results in,
𝑚𝑐𝑡∗ = −𝑣 + (𝜎 + 𝜑)𝑦𝑡∗− (1 + 𝜑)𝑎𝑡∗
( 59 ) To arrive at the natural level of output for the world, marginal cost plus the mark up for firms should equal zero. For the world this would mean that,
−𝜇 = −𝜐 + (𝜎 + 𝜑)𝑦̅𝑡∗− (1 + 𝜑)𝑎 𝑡∗
𝑦̅𝑡∗ = Σ𝑎 𝑡 ∗+ Α ( 61 ) where, Σ ≡ 1+𝜑 𝜎+𝜑 𝑎𝑛𝑑 Α ≡ 𝜐−𝜇 𝜎+𝜑 .
To link the marginal cost with the output gap, which is stated 𝑥𝑡∗ ≡ 𝑦𝑡∗− 𝑦̅𝑡∗, the natural level of
output will be subtracted and added from the right hand side of formula (59)
𝑚𝑐𝑡∗ = −𝑣 + (𝜎 + 𝜑)𝑦𝑡∗− (𝜎 + 𝜑)𝑦̅𝑡∗− (1 + 𝜑)𝑎𝑡∗+ Σ𝑎𝑡∗+ Α
( 62 ) 𝑚𝑐̂𝑡∗ = (𝜎 + 𝜑)𝑥𝑡∗
where in the last step was made use of the same assumption as for the domestic country, namely that the subsidy on labor is used to neutralize the mark up for firms, so 𝜐 = 𝜇. The dynamics of the foreign inflation are captured in the same way as formula (50),
𝜋𝑡∗ = 𝛽𝐸
𝑡{𝜋𝑡+1∗ } + 𝜆𝑚𝑐̂𝑡∗
( 63 ) Combining the last two formulas results in the Phillips curve for the whole world,
𝜋𝑡∗ = 𝛽𝐸𝑡{𝜋𝑡+1∗ } + 𝜅𝛽𝑥𝑡∗
( 64 ) where 𝜅𝛽 = (𝜎 + 𝜑)𝜆 .
3.5.2 The IS Curve
For the world as a whole, the Euler equation is the same as the Euler equation for the domestic country, given by the formula 𝑐𝑡∗ = 𝐸𝑡{𝑐𝑡+1∗ } −1
𝜎(𝑟𝑡 ∗− 𝐸
𝑡{𝜋𝑡+1∗ } − 𝜌). This is due to the
assumption that all countries have the same preferences. Therefore, if this Euler equation is combined with the goods market clearing condition for the world, the following formula is constructed, 𝑦𝑡∗ = 𝐸 𝑡{𝑦𝑡+1∗ } − 1 𝜎(𝑟𝑡 ∗− 𝐸 𝑡{𝜋𝑡+1∗ } − 𝜌) ( 65 )
𝑥𝑡∗ = 𝐸 𝑡{𝑥𝑡+1∗ } − 1 𝜎(𝑟𝑡 ∗− 𝐸 𝑡{𝜋𝑡+1∗ } − 𝜌) − Σ𝑎𝑡∗− Α + Σ𝑎𝑡+1∗ + Α ( 66 ) 𝑥𝑡∗= 𝐸 𝑡{𝑥𝑡+1∗ } − 1 𝜎(𝑟𝑡 ∗− 𝐸 𝑡{𝜋𝑡+1∗ } − 𝜌) − Σ(1 − 𝜌𝛽)𝑎𝑡∗ 𝑥𝑡∗= 𝐸 𝑡{𝑥𝑡+1∗ } − 1 𝜎(𝑟𝑡 ∗− 𝐸 𝑡{𝜋𝑡+1∗ } − 𝑟𝑟̅̅̅𝑡∗) 𝑟𝑟̅̅̅𝑡∗= 𝜌 − Σ(1 − 𝜌 𝛽)𝑎𝑡∗ ( 67 )
From the first to the second equation, the function of the productivity of the world was used, which is noted as an AR(1) function, 𝑎𝑡∗ = 𝜌𝛽𝑎𝑡−1∗ + 𝜀𝑡𝑎
∗
, where the error is on average zero.
3.5.3 Monetary Policy Rule
The foreign country has an own monetary policy in this model, and is therefore dependent on the foreign inflation and foreign output gap. As the foreign country is stated as the whole world, which is a closed economy, there is no difference between the domestic and foreign inflation of the foreign country.
