Investment Decision and Inflation Targeting in the

Noname manuscript No. (will be inserted by the editor)

Investment Decision and Inflation Targeting in the Brazilian Economy An analysis by Game Theory and Agent-Based Computational Economics Douglas S. Silveira · Silvinha P. Vasconcelos · Claudio R.F. Vasconcelos

August 3, 2018

Abstract The objective of this article is to understand in which conditions the Brazilian economy can find itself with a high level of investment, given the inflation target rate announced by the monetary authority. By doing so, we propose a dynamic game, assuming that agents have bounded rationality. To complement the equilibrium analysis of the game, a stochastic learning rule is introduced by an Agent Based Computational Economics (ACE) model. To achieve this goal, entrepreneurs and workers iteratively play a stage game to make investment decisions. Considering complementarity between investments, more than one Nash equilibria can emerge as a solution in the long term. In this sense, the conditions for successfully guiding the Brazilian economy toward the long-run equilibrium in which all players invest at the target inflation rate are: (i) the investment made by the entrepreneurs must be demand-creating innovation rather than cost-reducing innovation; (ii) a large fraction of entrepreneurs and workers currently investing is a necessary condition. Keywords Target inflation rate · Perfect-foresight and best-response dynamics · Demand-creating innovation JEL classification: E52, E58, E22, C73, C62.

Douglas S. Silveira Federal University of Juiz de Fora - Rua José Lourenço Kelmer, S/n - Martelos, Juiz de Fora - MG, 36036330. Tel.: +55 32 98865-0148 E-mail: [email protected] Silvinha P. Vasconcelos Federal University of Juiz de Fora - Rua José Lourenço Kelmer, S/n - Martelos, Juiz de Fora - MG, 36036330. E-mail: [email protected] Claudio R.F.Vasconcelos Federal University of Juiz de Fora - Rua José Lourenço Kelmer, S/n - Martelos, Juiz de Fora - MG, 36036330. E-mail: [email protected]

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Douglas S. Silveira et al.

1 Introduction The inflation-targeting (IT) mechanism has played a key role in macroeconomic stabilization in Brazil. In spite of large inflationary shocks, the inflation rate has been maintained at a low level. Aguir et al. (2015) concludes that IT in emerging market economies has been a more challenging task than in developed economies. The conduct of monetary policy has to build credibility and reduce inflation rate levels, and simultaneously deal with a greater vulnerability to shocks. Besides that, de Mendonça (2018) states that for an improvement in monetary policy transmission channels, signaling and transparency is a relevant issue for the central bank in order to anchor inflationary expectations. In fact, one task of the Central Bank of Brazil has been to build credibility as a monetary authority committed to price stability in the context of large inflationary shocks. Mukherjee and Bhattacharya (2011) states that changes in the short-term interest rate are related to the credibility of the monetary authority. There is a considerable set of variables capable of influencing investment decisions and the level of employment: real costs of capital, credit, exchange rate, wealth and expectations of entrepreneurs. As the fluctuations in real interest rates leads to significant changes in the costs of capital, it might reduce investments. In other words, faced with an increase in the interest rate and greater prices volatility, there is a deterioration in the entrepreneurs’ expectations and so they set back the investment decision. According to Fraga et al. (2003), this requires actions consistent with the IT framework combined with high levels of transparency and communication with the public. An important element for the slowdown in the rate of inflation in Brazil, as exposed by de Mendonça and Souza (2009), is the feasibility and credibility of the inflation targeting rate. The outcomes suggest that, despite small interest rate changes, when agents face a more credible environment, there is greater reliance on investing. Therefore, these conditions ensure a robust job creation process. de Mendonça and Lima (2011) presents an econometric analysis to determine the macroeconomic factors responsible for the investments made in Brazil under inflation targeting. A positive relationship between the credibility and investment is the main result. In words, the credibility of IT reduced the uncertainty in the economic environment and has led to an positive and statistically significant effect on private investment. In the face of what has been discussed so far, this paper aims to examine the necessary conditions for Brazil to maintain inflation within its limits, and discuss the transmission mechanisms able to generate incentives in order to achieve a growth path with a high level of investment in the economy. More precisely, we intent to investigate the necessary conditions needed to the Brazilian Central Bank to establish credibility on the inflation target rate. On this matter, there is an extensive literature that evaluate the effects of the inflation targeting system in Brazil. Arestis et al. (2011) examine some features of the Brazilian experience with IT regime by asking the question of whether it makes a difference in the fight against inflation whether a country has adopted IT or not. de Mendonca (2007) analyses the use of the basic interest rate after the adoption of inflation targeting in Brazil and the credibility of this monetary regime through two indexes that consider the Cukierman and Meltzer (1986) definition for credibil-

Investment Decision and Inflation Targeting in the Brazilian Economy

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ity. Céspedes et al. (2014) reviews the recent experience of a group of Latin American (IT) nations, finding that the restrictions on international capital movements and other non-conventional policy tools, especially changes in reserve requirements, have come into common use. In order to reach our results, we aim to develop a dynamic model based on Evolutionary Game Theory (EGT)1 , departing from Smith and Price (1973). Moreover, the selection from multiple equilibrium is done without considering the transition dynamics of the economy. Thus, we propose an EGT and an Agent Based Computational Economics (ACE) approach once it allows an inference about the current state and the transition dynamics of the Brazilian economy’s until the long-run equilibrium is reached. The transition mechanism of the Brazilian economy will be analyzed on the basis of both the best-response dynamics, as proposed by Gilboa and Matsui (1991), and the perfect-foresight dynamics, in line with Matsui and Matsuyama (1995). In words, we can understand the adaptive expectations as a dynamic best-response. On the other hand, the equilibrium under rational expectations corresponds to a perfectforesight dynamics. As far as we know, this paper is pioneer in applying EGT and ACE models in a combined manner to evaluate the Brazilian monetary policy issue. Strictly speaking, this is an approach that allows us to circumvent the problem of multiple long-run equilibria. In terms of possible contributions to the policymaker, it may provide a better understanding of the formation of expectations on the part of the economic agents. In this way, the Central Bank can measure how the inflation targeting can influence the investment decision by entrepreneurs and workers. In this sense, we propose a game in line with Asako et al. (2017), which is composed by two players that are facing a market under monopolistic competition: (1) entrepreneurs and (2) workers. Let each entrepreneur be the owner of a firm and suppose that he hires a worker to produce a good. In sequence, the stage game is dynamically played by entrepreneurs and workers. The available strategies are to (i) invest2 and (ii) not to invest. As in Gilbert (2006), these investments combines demand-creating and costreducing innovations. As a further matter, we consider that there exists complementarity between entrepreneurs’ and workers’ investments. It is in line with the discussion about the complementarity between physical and human capital investments discussed in Blundell et al. (1999) and with the capital-skill complementarity analyzed by Krusell et al. (2000). On the other hand, for simplicity, our study do not take into account the accumulation of physical and human capitals. 1 It is important to inform the reader that game theory models has been largely used in macroeconomics since the 80’s. For example, we can cite Vickers (1986), that proposes a game in order to capture the effects of signalling in a model of monetary policy with incomplete information. Morris and Shin (1998) demonstrates the uniqueness of equilibrium when speculators face a small amount of noise in their signals about the fundamentals and how they deal with the quantity of hot money in circulation and the costs of speculative trading. A global game modeling approach was proposed by Carlsson and Van Damme (1993), Morris and Shin (1998), and Katagiri et al. (2016). In this type of game, an equilibrium is determined solely on the basis of signals about future fundamentals regardless of the current state of the economy. 2 As workers incur costs to acquire general or firm-specific skills or make more effort to increase demand for goods or reduce production costs, it is reasonable to interpret this situations as workers’ investments.

