University of Amsterdam
MSc in Economics – Monetary Policy and Banking Supervisor: Dr. Christian A. Stoltenberg
Student number: 10605169
Francisco Pais de Sousa
Wage Rigidity In Portugal – A DSGE Approach
July 2014
Abstract:
In this thesis I study the impact of wage rigidity on the Portuguese business cycle fluctuations. I do so by building on a small open economy DSGE model and data on public sector careers and minimum wage negotiations in Portugal. My findings suggest that Portugal has a high level of wage rigidity in the European context. Especially, the public sector careers and minimum wage negotiations denote a very lengthy process of wage adjustment. High wage rigidity implies less responsive wage inflation. In the event of a monetary contraction, this results in a lesser decrease of price inflation in Portugal.
Contents
1-‐ Introduction……… 3 2-‐ Literature Review ………... 5 3-‐ Model .……… 7 3.1 – Households ……….…...7
3.2 -‐ Terms of Trade and International Risk Sharing ………11
3.3 – Firms ……….………..12
3.4 -‐ Foreign Country ………...15
3.5 – Equilibrium ………...16
3.6 -‐ Log-‐linearized Equations and Monetary Policy Rule ……….18
4-‐ Portuguese Labour Market ………...23
5-‐ Wage Rigidity Parameter ……….26
6-‐ Impulse Response Analysis and Policy Implications ………...30
6.1 – Portuguese level of wage rigidity compared to wage flexibility …... 31
6.2 – Portuguese economy as a whole compared to two of its sectors….. 35
7-‐ Conclusion ……….36 8-‐ References .………....38 9-‐ Appendix ……….41
1 -‐ Introduction
Nominal wage rigidity is a structural characteristic of the Portuguese economy. It is historically connected to the period that followed the democratic revolution of 1974, during which workers generally improved their contractual conditions, especially those with permanent contracts. The phenomenon of wage rigidity is a divisive topic in Portuguese politics, as left wing parties see the conquered rights with good eyes while right wing parties refer to it as one of the main weaknesses of the Portuguese economy.
The last three years made the debate on labour market rigidity become a hot topic in the Portuguese society. Since the 17th of May 2011, Portugal has been
under financial assistance from the so-‐called Troika (European Commission, European Central Bank and International Monetary Fund). During this period of influence, all these institutions have repeatedly stated that Portugal should increase the flexibility of its labour market. Furthermore, the ruling prime minister of Portugal, Pedro Passos Coelho, has been publicly in favour of greater labour market flexibility since he started to run for office in 2010. Aligned with this, the push for reforms in the labour market has been visible, despite the well-‐ established power of the trade unions.
In 2011 the World Economic Forum ranked Portugal 111th out of 142 countries
with respect to ‘flexibility of wage determination’. In 2013, Portugal changed its position to 105th out of 148 countries.
Wage rigidity introduces the hypothesis that the wage setting process is not perfectly flexible. Assuming a flexible wage setting process, households optimally alter the price of the labour they supply whenever the state of the economy requires it. Therefore, wage rigidity questions the ability of every household to adapt instantly to shocks in the economy via changes in the price of their labour. This causes nominal wages to be less responsive to the economy fluctuations. Assuming sticky prices, the bigger the degree of wage rigidity, the less responsive to the economy fluctuations real wages are. Moreover, the evolution of wages and prices are connected to each other. Wage rigidity might cause wage and price inflation to be persistently far away from their long-‐term values. This generates inefficient allocations of resources and reduces welfare. That being said, rigid wages increase the importance of inflation stabilization.
In order to analyze the role played by the Portuguese labour market nominal wage rigidity in the Portuguese business cycle, I use a Dynamic Stochastic General Equilibrium small-‐scale small open economy model. The model combines the small open economy model by Walsh (2010) with the Calvo-‐type staggered wage setting as in Erceg, Henderson and Levin (2000) and Galí (2008). Also, I provide a short description of the Portuguese labour market and highlight the importance of the minimum wage discussions and the public sector careers for the study of wage rigidity in Portugal. I build on established literature and data on the public sector careers and minimum wage negotiations to construct three different wage rigidity parameters. One denotes the labour market rigidity of the Portuguese economy as a whole. The second denotes the rigidity
concerning the minimum wage negotiations and the third refers to the nominal wage rigidity of the Portuguese public sector careers. Finally, using the DSGE model, I discuss the impact of different levels of wage rigidity on the transmission of shocks in an economy. I do so by comparing the three constructed wage rigidity parameters with the scenarios of a standard level of labour market flexibility and of total flexibility. From these comparisons, I draw policy implications of wage rigidity for the Portuguese economy.