𝑟𝑡∗= 𝜑
𝜋𝜋𝑡∗+ 𝑥𝑡∗
4. The Model: Inside a Monetary Union
The second model is for a country inside a monetary union. Most of last model can be used, as preferences and therefore also the optimal conditions are still the same. There are not many changes to the previous model, but those that are made are fundamental, and therefore change the results in an important way. By analyzing the differences resulting from being inside a monetary union, the main point is that some of the definitions change. Two definitions that change property, are the law of one price and the definition of the real exchange rate. Nominal exchange rate does not change and is therefore always equal to one in the model. The law of one price in this model implies 𝑃𝑖,𝑡 = 𝑃𝑖,𝑡𝑖 , meaning that the price of a good is exactly the same in the domestic country compared to
country i. The real exchange rate changes to 𝒬𝑖,𝑡 ≡𝑃𝑡𝑖
𝑃𝑡. The log linearized versions of these formulas
are,
𝑝𝑖,𝑡 = 𝑝𝑖,𝑡𝑖 and 𝑞𝑖,𝑡 = 𝑝𝑡𝑖 − 𝑝𝑡
( 69 ) Applying these changes to the international risk sharing condition, gives the same result as for the previous model. Therefore, besides definitions there are no changes to the fundamentals of the model.
Mixing the law of one price with the unchanged definition of the terms of trade gives,
𝑠𝑡 = 𝑝𝑡∗− 𝑝𝐻,𝑡
( 70 ) 𝑠𝑡= 𝑠𝑡−1+ 𝜋𝑡∗− 𝜋𝐻,𝑡
where first differences are taken in the last step. Formula (35) of the first model, which describes the relation between the nominal exchange rate and the terms of trade, does not apply in this model anymore and is replaced by formula (70). The most important change in the model is the fact that the domestic country does not have an own central bank. Therefore, monetary policy is conducted on behalf of the whole world. The monetary policy rule for the world will be a Taylor rule, dependent on world inflation and the world output gap.
𝑟𝑡∗= 𝜑𝜋𝜋𝑡∗+ 𝑥𝑡∗
The interest rate used in the domestic IS curve is therefore also this world interest rate. The foreign part of this model, which can be seen as a closed economy, is exactly the same as in the previous model. The full model can be found in appendix C.
5. Welfare
In this part, the calculations for welfare will be shown. There are multiple ways to calculate welfare for an economy. The most straightforward one is to take the discounted sum of all future utilities for the households in the economy.
𝑊 = 𝐸0∑ 𝛽𝑡 𝑈(𝐶 𝑡, 𝑁𝑡) ∞
𝑡=0
This requires the consumption and labor for every period. To obtain this data, a simulation is done with the previous models until 𝑡 = 10000, repeated for a thousand times. As shown in the last two sections, the model used is based on deviations from the steady state. By definition:
𝑐𝑡 = log(𝐶𝑡) − log(𝐶̅)
in which, 𝑐𝑡 is the log deviation of consumption from the steady state, 𝐶𝑡 is the actual consumption
and 𝐶̅ is the steady state value of consumption. Rewriting this formula shows the necessary function for the actual consumption.
𝐶𝑡 = 𝐶̅𝑒𝑐𝑡
Just as consumption, labor is also stated as deviations from steady state, and therefore can be written in the same way as the previous functions for consumption. Since the simulated data are stated as deviations from the steady state, the steady state values for consumption and labor are needed to calculate the welfare for the economies.
In this model there are two rigidities, namely monopolistic competition and sticky prices. The employment subsidy is used to remove the friction caused by the monopolistic competition. In case the prices were flexible, there would be no frictions left and the economy would be perfect efficient. Labor then would be always on its steady state and therefore, to determine the steady state level of labor, an analysis can be made of the efficient state of the economy. This can be done by analyzing the social planners’ problem, which tries to maximize the public’s welfare with the restriction of the production function (15), the international risk sharing condition (20), and the market clearing condition (23). To simplify these formulas, we take 𝜎 = 𝜂 = 𝛾 = 1 as given. Therefore, the results
obtained are only valid under this condition. Applying this to formula (20) and (23) results respectively in, 𝐶𝑡 = 𝐶𝑡∗𝒬 𝑡 ( 72 ) 𝑌𝑡= 𝐶𝑡𝑆𝑡𝛼 ( 73 ) Combining 𝒬𝑡 = ℰ𝑡𝑃𝑡∗ 𝑃𝑡 , 𝑃𝐹,𝑡= ℰ𝑡𝑃𝑡 ∗, 𝑆 𝑡 = 𝑃𝐹,𝑡 𝑃𝐻,𝑡, and 𝑃𝑡 = 𝑃𝐻,𝑡 (1−𝛼)
𝑃𝐹,𝑡𝛼 , whereby this last function
only holds under the specified assumptions, results in 𝒬𝑡= 𝑆𝑡(1−𝛼). When merging this result with formula (72) and (73), a formula is obtained which describes the relation between domestic consumption, and domestic and foreign output. Hereby, we made use of the market clearing condition for the world, described by formula (24).