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In accordance with Acemoglu (1997), due to the presence of complementarity, the following outcomes are expected: all players invest or all players do not invest. In these equilibria, it is not guaranteed that the inflation rate converges to its target. Nevertheless, when all players are investing, inflation is expected to converge to its target. For achieving a long-run equilibrium at the target inflation rate, the following conditions are necessary: (i) entrepreneurs are willing to invest in demand-creating innovation rather than cost-reducing innovation. When entrepreneurs invests in costreducing innovation, the economy might suffer from deflation, once cost-reducing innovation leads to fierce competition among firms by decreasing prices and increasing their market shares. Therefore, when the investment is in cost-reducing innovation, in the long term, no player views the inflation target as credible; (ii) according to the analysis with the best-response dynamics, the share of entrepreneurs and workers investing are large enough. If only a few players are willing to invest in the current state of the economy, then the investment counterpart will decrease. As a consequence, the profits from its investment are expected to be lower. In addition, the economy continues in its long-term equilibrium trajectory, given by a scenario that combines lack of investment and absence of inflation. Under these circumstances, the inflation target is not credible and the target inflation rate is not achievable. These two conditions are needed. In this sense, policies to stimulate investments in demand-creating innovation are required to guide the economy successfully towards its long run equilibrium in which all players invest and the inflation rate is at its target. The remainder of this paper is organized as follows. Section 2 presents the game model baseline. Section 3 discusses the outcomes of the stage game. Section 4 evaluates the dynamics of the strategies using the EGT and ACE approaches. Section 5 concludes.

2 The Model In order to better delineate the discussion in the previous section, we will present the bases of the strategic decision on the investment to be made by entrepreneurs and workers, in view of the inflation targeting rate announced by the Central Bank. Thus, we extend the model proposed by Asako et al. (2017) for the Brazilian economy and complement our analysis with the application of an ACE algorithm. In this sense, there is a population of workers interacting strategically with a population of entrepreneurs. It can be represented in a continuum space of mass equal to one. One player from each population are randomly selected to play the stage game. In this sense, the two-period stage game is played iteratively for an infinite number of rounds denoted by τ ∈ [0, ∞). Table 1 shows the amount of investment chosen by the worker (entrepreneur) in human (physical) capital at time t0 , denoted by h (k) ∈ R+ . Firm chooses the level of production in t0 and t1 , which is given by the pair (x0 , x1 ) ∈ R2+ . Both entrepreneur and worker make their decisions on the investment at the same time (t0 ). Taking into account the available current information, the amount produced by the firm, x1 , in


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period t1 may vary. At the instant of time denoted by t1 , the entrepreneur rewards the worker a fraction β ∈ (0, 1) of the total profit earned by the firm after the production of the final good. The total real profit earned by the firm is represented by:

π=

p0 − υ0 1 p1 − υ1 x0 + x1 , γ 1 + r γ1

the pair (p0 , p1 ) ∈ R2++ correspond to the prices of the goods produced by the firm in t = 0, 1. The variable cost of production3 is given by (υ0 , υ1 ) ∈ R2+ . The price levels observed in the economy are (γ0 , γ1 ) ∈ R2++ and the real interest rate is represented by r ∈ R. Stage Game

Decision on investment (t0 )

Profits (t1 )

Entrepreneur

physical capital k

(1 − β )π

Worker

human capital h

βπ

Firm

production and sales x0

production and sales x1

Table 1: The Stage Game. At time t0 Entrepreneur and Worker invest in innovation activities. At time t1 firms total profit, π is paired between worker, which receive the payment (β π ) and entrepreneur, which earns the portion (1 − β )π of the Firm’s total profit π . The term γ1 remains unobservable until the end of t1 . Each player sets γ1 = γ˜1 . To calculate the value of γ˜1 players use the price level observed at t = 0, γ0 and the inflation target rate announced by the central bank. This assumption is in line with Sims (2003), being related to the implications of rational inattention issue. The agents are risk neutral. Thus, the payoffs earned by a representative entrepreneur and by a representative worker are given by (1 − β )π − k and β π − h, respectively. The demand functions for the good in t = 0, 1 are expressed as xt = dt − ε (pt − γt ). Note that if pt = γt , then dt ∈ R++ corresponds to the demand for the good, and ε ∈ R++ is the elasticity of demand with respect to the relative price (pt − γt ). In accordance with what has been exposed so far, equations (1) and (2) represents the inverse demand functions for the pair (p0 , p1 ), respectively:

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1 p0 = γ0 + (d0 − x0 ), ε

(1)

1 p1 = γ1 + (d1 − x1 ). ε

(2)

Other costs than the payments to the workers.

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We can linearly decompose these inverse demand functions into the price-competition effect (γ0 , γ1 ) and the goods-property effect (d0 /ε , d1 /ε ). By (2), the expected price in t = 1 when the stage game is in t = 0 is calculated as follows: 1 E[p1 ] = γ˜1 + (E[d1 ] − x1 ). (3) ε The worker’s and entrepreneur’s investments are assumed to be both compounded of demand-creating innovation and cost-reducing innovation. By doing so, the entrepreneurs face a continuous interval of innovation types ω ∈ [0, 1] to choose from. At the time t = 0, the pair (d0 , υ0 ), that represents the fundamental demand and the variable cost, respectively, are constants. On the other hand, when t = 1, (d1 , υ1 ) are random variables which distributions are conditional to the amount of investments k and h. In other words, we can say that (d1 , υ1 ) follow the probability density function (PDF) f (d1 |k, h) and g(υ1 |k, h), respectively. By it turns, the PDF’s f (d1 |k, h) and g(υ1 |k, h) may be conditional to the expected inflation rate γ˜1 /γ0 , once it may embrace some income effect and inter-temporal substitution effects on d1 and because υ1 may depend on γ˜1 . For simplicity, in the same way that Asako et al. (2017), we do not incorporate these effects explicitly. In addition, by assuming that when the amount of each of the possible forms of investment is greater than the respective threshold value k ∈ R+ or h ∈ R+ , the investment stochastically leads to a decrease in the variable cost and to an increase in the fundamental demand. Thus, the cumulative density function (CDF) satisfies the following constraints: F(d1 |k ∈ [0, k), h ∈ [0, h)) > F(d1 |k ∈ [k, ∞), h ∈ [0, h)), ∀d1 = F(d1 |k ∈ [0, k), h ∈ [h, ∞)) > F(d1 |k ∈ [k, ∞), h ∈ [h, ∞)), ∀d1 G(υ1 |k ∈ [0, k), h ∈ [0, h)) < G(υ1 |k ∈ [k, ∞), h ∈ [0, h)), ∀υ1