Two results from the construction of the three wage rigidity parameters are worth accentuating. Firstly, the Portuguese economy degree of nominal wage rigidity is very high in the European context. In practical terms, it takes significantly more than one year, on average, until a household gets the chance to reset wages. This is in line with the established literature on this topic and with the EPL Index by OECD. Secondly, the level of rigidity regarding the minimum wage negotiations and the public sector careers is even higher than that of the Portuguese economy as a whole. This is also aligned with the established literature on this topic because these two particular negotiations are associated with more pronounced influence of collecting bargaining and trade unions.
I highlight one main result from analyzing the impact of different wage rigidity levels on the transmission of shocks in a small open economy. It results from comparing the Portuguese level of wage rigidity with a flexible wage setting economy. The Portuguese economy denotes substantial wage rigidity compared to other countries. Consequently, wage inflation becomes less responsive. This implies that the decrease of price inflation is less pronounced after a monetary contraction in Portugal. Furthermore, I emphasize one main policy implication for the Portuguese economy. Wage rigidity increases the importance of inflation stabilization for an economy. As the Portuguese denotes a higher level of wage rigidity than many other economies, this implies that inflation stabilization becomes more important for the Portuguese economy compared to economies with more flexible wages.
In the next section I provide a brief overview of the related literature. Then I explain the small open economy DSGE model. In the following sections I analyze the Portuguese labour market and explore the theoretical concept of wage rigidity. In the end, I analyze the impact of wage rigidity in the transmission of shocks and draw policy implications from it.
2 -‐ Literature Review
In this section I provide a brief reference to the evolution of the studies on the topics covered by this thesis. In order to do so, I highlight relevant events and literature on the topics of macroeconomic modeling and wage rigidity.
Macroeconomic analysis relied on large-‐scale models during the 1960’s and 1970’s. These models incorporated economic theory in their equations. However, their coefficients were determined using time series analysis and were assumed to remain constant over time, which has raised much criticism. Robert Lucas (1976), in his famous “Lucas critique”, claimed that the economic parameters estimated by these models were not structural, but rather time and policy varying.
Rational decision-‐making and microeconomic foundations were brought by the seminal work of Kydland and Prescott (1982), which featured formation of expectations in a general equilibrium model. Firms and households were optimal and faced intertemporal maximization problems while a perfectly competitive economy with no kind of rigidities was assumed. Also, fluctuations were both endogenous and optimal for a given equilibrium of the variables of the model and, thus, policymaking was fruitless. Moreover, the main drivers of these aggregate fluctuations were technology shocks and money was not included in the model, as it was assumed to have no influence in the real economy. This model was adopted by many economists and was on the basis of the RBC (Real Business Cycle) Theory.
What followed was the New Keynesian model. It captured the microeconomics foundations of the RBC models and kept the DSGE framework but also introduced specific theoretical contributions that contrasted with the RBC models. According to Galí (2008), these contributions were three: Firstly, it was consistent with the Keynesian thinking, as it justified the existence of policymaking. This was done by modeling constraints in price and wage setting for the re-‐optimization of agents, such as Calvo pricing. Moreover, these same constraints to the re-‐optimization of each agent allowed money to interfere in the real interest rate. This was due to the fact that inflation did not react instantly to the changes in interest rate, making money become non-‐neutral in the real economy. Thirdly, markets were assumed to have imperfections, which were typically modeled by the Dixit-‐Stiglitz form of monopolistic competition.
As previously stated, the inclusion of rigidities in the DSGE models proved to be of irrefutable importance for justifying the business cycle fluctuations. Firstly, Calvo pricing was introduced to the goods market. Subsequently, Erceg, Henderson and Levin (2000) extended nominal Calvo-‐type rigidity to wages. This marked the inclusion of wage rigidity in macroeconomic modeling. Additionally, this work showed an important result for monetary policy in the context of nominal wage and price rigidities. It stated that monetary policy could not achieve the Pareto-‐optimal equilibrium that would occur under completely flexible wages and prices. This represents that monetary policy faced a trade-‐off in stabilizing the output gap, price inflation and wage inflation. For this reason
monetary policy rule should aim at maximizing welfare, given that it could not stabilize the output gap, wage inflation and price inflation at the same time.
Two works were important for the study of wage rigidity. Stiglitz (1984) called for the need of explaining wage rigidity. He stated that in order to explain the cyclical movements in wages, one must take into account the presence of asymmetric information, the limitations of enforcement mechanisms, the nature of insurance and restrictions on the degree of complexity of feasible contracts. Robert Solow (1998) enumerated the symptoms that define nominal wage rigidity. Some examples are high level and duration of unemployment benefits, high number of restrictions on hiring and firing and excessive influence of over-‐ protective trade unions.