𝐶𝑡= 𝑌∗𝑡𝛼𝑌𝑡(1−𝛼)
( 74 ) Combining this with the last restriction from the social planner, the production function, leads to,
𝐶𝑡= 𝑌∗ 𝑡 𝛼(𝐴
𝑡𝑁𝑡)(1−𝛼)
The welfare of the households depends on their utility function, which under the previous stated assumption, can be written as,
𝑈𝑡(𝐶𝑡, 𝑁𝑡) = log(𝐶𝑡) −
𝑁𝑡1+𝜑 1 + 𝜑
To find the steady state value of labor, the household welfare should be maximized under the condition that consumption equals the previous property. Also, given the fact that the domestic economy is infinitely small, the foreign variables can be seen as exogenous.
𝑑𝑊 𝑑𝑁𝑡= 𝜕𝑊 𝜕𝑁𝑡+ 𝜕𝑊 𝜕𝐶𝑡 𝜕𝐶𝑡 𝜕𝑁𝑡 = 0 𝑑𝑊 𝑑𝑁𝑡 = −𝑁𝑡𝜑+ 1 𝐶𝑡 (1 − 𝛼)𝑌∗ 𝑡 𝛼𝐴 𝑡 (1−𝛼)𝑁 𝑡−𝛼 = 0 𝑁𝑡= (1 − 𝛼) 1 1+𝜑 = 𝑁̅
This means that, under these specific conditions, labor has a steady state that depends on the openness of the economy and the elasticity of labor supply.
To obtain the steady state value for consumption, formula (74) is used. Combining this formula with the fact that technology equals one in the steady state and, due to symmetric preferences and production functions between all economies, 𝑌𝑡∗ = 𝑌𝑡, this formula results in 𝐶̅ = 𝑁̅. This implies
that the steady state values of labor and consumption are equal.
To compute the difference in welfare between two different scenarios, the permanent consumption increase that is needed to equalize the welfare in the different scenarios, is calculated. This is done by equalizing the welfare of the two scenarios including the increase in consumption for the scenario with the lower welfare.
𝐸0∑ 𝛽𝑡 ∞ 𝑡=0 [log((1 + Δ)𝐶1,𝑡) −𝑁1,𝑡 1+𝜑 1 + 𝜑] = 𝐸0∑ 𝛽 𝑡 ∞ 𝑡=0 [log(𝐶2,𝑡) −𝑁2,𝑡 1+𝜑 1 + 𝜑]
Therefore, the welfare gains are equal to the delta, and are noted as a percentage of a permanent increase in consumption. To obtain the right delta, an approach of trial and error is used. Since welfare is a relatively abstract concept, analyzing the changes in welfare in terms of permanent changes in consumption, will give a more useful and realistic value.
6. Data and Results
In this part, the used data will be explained and the results of the two different shocks will be analyzed. Firstly, the underlying values of the parameters will be discussed and their origin will be explained. Secondly, the two different shocks to the economy will be explained in two parts. In the first part, the domestic productivity shock will be analyzed, to see through which channels they affect the economy. The second part explains the foreign output shock. Finally, the welfare losses originating from these different systems will be examined and compared with each other to see what these losses are in terms of consumption. Also, changes in the features of the economy will be analyzed to see which features create more costs or benefits for joining a monetary union.