= G(υ1 |k ∈ [0, k), h ∈ [h, ∞)) < G(υ1 |k ∈ [k, ∞), h ∈ [h, ∞)), ∀υ1 We denote the mean of each distribution as follows: dI ≡ E[d1 |k ∈ [k, ∞), h ∈ [h, ∞)], dN ≡ E[d1 |k ∈ [0, k), h ∈ [0, h)],

υI ≡ E[υ1 |k ∈ [k, ∞), h ∈ [h, ∞)], υN ≡ E[υ1 |k ∈ [0, k), h ∈ [0, h)] The pair (dI , υI ) corresponds to the expected fundamental demand and the variable cost when both players decide to invest, respectively. Otherwise, when both players decide not to invest, we have the pair (dN , υN ). Following this intuition, it is trivial that when only one player invest more than the threshold, the expected fundamental demand and variable cost are expressed by:

αd dI + (1 − αd )dN = E[d1 |k ∈ [k, ∞), h ∈ [0, h)] = E[d1 |k ∈ [0, k), h ∈ [h, ∞)], αυ υI + (1 − αυ )υN = E[υ1 |k ∈ [k, ∞), h ∈ [0, h)] = E[υ1 |k ∈ [0, k), h ∈ [h, ∞)],


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in which the values assigned to αd and αυ must satisfy the constraints: αd dI + (1 − αd )dN > dN for dI > d0 ≥ dN ; and υN > αυ υI + (1 − αυ )υN > υI for υ0 ≥ υN > υI . For convenience, we assume that αd = αυ = α . By assigning a low value to α , a greater degree of complementarity between the investment made by each of the players is obtained. In this sense, we consider that α ∈ (0, 1/2). In the situation where only one of the players invests, the fundamental demand reacts with a rather timid growth, while the variable cost decreases marginally. It is only possible to observe substantial changes in these variables when both players invest. Our analysis consider that dI and υI are exogenous4 . In the next section, we will briefly present the equilibrium conditions and some parameters that involve the strategic interaction between the entrepreneur and the worker. The effects of each type of innovation on the inflation rate are discussed.

3 The Equilibrium Conditions of the Game In order to find the players’ best response, we will analyze separately the optimization problem of the entrepreneurs and the workers. The problems of each entrepreneur and each worker are consistent with maximization problem of indirect utility functions derived from log utility5 .

3.1 Entrepreneur’s Best Response A representative entrepreneur faces the following maximization problem: max (1 − β )

x0 ,x1 ,k

h p −υ i p −υ 1 1 0 1 0 E x1 − k. x0 + γ0 1+r γ˜1

As the optimal values attributed to x0 , x1 , and k can be defined independently, it is possible to separate this problem into three parts. First, the value of x0∗ (optimal) is determined so that x0∗ ≡ arg maxx0 p0γ−0υ0 x0 , which implies in equation (4): 1 1 υ0 arg max 1 + x0 = [d0 + ε (γ0 − υ0 )]. (d0 − x0 ) − x0 εγ0 γ0 2

(4)

Substituting (4) into (1) we have: p∗0 =

1 d0 + γ0 + υ 0 . 2 ε

4 Asako et al. (2017) deeply argue this issue as well as each entrepreneur’s innovation selection and the equilibrium outcomes of the stage game. For the reader interested in the derivations of the maximization problems of workers and entrepreneurs, please refer to the appendix section of the aforementioned paper. 5 A detailed discussion is presented in Asako et al. (2017).

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Assuming an equilibria in which all players from each population adopt the same strategy, we have that p∗0 = γ0 must hold. Therefore, p∗0 = γ0 = (d0 + ευ0 )/ε . Knowing that γ˜1 is exogenously determined and (d1 , γ˜1 ); (υ1 , γ˜1 ) are independent, the optimal amount (x1∗ ) produced in t1 is given by: arg max E x1

h p −υ i 1 1 x1 . γ˜1

Which yields equation (refeq.5) 1 E[υ1 ] 1 x1 = [E[d1 ] + ε (γ˜1 − E[υ1 ])]. (E[d1 ] − x1 ) − arg max 1 + x1 ε γ˜1 γ˜1 2

(5)

By inserting the result of (5) into (2), we have: E[p∗1 ] =

1 E[d1 ] + γ˜1 + E[υ1 ] . 2 ε

Notice that although k assumes continuous values, its optimal choice is binary, i.e., k ∈ (0, k). When k = 0 it strictly dominates any given k ∈ (0, k), as the level of investment does not affect d1 and the entrepreneur has to pay k. Likewise, k ∈ k, ∞ is strictly dominated by k = k. Thereby, considering the binary choice k ∈ (0, k) is sufficient. Analogously, each worker faces the choice h ∈ (0, h). Let π1 be the real profit of the firm at time t = 1, we have: E[π1 |k, h] ≡

h p −υ i 1 1 1 1 E x1 = (ε γ˜1 + E[d1 |k, h] − ε E[υ1 |k, h])2 1+r γ˜1 4γ˜1 (1 + r)ε

To facilitate the economic understanding, we will define A and B as below: A ≡ E[π1 |k = k, h ∈ [0, h)] − E[π1 |k = 0, h ∈ [0, h)] =

α [(dI − ευI ) − (dN − ευN )] [2ε γ˜1 + α (dI − ευI ) + (2 − α )(dN − ευN )], 4γ˜1 (1 + r)ε

B ≡ E[π1 |k = k, h ∈ [h, ∞)] − E[π1 |k = 0, h ∈ [h, ∞)] =

(1 − α )[(dI − ευI ) − (dN − ευN )] [2ε γ˜1 + (1 + α )(dI − ευI )+ 4γ˜1 (1 + r)ε + (1 − α )(dN − ευN )].

The expression A provides the increase of firm’s expected profit as an effect of the entrepreneur’s investment when h ∈ [0, h) - which is positive if (1 − β )A > k and negative if (1 − β )A < k. The value of B represents that increase when h ∈ [h, ∞) -


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which is positive if(1 − β )B > k and negative if (1 − β )B < k. Assume α ∈ (0, 1/2), we have B > A by the following expression: B>A ⇔

1−α α

>

2ε γ˜1 + α [(dI − ευI ) − (dN − ευN )] + 2(dN − ευN ) 2ε γ˜1 + α [(dI − ευI ) − (dN − ευN )] + [(dI − ευI ) + (dN − ευN )]

as the left side of the inequality is greater than one and the right side is less than one. For (1 − β )A < k < (1 − β )B, the entrepreneur will choose k = k if the worker chooses h ∈ [h, ∞). Otherwise, the entrepreneur sets k = 0. There is still the possibility for entrepreneurs to adopt a mixed strategy in which k = k with probability σe ∈ [0, 1] and k = 0 otherwise. Similarly, the worker may choose a mixed strategy in which h = h with probability σw ∈ [0, 1] and h = 0 otherwise. The entrepreneur’s expected payoff in mixed strategies is given by: (1 − β )(σw E[π1 |k = k, h = h] + (1 − σw )E[π1 |k = k, h = 0]) − k,

(6)

and the expected payoff in mixed strategies when there is no investment is (1 − β )(σw E[π1 |k = 0, h = h] + (1 − σw )E[π1 |k = 0, h = 0]).