Many other studies were done on the topic of nominal wage rigidity. Dickens, Gotte, Groshen, Holden, Messina, Schweitzer, Turunen and Ward (2006) provide a microeconomic approach to how wages change by analyzing the distribution of wage alterations. This study showed that the trade unions level of influence on wage bargaining is a significant factor for explaining the level of rigidity of a labour market. Haefke, Sonntag and Rens (2013) restricted the scope of study of nominal wage rigidity to new hires. The results demonstrated that for this particular group of workers there was little evidence for wage stickiness, as wages responded almost one-‐to-‐one to changes in productivity. This draws attention to the importance of the length of the relationship between employer and employee for the study of nominal wage rigidity.
Furthermore, many are the studies regarding the consequences of nominal wage rigidity. Nickell (1997) discussed how higher levels of wage stickiness in Europe might contribute for higher levels of unemployment compared to North America. Doing so, this work contributed for demonstrating important impacts of wage rigidity: lower labour mobility and higher unemployment. Moreover, Messina, Caju, Duarte, Izquierdo and Hansen (2010) showed again that nominal wage rigidity has a negative impact on employment. Also, this study introduced the possibility that that nominal wage rigidity might affect labour productivity positively.
Finally, labour market dynamics and levels of wage rigidity are also studied on a national scale. In the specific case of Portugal, nominal wage rigidity is a structural characteristic of the economy, as Alexandre, Bação, Silva and Portela (2010) pointed out. Cardoso and Portugal (2005) argued that firms respond to the Portuguese labour market rigidity by creating ‘wage cushions’. Firms do so by hiring employees with wages above those set in the sectorial bargaining, which gives room to decrease wages when needed. Nonetheless, Portugal features low job creation and destruction, which according to Portugal and Blanchard (2001) is mainly caused by high employment protection.
3 -‐ Model
In this section I present the model that I use in the next stages of this thesis. As stated in the introduction, it is based on the small open economy model as in Walsh (2010) and Galí (2008), featuring sticky prices. I augment the model with wage stickiness following the works of Erceg, Henderson and Levin (2000) and Galí (2008).
The structure of the model economy consists of three different sectors, households, intermediate goods firms and final goods firms, each of them optimizing their respective objective function. Also, both labour and intermediate goods markets face monopolistic competition and sticky prices and wages. I close the model with a Taylor-‐type monetary policy rule.
The ‘Appendix’ section contains the full list of variables and parameters mentioned in the model.
I follow Walsh (2010) for the approach and order of description of the model. In order to do so, I focus mainly on chapter nine of his 2010 book. For the ‘Firms’ section I also follow its chapter eight.
3.1 -‐ Households
The economy contains a continuum of infinitely lived households, indexed by 𝑘. Each household aims at achieving an optimal bundle, 𝐶!,!, of home produced goods, 𝐶!!, and foreign produced goods, 𝐶
!!. Given that every household has constant elasticity of substitution (CES) over foreign and home goods, each of them faces the following minimization problem:
(1) 𝑚𝑖𝑛!!!! 𝑃!!𝐶 !! + 𝑃!!𝐶!! subject to (2) 𝐶!,! = (1 − 𝛾)!! 𝐶!,!! !!! ! + (𝛾)!! 𝐶 !,!! ! !!! ! !!! .
In words, households minimize total expenditures given the price of each type of goods, 𝑃!! for home goods and 𝑃
!! for foreign goods, taking into account the restriction on the composition of consumption. This restriction depends on the degree of openness of the economy, 𝛾, and the price elasticity of substitution between home and foreign goods, 𝑎.
This optimization problem leads us to the following first order condition: (3) 𝐶!! 𝐶!! = (1 − 𝛾) 𝛾 𝑃!! 𝑃!! !! ,
which gives the intuitive result that the home country’s relative demand for home and foreign produced goods, 𝐶!! 𝐶
!! , depends on their relative price (𝑃!! 𝑃
!!). Also, the bigger the degree of openness is, the greater the weight of consumption of foreign goods in the total consumption.
Combining the first order conditions with respect to 𝐶!! and 𝐶
!!and applying the restriction on the composition of consumption described in (2), I reach the following expression: (4) 𝑃!= 1 − 𝛾 𝑃!!!!! + 𝛾 𝑃 !!!!! ! !!! .
The above equation provides the Consumer Price Index (CPI).
Simultaneously, each household k aims at maximizing its lifetime utility function taking into account a sequence of budgets constraints:
(5) 𝑈 = 𝐸! 𝛽!𝜃!! 𝐶!,!!!!!! 1 − 𝜎− 𝑁!,!!!!!! 1 + 𝜂 ! !!! subject to (6) 𝐸! 𝛽!𝜃!! 𝑊!,!𝑁!,!!! + 𝑅!!!𝐵!,!!!!!− 𝑃!!!𝐶!,!!!− 𝐵!,!!! = 0 ! !!! .
As in Walsh (2010), each period’s utility depends positively on the total consumption of foreign or home produced goods, 𝐶!,!, and negatively on the amount of labour supply by each household 𝑘, 𝑁!.