6.1 Calibration
This part shows which values are given to the different parameters. The calibration has been done for the Netherlands, and the assumption has been made that the European Monetary Union is the rest of the world. This assumption is made to simplify the model and is not unreasonable, with 57 percent of Dutch exports going to members of the European Monetary Union (Ramaekers, 2012). The time periods are in quarters of a year, and the assumption is made that 𝜎 = 𝜂 = 𝛾 = 1. This equation assumes that consumption enters the utility in log form, the households are indifferent between domestic and foreign goods, and the households are also indifferent between goods produced in different countries. The values of the rest of the parameters are mostly equal to the ones set by Gali and Monacelli (2005). This means that by setting the mark up for firms (μ) equal to 1.2, the elasticity of substitution of goods within the domestic market (ε) equals 6. This means that the mark-up for firms equals 20 percent, which is a common value also used by Bernanke, Gertler and Gilchrist (1998), and Christensen and Dib (2008). As noted in Mastrogiacomo et al (2013), and Bargain et al (2012), the Frisch elasticity of labor supply for the Netherlands can be approximated as 1/3, just as in the paper of Gali and Monacelli (2005), implying a 𝜑 of 3. This elasticity of labor supply measures the substitution effect of the wage rate on the labor supply, and differs between certain groups in society. In the Netherlands, men who are in a relation (0.14 with children, 0.07 without children), have a much lower elasticity of labor supply compared to women (0.5 with children, 0.27 without children). Singles (0.39 for men, 0.47 for women) on the other
hand, have a much higher elasticity compared to couples (Mastrogiacomo et al, 2013). On average, an elasticity of 1/3 is representative for the Dutch society as a whole. The parameter that captures if a firm is not allowed to change prices is θ, and is set to 0.75, meaning that firms have a chance of 25 percent to change their price in that particular period. As a period equals a quarter of a year, this implies that firms, on average, change their prices once a year. According to a survey of Hoeberichts and Stokman (2004), the price stickiness of Dutch firms depends on the degree of competition within that sector, and therefore different sectors should have different values of θ. Of the most competitive firms, more than 40 percent change their prices at least twice a year, and for firms dealing with lower competition, this value equals 20 percent. Since there are no different sectors within this model, an average should be taken. As on average most firms change their prices less than twice a year, a single price change a year would be representable for all the firms in the Netherlands. A common value of 0.99 is given to β, and due to the fact that parameters are measured per quarter, this implies a risk free interest rate of 4 percent a year. For the Taylor rule set monetary policy rule, a value is used of 1.5 for 𝜙𝜋, what captures the weight given to the inflation. The
openness of the economy, captured by α, is around 0.7 for the Netherlands. This is taken from data of the imports over GDP of the Netherlands.
The two different shocks in the model are the domestic and foreign productivity shock, which are both given by an AR(1) function. Both the parameters of the first auto regression and the standard errors of the shocks, are calibrated on data between 1995 and 2015. Regressions on labor productivity have been made for the Netherlands and the European Union as a whole, measured as GDP per hour worked. To make the data stationary, the differences of the log values of the productivity data have been used. This gave a value of 0.57 for the first auto regression parameter for the Netherlands, with a standard error of 0.185, and a value of 0.66 for the European Union as a whole, with a standard error of 0.243.
𝑎𝑡 = 0.57(0.185)𝑎𝑡−1+ 𝜀𝑡𝑎 𝑤ℎ𝑒𝑟𝑒 𝜎𝑎 = 0.0057339 𝑎𝑡∗ = 0.66(0.243)𝑎 𝑡−1 ∗ + 𝜀 𝑡𝑎 ∗ 𝑤ℎ𝑒𝑟𝑒 𝜎𝑎∗ = 0.0042687
Correlation between the two different shocks has been valued at 0.7665. This correlation tends to be quite high, but is explainable due to the high openness within the European Union, and the Netherlands in particular.
6.2 Mechanism after a shock
The two different shocks are to domestic and foreign productivity, and capture the above mentioned values for the parameters. The shocks are executed with a positive value of 1, which is called a unity shock. The best way to interpret the results, is to see the deviations from steady state as changes as a percentage of the shock. Therefore, a shock that increases the inflation to 0.5, illustrates that the inflation will increase with 50 percent of the value of the shock. This has been done to make the shocks more visible.
6.2.1 Domestic productivity shock
The first thing to notice, is that both for the country within as for the country outside of a monetary union, nothing changes for the foreign country. As the domestic country is one of infinite countries, their changes will not affect the foreign output or inflation. After a unit shock to the domestic productivity, it is obvious that the domestic production increases, and therefore also their consumption. However, due to the fact that households maximize their utility, the increased consumption makes them substitute labor for leisure, and therefore production increases less than one to one with the increased productivity. The natural level of output on the other hand, increases exactly one to one with the productivity and causes the output gap to decrease. This decrease in the output gap is caused by a lower demand for consumption compared to the natural demand. This also causes the domestic prices to decrease below their steady state level. From here on, the different systems will come into play. Figure 1 shows the impulse response function for both the country outside a monetary union and for the country within a monetary union.