(7)

The entrepreneur invest if and only if (6) > (7), i.e., (1 − β )[(1 − σw )A + σw B] > k. By this relation, we can find an optimal value for σw , so that:

σw∗ ≡

k/(1 − β ) − A B−A

Thus, the entrepreneur invests if and only if σw > σw∗ . Note that σw∗ ∈ (0, 1), once (1 − β )A < k < (1 − β )B. On this occasion, σe (σw ), which represents the best-response correspondence, is given by  1 if σw > σw∗  σe (σw ) = ∈ [0, 1] if σw = σw∗  0 if σw < σw∗ .

Therefore, the the best response in terms of k∗ for the entrepreneur is:  k if k < (1 − β )A  k∗ = σe (σw ) if (1 − β )B > k > (1 − β )A  0 if k > (1 − β )B. 3.2 Worker’s Best Response A representative worker faces a problem given by max β h

h p −υ i p −υ 1 1 0 1 0 E x1 − h. x0 + γ0 1+r γ˜1

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When t = 0 the term β p0γ−0υ0 x0 could be ignored since it is not affected by h. For a given amount of investment made by the workers, if k ∈ [0, k), then the expected increase in firms’ profit is denoted by: E[π1 |k ∈ [0, k), h = h] − E[π1 |k ∈ [0, k), h = 0] = A; when k ∈ [k, ∞), we have E[π1 |k ∈ [k, ∞), h = h] − E[π1 |k ∈ [k, ∞), h = 0] = B. When k ∈ [0, k) and if β A > h the increase in the worker’s expected payoff is positive. On the other hand, if β A < h, it is negative. Following this intuition, if k ∈ [k, ∞), the increase is positive if β B > h and is negative if β B < h. If we consider that β A < h < β B, the worker’s best response is conditional to the entrepreneur’s mixed strategy. Let σe∗ be defined as

σe∗ ≡

h/β − A , B−A

and given that β A < h < β B it is trivial to see that σe∗ ∈ (0, 1). In this sense, we have the following best-response correspondence σw (σe ) for the representative worker:  1 if σe > σe∗  σw (σe ) = ∈ [0, 1] if σe = σe∗  0 if σe < σe∗ . Therefore, the best response in terms of h∗ for the worker is:  h if h < β A  h∗ = σw (σe ) if β B > k > β A  0 if h > β B.

3.3 The Nash Equilibria

Nash Equilibria h < βA A < h/β < β k < (1 − β )A A < k/(1 − β ) < β k > (1 − β )B

(k, h) (k, h) (0, h)

(k, h) (k, h), (0, 0), (σe∗ , σw∗ ) (0, 0)

h > βB (k, 0) (0, 0) (0, 0)

Table 2: The Nash Equilibria outcomes Making the conjecture on the best response of each of the players we summarize in table 2 the Nash equilibria of the game6 . When workers and entrepreneurs decide 6 The proof of each equilibrium is beyond the scope of this article, but can be found in Asako et al. (2017).


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to invest (A and B) and the marginal increases in the expected payoffs are sufficiently high, the best response for both players is to invest. On the other hand, if the value of both investments are low, players do not invest. If A < k/(1 − β ) < B and A < h/β < B, then a multiple equilibria solution arises and the complementarity between both player’s investments is critical to determine the outcome of the game.

3.4 The Effects of Innovation on the Inflation Rate at the Equilibrium To evaluate the observed effect of each type of innovation on the inflation rate we must recall that (d1 , υ1 ) = (dI , υI ) is the equilibrium outcome for the pair (k∗ , h∗ ) = (k, h) and that (d1 , υ1 ) = (dN , υN ) is the equilibrium outcome for the pair (k∗ , h∗ ) = ˜ (0, 0). So, (γ1 +dI2/ε +υI ) is the expected price level in the first equilibrium. In the sec˜ ond equilibrium, we have that (γ1 +dN2/ε +υN ) . For the case in which workers and entrepreneurs invest in the economy, the expected price level should be equivalent to that in the situation where the expectations are rational: dI + ευI . ε

γ˜1 =

The expected price level in the economy for the case in which players do not invest is:

γ˜1 =

dN + ευN . ε

Taking into account the possibility of players to choose different strategies, γ˜1 should be defined in the range dN + ευN dI + ευI dN + ευN dI + ευI , max . γ˜1 ∈ min , , ε ε ε ε In this sense, the expected inflation rate between t = 0 and t = 1, (γ˜1 /γ0 ) − 1, corresponds to γ˜1 dN + ευN dI + ευI dN + ευN dI + ευI − 1, max −1 . − 1 ∈ min , , γ0 d0 + ευ0 d0 + ευ0 d0 + ευ0 d0 + ευ0 Which leads us to the following remark: Remark 1 If and only if dI − d0 > ε (υ0 − υI ), the equilibria with a positive inflation and with a zero or negative inflation rate exist. Inflation is observed in the equilibrium in which both players invest, given by (k∗ , h∗ ) = (k, h). The zero inflation rate occurs when (k∗ , h∗ ) = (0, 0),i.e., when both players do not invest. When dI − d0 ≤ ε (υ0 − υI ), there is no inflation in either of the equilibria. If and only if dI − d0 > ε (υ0 − υI ) a positive inflation rate is sustainable as an equilibrium.

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In order to better understand how the types of innovation and the target inflation rate determine equilibrium conditions, let us suppose the following cases: −d0 Case I: If firms choose demand-creating innovation, then υd01 − υ1 > ε holds. At the ∗ ∗ equilibrium with k = k and h = h a positive inflation rate is observed. Thus, the target inflation rate is assumed to be credible; 0 Case II: If firms choose cost-reducing innovation, then υd10 −d −υ1 < ε holds. Therefore,we do not observe inflation in both equilibria. Thus, no player views the target inflation rate as credible. In other words, when entrepreneurs invest in demand-creating innovation in place of cost-reducing innovation, the high value of dI guarantees the inequality dI − d0 > ε (υ0 − υI ) and the announced target inflation rate is said to be credible. Otherwise, when entrepreneurs choose the cost-reduction innovation, the low value of υI guarantees that dI − d0 < ε (υ0 − υI ) and no players views the target inflation rate as credible. This is because in this case the upper bound of γ˜1 is equal to zero. The condition dN + ευN ≤ d0 + ευ0 always holds from the assumptions dN ≤ d0 and υN ≤ υ0 . In such case, the policymaker must stimulate investments in demand-creating innovation, such as structural reforms, tax incentives and subsides.