To compare the utility of different combinations of labour and consumption over time, households discount the future by the factor 𝛽. Also, I follow Erceg, Henderson and Levin (2000) by applying the Calvo price setting mechanism to the labour market. I then assume that wage contracts are not revised until
households are given the chance to adjust them. Merely 1 − 𝜃! ∈ 0,1 of the total number of households is allowed to reset their wage. Being so, the utility function is augmented by the probability of the household not being able to adjust its wage, 𝜃!, which influences not only wage setting but also consumption and investment decisions.1
𝜎 represents the elasticity of intertemporal substitution while 𝜂 denotes the inverse of wage elasticity of labour supply.
The optimization must fulfill the restriction imposed by (6): the discounted sum of the subtraction of one period’s expenditure (via consumption, 𝑃!!!𝐶!,!!!, or investment in bonds, 𝐵!,!!!) from one period’s earnings (via labour, 𝑊!,!𝑁!,!!!, or internationally traded bonds from the period before, 𝑅!!!𝐵!,!!!!!) must equal zero. 𝑊!,! denotes the nominal wage earned for each unit of labour. This restriction introduces a ‘no-‐Ponzi game’ condition for households.
The labour market is defined by the rules of monopolistic competition. Each household 𝑘 faces demand for its specific differentiated type of labour. This demand function, which is obtained from the intermediate goods producing firms’ cost minimization problem, is as follows:
(7) 𝑁!,! = 𝑊!,! 𝑊! !!! 𝑁!.
Since 𝜀!, the inverse of wage elasticity of substitution between different types of labour, is greater than zero, the above equation presents demand as negatively dependent on the relative wage demanded by the household, (𝑊!,! 𝑊!). Also, it depends positively on the amount of aggregated labour supplied in period 𝑡, 𝑁!.
I introduce the restriction of fulfilling the labor demand, (7), to the maximization problem of (5) subject to (6). Doing so and combining the optimality conditions for the nominal wages of household 𝑘 in period 𝑡, 𝑊!,!, and the respective level of consumption, 𝐶!,!, yields: (8) 𝐸! 𝛽!𝜃 !! 𝑁!,!!!! 𝑁!!! 𝜀! 𝜀! − 1 = 𝐸! 𝛽!𝜃!! ! !!! ! !!! 𝐶!,!!!!! 𝑊!,! 𝑃!!! 𝑁!,!!! .
This equation presents the first order wage setting condition for each period 𝑡. On the left hand side, it gives the expected value at period 𝑡 of the infinite sum of the discounted present and future period’s utility obtained from an additional unit of labour supplied. In order to verify optimality, it shall match the right hand side of the equation, which represents the expected value at period 𝑡 of the infinite sum of discounted present and future marginal disutilities obtained from an additional unit of labour supplied.
1 I will further explore this fact in the next sections.
When this condition is verified, the wage setting function meets a specific value. Since the marginal benefit subtracted by the marginal loss of supplying labour equals zero, setting the wage at period 𝑡 above or below the level set by (8), given the assumed restrictions, would be non-‐optimal.
I re-‐arrange (8) to have only one expectations operator and to get labour, 𝑁!!!, and the marginal utility of consumption, 𝐶!,!!!!! , as common factors. This yields: (9) 𝐸! 𝛽!𝜃 !!𝑁!!!𝐶!,!!!!! 𝑊!,! 𝑃!!! − 𝜀! 𝜀!− 1𝑀𝑅𝑆!,!!! ! !!! = 0,
where 𝑀𝑅𝑆!,!!!stands for marginal rate of substitution between consumption and labour at period for household 𝑘 at period (𝑡 + 𝑖).
Since this thesis focuses on the role played by wage rigidity in the fluctuations of the business cycle, it is worth further exploring (9) focusing on the value of 𝜃!. If it equals zero, the labour market is defined by perfect wage flexibility. Mathematically, that implies that 𝜃!! is equal to zero for every 𝑖 from one to infinity and equal to one in the current period 𝑡. In this case, the equation simplifies to: (10) 𝛽𝑁!𝐶!,!!! 𝑊!,! 𝑃! − 𝜀! 𝜀!− 1𝑀𝑅𝑆!,! = 0.
Assuming the scenario where 𝛽 > 0, 𝑁! > 0 and 𝐶!,!!! > 0, the household 𝑘 sets the real wage, (𝑊!,! 𝑃!), as a mark-‐up, (𝜀! (𝜀!− 1)), over the current period marginal rate of substitution between labour and consumption, 𝑀𝑅𝑆!,!. This way, the wage setting decision of household 𝑘 in period 𝑡 is no longer forward looking, as (10) documents.