Firstly, the country outside a monetary union will be explained, where the target is to stabilize domestic inflation and the output gap. To neutralize the decrease in output gap and domestic inflation, the central bank will decrease the interest rates. This causes output to increase, and therefore the output gap to grow towards its steady state. Since productivity returns to its steady state value after a shock, this causes the natural level of output to decrease, and decreases the output gap even more. Also, domestic inflation will increase due to the lower interest rates. The lower interest rates have a different effect as well, namely a decrease in the demand of the domestic currency, since the country becomes less attractive to store international funds. Therefore, the
currency will have a large depreciation at the moment the central bank decreases their interest rates. This depreciation will cause the domestic products to decrease in price compared to the foreign products, and therefore the exports and the terms of trade will be increased. This will lead to an even bigger push to the domestic production. The depreciation of the currency also causes the
imported inflation to increase on impact, and as foreign inflation counts for most of the CPI inflation, this outgrows the domestic inflations, and CPI inflation increases. As the interest rate of the central bank slowly goes back to the steady state, the domestic currency starts to appreciate. Eventually, the lower interest rates have brought inflation and output gap back to the steady state, with lower prices than before. These lower prices make the domestic currency more valuable than the foreign currency, and overall appreciated compared to the steady state before the shock.
For the country inside a monetary union, monetary policy is conducted on behalf of the whole currency area. The central bank of the area will therefore not alter its interest rates, due to the fact that the variables of the whole area do not deviate from the steady state. The only factor that stabilizes domestic inflation and output gap is the terms of trade. A decrease of the domestic prices leads to an increase of the terms of trade and export, and therefore gives a push to the domestic production. As the effect of the increased productivity decreases over time, domestic inflation and output gap go back to their steady state values, although this takes more time compared to situations in which the interest rates are lowered. One key value, the CPI inflation, is stabilized much better compared to the first situation. The reason behind this is the fact that there is no devaluation of the currency, which otherwise causes a high imported inflation.
As can be seen in figure 1, the consumption of the domestic country outside the monetary union increases more after the domestic productivity shock. This is due to the fact that output will not only increase because of the increased productivity, but the central bank also stimulates the economy by lowering the interest rates. This stimulation causes labor to increase, which is higher for a country outside the monetary union. For a negative domestic productivity shock these values are exactly the opposite, and therefore we can state that consumption is better stabilized inside a monetary union and labor outside a union. As noted before, an important finding is the fact that for countries with a high openness, being inside a monetary union has a better stabilizing effect on the CPI inflation. Domestic inflation on the other hand is better stabilized if a country has its own central bank.
6.2.2 Foreign productivity shock
If a unit productivity shock hits the foreign country, the first effects are on the world variables. As the foreign country can be seen as a closed economy, and the domestic variables do not influence its values, the state of the domestic country does not matter. If productivity increases, this will cause output to increase, but due to the substitution from labor to leisure, this increase will be less than the increase of the natural level of output. Therefore, there will be a decrease in the output gap, and demand falls below the natural demand, what leads to an inflation below the steady state level. To fight the lower inflation and output gap, the central bank will decrease the interest rates. This causes inflation and output to increase, back to the steady state level. The changes in the foreign variables do have an effect on the domestic variables. These effects however, depend on their system.
For a country outside a monetary union, there are not much changes. The decreased interest rate of the world, compared to the domestic economy, increases the demand of the domestic currency, and therefore it appreciates. Foreign goods become even cheaper, which causes the CPI inflation to drop. The decrease of the foreign prices leads to an increase in imports, and therefore the domestic country consumes more than in the steady state. The domestic country keeps producing the same, and therefore the output gap and the domestic inflation stay the same as well.
The second scenario has a different effect for a country inside a monetary union. As the central bank of the monetary union lowers the interest rates, this affects the domestic economy. Output increases and at impulse the domestic inflation increases as well. Due to the higher output, domestic prices decrease and this effect overtakes the increase of inflation, whereby domestic inflation decreases. Both the domestic and foreign prices decline. This decline is worse for foreign prices, and therefore causes a decrease in the terms of trade of the domestic country. The country will import more and export less, whereby domestic consumption increases and domestic production starts to decrease. Due to the high openness of the country, CPI inflation decreases on average, meaning that the foreign inflation overtakes the decrease in domestic inflation.
The movements in figure 2 show that almost all variables of the country that is not part of a monetary union, are better stabilized after a foreign productivity shock hits. This is due to the fact that a perfect stable country within a monetary union will have a lower interest rate after a foreign
variable whereby stability improves after joining a monetary union is the CPI inflation. So, an important and surprising insight in the choice of joining a monetary union, is the fact that countries with a high trade openness have a more stable CPI inflation when being part of this union.