3.5 Decision on Innovation Type Let us suppose that entrepreneurs have decision-making power over the innovation type to choose. Thus, for convenience, we assume that w reflects a binary choice, so that w ∈ 0, 1 . If w = 1, then the entrepreneur invest in demand-creating innovation. Otherwise, the entrepreneur invest in cost-reducing innovation. For each investment d ∗ represent the expected fundamental demand. υ ∗ corresponds to the variable cost. If dI = d ∗ > dN and υI = υN for w∗ = 1 as well as if dI = dN and υI = υ ∗ < υN for w∗ = 0, then d ∗ and υ ∗ may correspond to dI and υI in the baseline model. The optimal value for w is obtained from the firm’s expected profit in t = 1, given by E[π1 |k, h]. It is demonstrated as follows.

Corollary 1 Given the expected demand-creation effect (d ∗ − dN ) and the expected cost-reduction effect (υN − υ ∗ ), the entrepreneurs’ decision with regard to w∗ ∈ 0, 1 is given by 1 if d ∗ − dN > ε (υN − υ ∗ ) w∗ = 0 if d ∗ − dN < ε (υN − υ ∗ ). Proof . When w∗ = 1, we have: 1 ∗ 2 ∗ ˜ E[π1 |w∗ = 1] = 4γ˜ (1+r) ε [ε γ1 + σw (d − ευN ) + (1 − σw )(α d + (1 − α )dN − ευN )] 1

When E[π1

w∗

|w∗

= 0, we have:

1 2 ∗ ∗ ˜ = 1] = 4γ˜ (1+r) ε [ε γ1 + σw (dN − ευ )+(1− σw )(dN − αευ −(1− α )ευN )]

Thus, E[π1

1

|w∗ = 1] > E[π

∗ ∗ ∗ 1 |w = 0] holds if and only if [(d −dN )− ε (υN − υ )](σw +

(1 − σw )α ) > 0. Under the assumption that α > 0, if (d ∗ − dN ) > ε (υN − υ ∗ ) holds, we finally have that E[π1 |w∗ = 1] > E[π1 |w∗ = 0].


13

When the elasticity of demand (ε ) increases, firms are more prone to cost-reducing innovations. Otherwise, firms decide to invest in demand-creating innovation. This happens because if the demand for a good is elastic, a reduction in price causes revenue to increase - which provides an incentive for the entrepreneurs to makes cost reduction. When demand is inelastic, entrepreneurs may increase prices and observe an increase in demand for their products through demand-creating innovation, as presented by Kamien and Schwartz (1970) and Spence (1975). When ε is low7 , remark 1 is satisfied and entrepreneurs have an incentive to choose demand-creating innovation. Otherwise, policies to raise the value of d ∗ could hold the condition given by d ∗ − dN > ε (υN − υ ∗ ).

3.6 The Output Growth Rate To measure the performance of the economy we will use the output growth rate g≡

x1 εγ1 + d1 − ευ1 −1 = x0 εγ0 + d0 − ευ0

(8)

Considering of h a sufficientlyilarge number entrepreneurs and workers, the range of γ1 ≡ γ˜1 ∈ dN +εευN , dI +εευI , γ0 = d0 +εευ0 and d1 − ευ1 ∈ [dN − ευN , dI − ευI ]. Replacing this values in (8) g∈

hd

N

d0

− 1,

i dI −1 . d0

Thus, we can see that the performance of the economy in the equilibrium (k∗ , h∗ ) = (k, h) is equal to [(dI /d0 ) − 1], which is greater than the equilibrium (k∗ , h∗ ) = (0, 0) given by [(dN /d0 ) − 1]. So, we have the following Remark: Remark 2 When both players invest in the equilibrium, i.e., (k∗ , h∗ ) = (k, h), g is greater than the output growth rate in the equilibrium where neither player invests, i.e., (k∗ , h∗ ) = (0, 0). Let the sum of workers’ and entrepreneurs’ surplus be equal to the social surplus, Π ≡ π1 (k, h) − k − h. In the equilibrium (k∗ , h∗ ) = (k, h) the social surplus is higher than that in which (k∗ , h∗ ) = (0, 0). This condition holds unless the total investment cost (k + h) overcomes the potential increase in demand (dI − dN ) or the potential decrease in variable cost (υN − υI ). Adequately, to the extent that investments are made for the purpose of stimulating the demand-creating innovation, the equilibrium (k∗ , h∗ ) = (k, h) improves the economic environment. Hereupon, the policymakers should guarantee the conditions in order to guide the economy toward this equilibrium. 7

A sufficiently low vale of ε is a necessary condition for positive inflation in a equilibrium

14


4 Evolutionary Game and Agent Based Computaional Economics (ACE) According to Smith and Price (1973), evolutionary games imply a convergence to the dominant long-run equilibrium. In this equilibrium, achieved after a period of dynamic interaction, players must have adopted an evolutionary stable strategy (ESS) that is a strategy where players have no incentive to abandon, unless some external force disturbs the underlying conditions of the game. Then, if classical game theory can be defined as the science that studies strategic behavior, with the theory of evolutionary games it takes a step forward, since we now have the science that studies the robustness of strategic behavior. In order to achieve this robustness, in subsection 4.1, we present the Replicator Dynamics (RD) analysis, obtained from a non-linear differential equation system, and complement the study with an Agent Based Computational Economics (ACE) model, presented in subsection 4.2, which implements a stochastic component in the evolutionary dynamics of the game. 4.1 The Replicator Dynamics In an evolutionary game it is assumed bounded rationality, a large population, n, of players (n → ∞) and an implicit recognition that agents learn. Every period, a player is randomly matched with another player and they play a two-player game. Each agent is assigned a strategy and they cannot choose it. In other words, they are “programmed” to play a strategy in the initial period (t = 0) and it may not maximize their utility function. However, the systematic interaction with other agents will lead them to modify or update their behavior over time by choosing a given strategy. Thus, one player can imitate other players’ strategies. Following this intuition, we evaluate the specific case whereupon (1 − β ) < k < (1 − β )B and β A < h < β B hold. As there is uniqueness of the Nash equilibrium in other cases, it converges to the Evolutionary Stable Strategy (ESS), as appointed in Friedman (1991). So, in this section the evolutionary dynamics of players’ strategies between investment and non-investment is evaluated by using the perfect-foresight dynamics in the same approach used by Gilboa and Matsui (1991) and Matsui and Matsuyama (1995). By doing so, players are assumed to be forward-looking. The best-response dynamics approach is also applied. Unlike the previous dynamics, players are assumed to be backward-looking and foresee that the same situation prevails in the future. Vega-Redondo (1997), Kaneda (2003) and Oyama (2009) combined the evolutionary game theory approach and the perfect-foresight dynamics in order to justify the uniquely absorbing and globally accessible equilibrium. Here, we focus on the best-response dynamics because, according to Asako et al. (2017) there is a lack of understanding on how an economy, given its initial condition8 will evolve to the longterm equilibrium. Our approach follow the literature that compares results under rational and adaptive expectations. The first one has been largely used, although it is sustained by the strong assumption that economic agents uses all the set of information available in 8

The initial state corresponds to the period of the stage game right after the implementation of a policy.