However, under a circumstance where there is any rigidity in wage setting, 𝜃! is a positive number. In general terms, the bigger 𝜃! is, the less likely it is that a household gets the chance to reset its wage in future periods. As households are perfectly rational, a higher 𝜃! implies that the future has a greater weight in current period wage setting.
The combination of the optimality conditions with respect to bonds holdings, 𝐵!,!, and consumption 𝐶!,!, yields:
(11) 𝐶!,!!! = 𝐸 ! 𝛽𝜃! 𝑃! 𝑃!!! 𝑅!!!𝐶!,!!!!! .
The above equality is the so-‐called Euler equation. It relates present consumption to future consumption through the discount factor, 𝛽 , the probability of the household not being able to reset its wage, 𝜃!, and the real interest rate, 𝑃! 𝑃!!! 𝑅!!!.
This equation provides the channel through which monetary policy affects inflation, as it contains the interest rate, 𝑅!!!, and the inverse of the inflation rate
𝑃! 𝑃!!! linked by the equality.
3.2 -‐ Terms of Trade and International Risk Sharing
I assume that the law of the one price holds. This implies the following:
(12) 𝑃!! = 𝑆!𝑃!∗,
where 𝑃!!denotes the price of foreign produced goods in terms of domestic currency, 𝑃!∗ the price of the same foreign produced goods but expressed in foreign currency and 𝑆! the nominal exchange rate.
Also, I assume complete exchange rate pass through:
(13) 𝑃! = 𝑆!𝑃!∗.
This implies that import prices have a one-‐for-‐one response to changes in the exchange rate. 𝑃! is the CPI as derived in (4).
Following Walsh (2010), I treat the rest of the world as a large closed economy. For that reason, I do not distinguish between the price level of home and foreign produced goods.
Terms of trade, ∆!, and the real effective exchange rate, 𝑄!, are defined as follows: (14) ∆!= 𝑃!! 𝑃!! = 𝑆!𝑃!∗ 𝑃!! and (15) 𝑄! = 𝑃!! 𝑃! = 𝑆!𝑃!∗ 𝑃! = 𝑃!! 𝑃! ∆!.
Foreign households face the same constrained optimization problem that domestic households do.2 Assuming foreign households have access to the same
internationally traded bonds, this yields the foreign households Euler equation:
(16) 𝐶!∗ !! 𝑆!𝑃!∗ = 𝐸! 𝛽 𝑅!!! 𝑆!𝑃!∗ 𝐶!!!∗ !!
The left hand side represents the foreign household disutility from obtaining an extra unit of domestic currency to buy a bond that pays 𝑅!!! in the next period. In order to verify the optimality condition this must match the expected discounted marginal utility obtained from buying that same extra unit of currency for investing in the bond. The earnings are then converted to foreign currency again.3
(16) provides a twofold channel. Firstly, it relates domestic consumption to foreign consumption. Secondly, since I assume the rest of the world to be a closed economy, foreign consumption equals foreign output, which implies a link between domestic consumption and foreign output.
3.3 -‐ Firms
There are two categories of firms operating in this model economy: composite final consumption goods producing firms and intermediate goods producing firms. The latter hires differentiated labour to produce differentiated goods, while the former buys these intermediate goods as the only input to produce homogeneous composite final consumption goods.
These final goods are then consumed by households as home produced goods, (𝐶!,!! )4. Because I assume the rest of the world to be a closed economy, these
goods can only have the domestic market as destination.
The final consumption goods market is defined by perfect competition because the many firms that are part of it sell the same completely homogenous product.
The production function of a representative final composite consumption goods producing firm is as follows:
(17) 𝑌!= 𝑌!,! !!.!!! !!,! ! ! 𝑑𝑗 !!,! !!,!!! .
As the integral suggests, the final goods firms assemble inputs from a continuum of intermediate goods firms indexed by 𝑗 ∈ (0,1). With them, each final goods firm creates its share of the total final good production, 𝑌!.
𝑌!,! denotes the production of intermediate goods by firm j at period t, whereas 𝜀!,! signifies the inverse of price elasticity of substitution between the differentiated intermediate goods. It is valuable to stress the presence of the index 𝑡 in 𝜀!,!. This comes from the fact that, contrarily to the case of the inverse
3 As in Wash (2010), I assume homogeneous foreign households and do not introduce an index
for them.
of wage elasticity of substitution between different types of labour, 𝜀!5, this
elasticity varies over time (indexed by 𝑡). According to Galí (2008), this implies a time varying price mark-‐up and serves as a source of a cost-‐push shock in the linearized model.
In line with Galí (2008), I follow two simplifying assumptions: final consumption goods firms only use home produced intermediate goods as inputs and home intermediate goods are not exported. This limits trade to final consumption goods.
Each firm 𝑗 solves its cost minimization problem, which yields: (18) 𝑌!,! = 𝑃!,! ! 𝑃!! !!!,! 𝑌!.
This equation determines the demand schedule for each firm 𝑗. It depends negatively on the relative price of the relative price of the firm’s goods, 𝑃!,!! 𝑃
!! , and positively on the production of the final good, 𝑌!.
Intermediate goods firms supply differentiated goods in the context of monopolistic competition. Accordingly, they hire all available differentiated types of labour. That said, the production function of the representative intermediate goods firm 𝑗 is as follows:
(19) 𝑌!,!!! = 𝑒!!𝑁!,!!!,
where 𝜀! denotes the common technology parameter for every intermediate goods firm 𝑗. It depends positively on that same parameter and, logically, on the number of labour units hired.
Cost minimization by a representative firm 𝑗, shows that real marginal costs 𝑀𝐶!,! are equal for every intermediate goods firm. For that reason, I drop the index 𝑗 and define 𝑀𝐶! as:
(20) 𝑀𝐶! = 𝑊! 𝑃!! 𝑒!! .
As stated in the literature review, following the works of Calvo (1983), price rigidity was established as a common practice in New Keynesian models. Accordingly, the works of Walsh (2010), Galí (2008) and Erceg, Henderson and Levin (2000), on which this models builds, feature price rigidity.
I assume that intermediate goods firms adjusting prices each period is not a certain event. It is associated with a constant probability over time (1 − 𝜃!). This implies that, on average, 𝜃! of the intermediate firms do not get the change to reset their prices.
Intermediate goods firms are profit-‐maximizers as in many other models. However, the introduction of price rigidity changes the dynamics of the optimal decision-‐making for firms. When these firms set prices they take into account the fact that might not be able to do so in the next period. This forces firms to take into account future periods. More rigid prices (larger 𝜃!) imply that future periods have greater importance for firms while setting prices for the current period.
A representative intermediate goods firm 𝑗, when given the opportunity, sets its price, 𝑃!,!!, by solving the following problem:
(21) 𝜛!,! = 𝐸! 𝜃!!χ!,!!! 𝑃!,!! 𝑃!!!! 𝑌!,!!!− 𝑌!,!!!𝑀𝐶!!! ! !!! subject to (22) 𝑌!,! = 𝑃!,! ! 𝑃!! !!!,! 𝑌!.
In words, the firm maximizes its profits, 𝜛!,!, constrained by the demand schedule for its good, (22) and (18), with respect to its price. 𝜒!,!!! denotes the real stochastic discount factor.6
The solution for this problem yields: (23) 𝑃!,!! 𝑃!!!! = 𝜀!,! 1 − 𝜀!,! 𝐸! 𝛽!𝜃 !! 𝑌!!!!!! 𝑃!!! ! 𝑃!! !!,!!! 𝑀𝐶!!! ! !!! 𝐸! 𝛽!𝜃 !! 𝑌!!!!!! 𝑃!!!! 𝑃!! !!!!,!!! ! !!! .
This equation gives the first order condition price-‐setting rule.
Because firms face the same problem and the same price rigidity level, every firm that can adjust its price will do so the same way, as long as they have the same
6 The stochastic discount factor, 𝜒
!,!!!, is equal to 𝛽! !!!!! !
!!
real marginal costs7. As (20) shows, that is always the case, which implies that
when given the chance, every firm resets their prices to same level.
Final goods firms maximize their profit with respect to their output, which gives the following price-‐setting rule:
(24) 𝑃!! !!!! = (𝑃 !,!!)!!!!𝑑𝑗. ! !
As stated above, when firms are given the chance to reset their price they all do it the same way because they face the same marginal costs.
I combine this fact with the premise that each period, on average, only (1 − 𝜃!) ∈ (0,1) of the firms is able to adjust their prices, while the other 𝜃! ∈ (0,1) is not. This allows me to derive the price of home produced goods in period 𝑡, 𝑃!!: (25) 𝑃!! !!!!,! = 1 − 𝜃 ! 𝑃!,!! !!!!,!+ 𝜃! 𝑃!!!! !!!!,!,
where 𝑃!,!! denotes the price set by the firm 𝑗, the same as any other price set in period 𝑡.
3.4 -‐ Foreign Country
Because I assume that the rest of the world is a large closed economy and that the model economy can only import final goods from it, I only need to define the level of domestic consumption of foreign goods,𝐶!!∗.
In order to do it, I assume homogenous preferences across the world. This yields:
(26) 𝐶!!∗ = 𝛾Δ
!𝑌!∗,
where 𝑌!∗ denotes the level of foreign output.
Finally, the assumption that the rest of the world is a large closed economy, allows me to conclude that the foreign trade is irrelevant to the level of foreign output and income. As there is no capital or investment, the level of foreign income or output is, approximately, the same as foreign consumption, 𝐶!∗.