15

order to build their expectations. We highlight and follow the intuition of Woodford (2002), Mankiw and Reis (2002) and Sims (2003), which evaluate the macroeconomic environment by considering the fact that agents may deviate from rational expectations. On this matter, knowing that the target inflation rate could not be achievable in the long term, there is a clear difference between the perfect-foresight and best-response dynamics behaviour for guiding the economic agents into the equilibrium. When considering the perfect-foresight approach, the long term equilibrium is never affected by the current state of the economy. On the other hand, the current state of the economy as well as the equilibrium conditions are affected when considering the best-response dynamics. In order to derive the ESS, we first determine πH , πM and πL as follows: 1 (ε γ˜1 + dI − ευI )2 , 4γ˜1 (1 + r)ε 1 πM = [ε γ˜1 + α (dI − ευI ) + (1 − α )(dN − ευN )]2 , 4γ˜1 (1 + r)ε 1 πL = (ε γ˜1 + dN − ευN )2 , 4γ˜1 (1 + r)ε

πH =

πH corresponds to the expected total profit when workers and entrepreneurs decide to invest. If only one player decide to invest, the expected total profit is given by πM and if no player invests they receive πL . The payoff matrix of the stage game is presented in (9). NI(h = 0) I(h = h) I(k = k) (1 − β )πH − k; β πH − h (1 − β )πM − k; β πM (9) (1 − β )πL ; β πL NI(k = 0) (1 − β )πM ; β πM − h For simplicity, the payoff matrix presented in (9) is normalized to obtain (10) and we introduce the following parameters φ ≡ (πH − πM ) and θ ≡ (πL − πM ). This procedure will not affect the best response structure of the game and will simplify the analysis. Following the evolutionary algorithm, players are randomly matched and compete against each other’s and a one-shot game is played. The level of aggregate strategies of the populations do not change all at once. In fact, they continuously update their strategic behavior over time.

I(k = k) NI(k = 0)

I(h = h) NI(h = 0) (1 − β )φ − k; β φ − h 0; 0 0; 0 (1 − β )θ + k; β θ + h

(10)

Let sw ∈ [0, 1] be the share of workers using strategy I, so the share of workers adopting NI is 1 − sw . Analogously, for the population of entrepreneurs, the share of players using strategies I and NI are se ∈ [0, 1] and 1 − se , respectively. The observed variations in the proportion of players adopting each of the strategies reflects their evolutionary process within each population. Note that this relative frequency can be understood as the probability that a player will play a given strategy. Both the evolution of the game and the strategic behavior of the firms is conditioned to the fitness of their strategies. The fitness, according to Binmore and Samuelson (1992)

16


and Samuelson (2002), depends on the player’s payoff of a given strategy and on the relative frequency of the strategies observed in both populations. That is, players make decisions based on the expected utility of their payoffs. From the normalized payoff matrix (10) and considering (1 − β ) < k < (1 − β )B and β A< h < β B, there are simultaneously three Nash equilibria for the game: Θ NE = (k, h); (0, 0); (σe∗ , σw∗ ) . Using the values of πH , πM and πL the mixed-strategy equilibrium is given by ! − ) − ) h/ β − ( π π h/ β − ( π π M L M L (σe∗ , σw∗ ) = , . πH + πL − 2πM πH + πL − 2πM

Θ NE represents the set of Nash equilibria of the game. The following questions arise: departing from an initial game condition, which equilibrium will be reached? What, in fact, is the game played? Without loss of generality, to answer these questions, we first analyze the evolution and the robustness of players’ strategic behavior with the analytic solution of the RD system proposed by Taylor and Jonker (1978), which is a very general ordinary differential equation (ODE) system in evolutionary game theory. As presented in Hirth (2014), in a dynamic system, the growth rate s˙e /se equals the strategy I’s fitness e1 A(sw , 1 − sw )T less the average fitness (se , 1 − se )A(sw , 1 − sw )T of population of entrepreneurs, where s˙e = dse /dt and e1 = (1, 0) represents that  population choose the pure strategy  all entrepreneurs from the (1 − β )φ − k 0  be the payoff matrix of a representaI(k = k). Let A =  0 (1 − β )θ + k tive entrepreneur. After some trivial matrix algebra, the replicator dynamic equation for the population of entrepreneurs is s˙e = se ((e1 − (se , 1 − se ))A(sw , 1 − sw )T . Substituting the values of A (payoffs earned by the representative entrepreneur), in order to derive the dynamic replicator system for the population of entrepreneurs e and workers w: n h i h io s˙e = se (1 − se ) sw (1 − β )φ − k − (1 − sw ) (1 − β )θ + k (11) i h io n h s˙w = sw (1 − sw ) se β φ − h − (1 − se ) β θ + h

(12)

s˙e and s˙w represent the growth rate of the proportion of entrepreneurs and workers that adopt the first pure strategy I within each population. The stability of the system is a coordinate (se , sw ) ∈ [0, 1] × [0, 1], in which s˙e = s˙w = 0 is a necessary condition for the stationarity of (11) and (12). To check the stability of the points candidates for an ESS, i.e., an asymptotically stable steady state for the two-population game, we must the eigenvalues: Ω (se , sw ) =   use the Jacobian matrix (Ω ). Calculating d s˙e − λ d s˙e d s˙e d s˙e dse dsw  dse dsw  = 0. We finally  d s˙w d s˙w ; doing the determinant det(Ω − λ j ) = d s˙w d s˙w λ − dse dsw dse dsw √ have that λ1,2 = trΩ ± trΩ 2 − 4det Ω . For the stationary point to be asymptotically stable, the eigenvalues λ1,2 of the matrix (Ω ) evaluated at points that hold the condition s˙e = 0 and s˙w = 0 must have negative real parts. The analytical solution and the


17

phase diagrams, responsible for the evolutionary game dynamics, were made with the open source software Dynamo9 . To generate the outcomes summarized in Figure 1, two scenarios were created according to the limits of the Brazilian target inflation rate and its lower and upper bounds10 . The evolutionary dynamics for the case where the rate of inflation converges to its lower bound, is presented in Figure (1a). In this case, we consider that γ˜1 = [(dN + ευN )/ε ], i.e., no players are investing in t = 0. It implies that the economy is more likely to converge to the steady state in which players do not invest in the long term when the lower bound of the inflation rate is expected. Note that the system has five stationary points, but only two of them are evolutionary stable: (I, I) and (NI, NI). The coordinations (I, NI) and (NI, I) are unstable points and the interior solution (σe∗ , σw∗ ) is a saddle point. So, the equilibrium set, Θ NE , can be refined ESS by using the ESS concept and is given by: Θ = (I, I); (NI, NI) .