Mathematically, I assume equality instead of approximation:
(27) 𝐶!∗ = 𝑌
!∗
7 Following the logic of (10) in the section ‘Households’, (21) implies that when prices are
perfectly flexible, 𝜃!= 0 , the price set by firms is a mark-‐up over the current period marginal cost.
3.5 -‐ Equilibrium
In this section I assemble the equations that define the model. Below, I list these equations and verify the equilibrium condition.
The market clearing condition assures that the total output of a period is totally consumed, by domestic or foreign households in that same period. Therefore, it is defined as follows:
(28) 𝑌!= 𝐶!!∗+ 𝐶
!!.
This is added to the core of what was presented in the four sections before: The first section is ‘Households’. They have constant elasticity of substitution (CES) over foreign and home goods and choose between domestic and foreign final goods, (2) and (29). Also, they solve a constrained problem of minimization of expenditure, which yields an optimal relationship between foreign and domestic goods, (3) and (30). This allows me to determine the Consumer Price Index, (4) and (31).
Furthermore, I obtain the wage setting rule, (9) and (32), the Euler equation, (11) and (33), by solving the utility maximization problem. To this, I add the definition of general wage level, (34), and the marginal rate of substitution between labour and consumption, (35).
(29) 𝐶! = (1 − 𝛾)!! 𝐶!! !!! ! + (𝛾)!! 𝐶 !! ! !!! ! !!! (30) 𝐶!! 𝐶!! = (1 − 𝛾) 𝛾 𝑃!! 𝑃!! !! (31) 𝑃!= 1 − 𝛾 𝑃!!!!!+ 𝛾 𝑃 !!!!! ! !!! (32) 𝐸! 𝛽!𝜃!!𝑁!!!𝐶!,!!!!! 𝑊!,! 𝑃!!! − 𝜀! 𝜀! − 1𝑀𝑅𝑆!,!!! ! !!! = 0 (33) 𝐶!,!!! = 𝐸 ! 𝛽𝜃! 𝑃! 𝑃!!! 𝑅!!!𝐶!,!!!!! (34) 𝑊!!!!! = 1 − 𝜃 ! 𝑊!,!!!!!+ 𝜃!𝑊!!!!!!! (35) 𝑀𝑅𝑆! = 𝑁!!! ! 𝐶!!!!!.
Secondly, I model the ‘Terms of Trade and International Risk Sharing’. I define the terms of trade, (14) and (36), and the real effective exchange rate, (15) and (37). I add to this the relationship between total consumption and foreign consumption, (38). (36) ∆!= 𝑃! ! 𝑃!! = 𝑆!𝑃!∗ 𝑃!! (37) 𝑄!= 𝑃! ! 𝑃! = 𝑆!𝑃!∗ 𝑃! = 𝑃!! 𝑃! ∆! (38) 𝐶! = 𝜁𝑄!!!𝐶!∗.
Thirdly, I highlight the equilibrium conditions for the sector ‘Firms’. I define the production function, (17) and (39). From the maximization problem faced by firms I derive the marginal cost, (20) and (40), and the wage setting rule, (23) and (41). Given this, I can then conclude on the general price level, (25) and (42). (39) 𝑌!= 𝑌!,! !!.!!! !!,! ! ! 𝑑𝑗 !!,! !!,!!! (40) 𝑀𝐶!= 𝑊! 𝑃!! 𝑒!! (41) 𝑃!,!! 𝑃!!!! = 𝜀!,! 1 − 𝜀!,! 𝐸! 𝛽!𝜃 !! 𝑌!!!!!! 𝑃!!! ! 𝑃!! !!,!!! 𝑀𝐶!!! ! !!! 𝐸! 𝛽!𝜃 !! 𝑌!!!!!! 𝑃!!!! 𝑃!! !!!!,!!! ! !!! (42) 𝑃!! !!!!,! = 1 − 𝜃 ! 𝑃!,!! !!!!,!+ 𝜃! 𝑃!!!! !!!!,!.
Finally, I select the equilibrium conditions from the section ‘Foreign Country’. These are the relationship between foreign consumption of goods produced by domestic firms and foreign output, (26) and (43), and the principle that approximates foreign total consumption to foreign output, (27) and (44). I add to this the market clearing condition mentioned in the beginning of this section, (28) and (45). (43) 𝐶!!∗ = 𝛾∆ !𝑌!∗ (44) 𝐶!∗ = 𝑌!∗ (45) 𝑌!= 𝐶!!∗ + 𝐶 !!
Summing up, I have selected a set of equilibrium conditions to form a system of seventeen, (29) to (45), nonlinear difference equations.
The number of selected equations is the minimum enough to guarantee that the economic principles discussed in the last four sections are included in the system.