On the other hand, in Figure 1b we consider that γ˜1 = [(dI + ευI )/ε ], i.e., both players are investing in t = 0. In this case, the economy is more likely to converge to the steady state in which both players invest when the upper bound of the inflation rate is expected. As observed in the previous scenario, the set of the ESS is given by: Θ ESS = (I, I); (NI, NI) . The colors indicate the speed of the evolution along convergence paths. Red (blue) represent the highest (lowest) speed. As both phase diagrams show, the speed decreases when the current shares are closer to the equilibrium shares.

(a) Scenario I: Lower bound (2.75%)

(b) Scenario II: Upper bound (5.75%)

Fig. 1: Phase Diagrams. To generate the dynamics above, we use the following values: ε = 1, d0 = 100, dI = 400, dN = 100, υ0 = 9900, υI = 9800, υN = 9900, α = 0.45, β = 0.49, i = 0.07, k = 2.5, h = 2.5.The range of the expected inflation rate is [(γ˜1 /γ0 )] − 1 ∈ [0.0275, 0.0575]. 9

See Sandholm et al. (2012) http://www.ssc.wisc.edu/~whs/dynamo. The Central Bank of Brazil works with a interval given by [(γ˜1 /γ0 ) − 1] ∈ [0.0275, 0.0575]. Source: http://www.bcb.gov.br/Pec/relinf/Normativos.asp 10

18


On this matter, in fig.1, it is possible to infer that a coordination game is formed. In the long run, a rise in expected inflation may lead to an increase in the nominal interest rate. However, our results remain unchanged as long as it is assumed that the nominal interest rate is fixed for a while. Once the Brazilian economy embarks on the path toward the equilibrium with a positive inflation rate, it moves closer to the equilibrium over time. Therefore, even if the nominal interest rate rises and entrepreneurs and workers expected payoffs to decline in the long run, the economy should still evolve to the equilibrium with a positive inflation rate, because players expect the probability of successful coordination between entrepreneurs and workers to be high. Our result, in some way, differs from that found by Asako et al. (2017), where there is an evolutionary equilibrium for an economy with a zero or negative inflation rate.

4.2 The ACE Algorithm: Modelling the Economic System A stochastic component is implemented11 in the analysis of evolutionary equilibrium using the ACE method. In general, Agent Based Modeling has been largely used in the understanding of the evolution of cooperative behavior in social dilemmas, as can be seen in Chan et al. (2013) and Da Silva Rocha (2017). In this section, an algorithm to complement the evolutionary game and to guide the dynamic interaction among players is presented. We consider the competition in which the two populations are distributed in a well-mixed network. A well-mixed arrangement may be consistent with our framework, once we assume that all players have access to the same level of information and there is no local constraint capable of generating some exogenous noise in the communication between policymaker and entrepreneurs or workers. To implement the computational simulation, the intuition presented by Da Silva Rocha (2017) is followed. In this sense, at a time t = 0, we establish an initial proportion of entrepreneurs and workers, (se , sw ), that plays the (I) strategy. Evolutionary dynamics are introduced in sequence and at each Monte Carlo time Step (MCS), a Focal Agent i, which can update12 its strategy, is chosen randomly. This occurs simultaneously in both populations. Focal Agent i, in turn, plays against a random opponent from the rival population and starts the game presented in (10). Thus, Focal Agent i will obtain a payoff Vi . At the same time, an agent j is randomly chosen as a Reference Agent, which randomly plays against an opponent from the rival population and obtains a payoff V j . Notice that focal and reference players belong to the same population. The Focal Agent i compares Vi and V j to analyze the possibility of updating his strategy in two stages as stated ahead: (i) If Vi ≥ V j , focal player keeps his strategy; (ii) If Vi < V j , focal player might update his strategy to the one adopted by reference player with a probability given by the variable η :

η= 11

V j −Vi max.payo f f − min.payo f f

(13)

To implement the algorithm, we made use of the Java programming language. The update mechanisms used in ACE models are synchronous and asynchronous. Here, we use the second, since it allows the overlapping generations interactions. See Hauert (2002) and Chan et al. (2013). 12


19

The maximum and minimum payoffs are obtained from the game matrix in 10. By doing this procedure, we guarantee that η ∈ (0, 1). To establish a decision criterion whether Focal Agent i updates his strategy or not, we use a random number generator named rnd ∈ (0, 1). In this way, a stochastic component on the dynamics of the game is implemented. Focal Agent i compares rnd with the probability η , so that: (iii) If η ≥ rnd, Focal Agent i updates its strategy and imitates the Reference Agent j; (iv) If η ≤ rnd, Focal Agent i does not update its strategy. Population of Entrepreneurs

Population of Workers

1

1

0.9

0.9

0.8

0.8 0.7

Invest Not Invest

0.6

Frequency

Frequency

0.7

0.5 0.4

0.5 0.4

0.3

0.3

0.2

0.2

0.1 0

Invest Not Invest

0.6

0.1

0

5

10

15

20 25 30 Monte Carlo Step

35

40

45

0

50

0

5

10

15


35

40

45

50

Fig. 2: ACE for Scenario I: lower bound (2.75%). The initial share of entrepreneurs and workers playing the strategy I is given by(se , sw ) = (0.84, 0.84).

At every MCS, randomly selected individuals from both populations have the opportunity to, change strategy at least once,on average, comparing their payoffs with the Reference Agent j. We say that, on average, individuals can update strategies once, because within a MCS the same player may be invited to play many times, and other players may not, since the process of players’ selection is random. Thus, when all players in both populations, have the opportunity to update their strategies,on average, a MCS is completed and a new MCS starts in order to repeat the dynamics of the game. This procedure characterizes the ACE model.

Population of Entrepreneurs 1 0.9

0.8

0.8

0.7

0.7

0.6

Frequency

Frequency

Population of Workers 1 0.9

Invest Not Invest

0.5 0.4

0.6

0.4

0.3

0.3

0.2

0.2

0.1 0

Invest Not Invest

0.5

0.1

0

5

10

15

20

25

30

Monte Carlo Step

35

40

45

50

0

0

5

10

15


35

40

45

50

Fig. 3: ACE for Scenario I: lower bound (2.75%). The initial share of entrepreneurs and workers playing the strategy I is given by(se , sw ) = (0.87, 0.87).

20


In order to present the outcomes obtained by the ACE algorithm, we consider the same inputs used to generate the phase diagrams in Figure 1. Thus, we can show through Figures (2) and (3) that in scenario I, in order for the equilibrium to be reached with both players investing, it is necessary that at time t = 0, the share of entrepreneurs and workers willing to invest to be greater than or equal to (se , sw ) = (0.87, 0.87). Otherwise, the equilibrium obtained is with both players not investing. Population of Entrepreneurs


1

1

0.9

0.9

0.8

0.8 0.7 Invest Not Invest

0.6

Frequency

Frequency

0.7

0.5 0.4

0.5 0.4

0.3

0.3

0.2

0.2

0.1 0

Invest Not Invest

0.6

0.1

0

5

10

15


35

40

45

0

50

0

5

10

15


35

40

45

50

Fig. 4: ACE for Scenario II: upper bound (5.75%). The initial share of entrepreneurs and workers playing the strategy I is given by(se , sw ) = (0.24, 0.24).

Population of Entrepreneurs


1

1

0.9

0.9

0.8

0.8 0.7 Invest Not Invest

0.6

Frequency

Frequency

0.7

0.5 0.4

0.5 0.4

0.3

0.3

0.2

0.2

0.1 0

Invest Not Invest

0.6

0.1

0

5

10

15


35

40

45

50

0

0

5

10

15


35

40

45

50

Fig. 5: ACE for Scenario II: upper bound (5.75%). The initial share of entrepreneurs and workers playing the strategy I is given by(se , sw ) = (0.23, 0.23). By Figures (4) and (5) we can observe that in scenario II, in order for the equilibrium to be with both players investing, it is necessary that at time t = 0, the share of entrepreneurs and workers willing to invest is greater than or equal to (se , sw ) = (0.24, 0.24). Otherwise, if this share is less or equal to (se , sw ) = (0.23, 0.23), the equilibrium reached is with both players not investing. From what we presented here, it is possible to visualize that for a level of inflation that corresponds to the upper bound of the inflation target, a smaller share of players that are willing to invest is necessary to guide the economy to the steady state. In addition, we can see that the


21

results obtained by the ACE give more robustness to those obtained by the RD. In the next subsection, we will present some theoretical derivations to support the results of the simulation exercise.

4.3 The Policy Channels: Effects of the Target Inflation Rate To examine the signaling and transmission mechanisms of monetary policy, we evaluate the effects of the target inflation rate. To explore the policy channels for reducing the nominal interest rate, i, and for announcing the target inflation rate,γ˜1 /γ0 , we decompose the real interest rate of the Brazilian economy into its nominal interest13 and the expected inflation rates: 1+r =

1+i γ˜1 /γ0

In this way, by considering the medium term, i is the lowest rate possible to maintain inflation between the limits of the target announced by the Central Bank. The expectation of rising prices by the entrepreneurs leads to an increase in the level of equilibrium prices. This drives to an increase in expected profit. The comparative statics of γ˜1 are presented in Corollary 2. Corollary 2 (a) A and B increase with γ˜1 . (b) The share of players investing in the economy, sI , increases with γ˜1 . Proof .(a) can be easily deduced by the definition of A and B given at the beginning of section 3. (b) πH >πM >πL and ∂∂πγ˜H > ∂∂πγ˜M > ∂∂πγ˜L > 0 hold. In addition, α ∈ (0, 1/2) 1 1 1 and πH , πM and πL are convex. Thus, the following inequalities hold:

∂ πH ∂ πM ∂ πM ∂ πL − > − >0 ∂ γ˜1 ∂ γ˜1 ∂ γ˜1 ∂ γ˜1 Thus,

∂ σe∗ ∂ γ˜1

< 0 and

∂ σw∗ ∂ γ˜1

< 0 are obtained, and sI increases with γ˜1 .

From this corollary, we observe that the increase in γ˜1 makes the evolutionary dynamics of equilibrium more likely to be the one in which both entrepreneurs and workers invest. Put another way, through the increase in (A + B) under the perfectforesight dynamics and through the decrease in σe∗ and σw∗ under the best-response dynamics, the economy operates with an inflation rate closer to the upper bound of the target announced. 13 For the numerical simulations, we use the values provided by the Brazilian Central Bank: https://www.bcb.gov.br/Pec/Copom/Port/taxaSelic.asp.

22


4.4 The Willingness to Invest Here, we try to evaluate the possible incentives that can be given to the players, so that they invest more in the economy, although the signaling of the Central Bank is to seek an inflation rate closer to its lower bound. To achieve our purposes, we depart from the following question: do players always expect the Brazilian Central Bank to meet the inflation target? In our analysis, the agents are rational, i.e., for the sake of credibility, they expect that the price level to be reached in t + 1, γ˜1 , will be feasible if and only if its value belongs to the interval established as credible. On the other hand, if players perceive uncertainty about γ˜1 , the policymaker announcement may have a minor effect. Through the presented results, the effect of an increase in γ˜1 keeps the payoff structure of the game more favorable to those players who are willing to invest. However, there is a limit for γ˜1 to have an effect on rational agents. Therefore, policy is not always efficient in stimulating investment in the economy. in addition, the current state of the economy is of uppermost importance in analyzing policy effectiveness, especially when considering that players are backward-looking. As the game is dynamic, and there is a rule of learning and updating beliefs, the balance is said to be "path dependent". In words, the fraction of players who invested in the most recent rounds exert great influence on the equilibrium outcome. In contrast, the current state does not matter if players are forward-looking. Consider that for a given initial condition, the relative frequency of entrepreneurs and workers willing to invest (se , sw ) are already near the lower bound inflation rate equilibrium. To increase the level of investment by agents, γ˜1 should be raised to the upper bound inflation rate equilibrium. On this matter, the following policy measures could be implemented. First, government may grant subsidies or tax incentives in order to expand sI by reducing the cost of investment, k. Second, initiatives to synchronize interests between workers and entrepreneurs, with the objective of reducing h. By doing so, if workers expect that they will be in the workforce for a sufficiently long period, the greater will be their incentives to invest in human capital. Another potential measure is the so-called networking innovation activities, i.e., the promotion of collaboration among industry, academia, and the government leads to a reduction in investment costs and an increase in expected returns. Several studies, like Nelson and Shaw (2003), Zucker et al. (1998) and Adams et al. (2003), show that it can play an important role in increasing the level of innovation.

5 Conclusion In this paper we proposed an ACE model to evaluate in which conditions of equilibrium the Brazilian economy could find itself with a high level of investment, given the inflation target rate announced by the monetary authority. Our approach consists of an iteratively played stage game among entrepreneurs and workers, which assuming the presence of complementary between the investments decisions implies the existence of two equilibria.


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Therefore, ACE learning models contributions is twofold: (a) a refinement of dynamic equilibrium analysis through numerical simulation to determine the trajectory of the level of investment in the economy for a given inflation targeting rate; (b) a more precise dimensioning of the proportion of agents willing to invest in the creation of demand rather than cost reduction, optimizing the implementation of the economic policy by the central planner. In one of the balances, both entrepreneurs and workers invest. In this case, there are two conditions necessary to ensure that the economy converges to the inflation target: (i) entrepreneurs should signal that investment is a demand creation; (ii) the proportion of workers and entrepreneurs investing at the current moment should be large enough. If these conditions are not observed, policies need to be developed to drive the economy into such favorable conditions. On the other hand, according to the proposed model, there is a balance in which players coordinate their actions so that there is no investment in the economy. In this sense, a important extension to the model for future researches is to consider the accumulation of physical and human capital in the economy. Settle these issues are not a trivial task from the evolutionary game theory perspective, but it expands the power of analysis and understanding of the ideal conditions to ensure that the inflation rate converges to the announced target inflation rate.

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