Nineteen variables are used: 𝐶!, 𝐶!!, 𝐶
!!, 𝑃!, 𝑃!,!, 𝑅!, 𝑊!,!, 𝑊!, 𝑃!!, 𝑃!!, 𝑁!, 𝑀𝑅𝑆!, ∆!, 𝑄!, 𝐶!∗, 𝑀𝐶
!, 𝑌!, 𝑌!∗, 𝐶!!∗. Also, a technology parameter, 𝜀!, is used.
The mathematical equilibrium of a system of equations requires the number of equations to be equal to the number of variables. However, the system of nonlinear equations has twenty variables, but only seventeen equations.
For that reason, I introduce three different equations. The first two are the equations that define foreign output and the log of the technology parameter as AR(1) processes. The third is a monetary policy rule that targets price inflation through interest rate changes.
I will present these three equations as well as the log-‐linearized equations that allow me to solve the model in the next section.
3.6 -‐ Log-‐linearized Equations Around The Steady State and
Monetary Policy Rule
I proceed with log linearizing the above selected equilibrium equations. As in Walsh (2010), I explain the most important economic dynamics of the model through obtaining the three basic equations that define this model: the small open economy forward-‐looking New Keynesian IS curve, the small open economy New Keynesian Phillips curve and the small open economy wage Phillips curve. Moreover, I present the simplified version of the model with which I will work in the following sections. Finally, I close the model by defining the monetary policy rule and I comment the choice of this rule.
I log linearize the equilibrium conditions presented in the previous section around the zero wage and price inflation steady states. This turns the below variables into percentage deviations from the steady state.
After this, I assume the existence of four different shocks: a cost-‐push shock, a monetary policy shock, a technology shock and a foreign output shock.
With this combination of log linearized equations and shocks, I build the following three basic equations and a simplified version of the model.
I combine the log-‐linearized equations of the goods market clearing condition, (24) and (45), the definition of terms of trade, (14) and (36), the real effective rate definition, (15) and (37), the optimal international risk sharing condition, (38), the approximation of foreign total consumption to foreign output, (27) and (44), and the Euler equation, (11) and (33), to obtain the small open economy forward-‐looking New Keynesian IS curve:
(46) 𝑦! = 𝐸!𝑦!!!− 1 𝜎! 𝑟!− 𝐸!𝜋!!!! + 𝛾 1 − 𝜉 𝜎! 𝐸!𝑦!!!∗ − 𝑦!∗ , 𝑔𝑖𝑣𝑒𝑛 𝑡ℎ𝑎𝑡: 𝜎! = 𝜎 1 − 𝛾(1 − 𝜉); 𝜉 = 𝑎𝜎 + 𝑎𝜎 − 1 1 − 𝛾 .
Like other IS curves, this equation depicts the behavior of the aggregated supply sector. Accordingly, it shows the typical negative relationship between one period’s output and the respective interest rate that bonds yield. The higher the latter is, the more the former shrinks, as households start to invest more and delay consumption for the future.
However, this IS curve includes some particular features. It is forward-‐looking because it takes into account the expectations for the future of certain variables, such as the growth of foreign income.
The open economy feature can be seen by the presence of the degree of openness of the economy, 𝛾. Put simply, this is due to the fact that domestic goods face the competition of foreign goods even though the rest of the world is assumed to be a closed economy, which has further implications in the domestic economy.
I combine the log-‐linearized equations of the general price level of home produced goods, (25) and (42), and the pricing rule condition, (23) and (41), to obtain the small open economy New Keynesian Phillips curve.
(47) 𝜋!! = 𝛽𝐸 !𝜋!!!! + 𝜅!𝑚𝑐!+ 𝑧! 𝑔𝑖𝑣𝑒𝑛 𝑡ℎ𝑎𝑡: 𝜅! =(1 − 𝛽𝜃!)(1 − 𝜃!) 𝜃! ; 𝑚𝑐! = 𝑤!− 𝑝!!− 𝜀!; 𝜔!! = 𝑤!− 𝑝!!; 𝑧!~ 𝑁 0, 𝜎!! .
𝑧! follows a normal distribution with mean zero and a given variance. It is a cost-‐ push shock that results from the mark-‐up over the real marginal cost, (𝜀!,!) (1 − 𝜀!,!), that determines relative pricing in (23) and (41).
Home price inflation, 𝜋!!, depends positively on future home price inflation. Also, it depends on the difference between real production wage8, 𝜔
!!, and the representative of the marginal product of labour, 𝜀!. The intuition behind it is that firms manage their profit margins. If, ceteris paribus, the real production wage increases, firms adapt their margin by increasing prices. In case the marginal product of labour increases firms can decrease their prices because they can now use less labour units per production unit. This way, they can maintain the profit margin and increase sales.9
8 I define real production wage as 𝑊
! 𝑃!! .
9 Iterating (47) forward yields: 𝜋
! ! = 𝜅
! !!!!𝛽!𝑚𝑐!!!. Inflation is equal to the discounted sum of
