In this chapter I proposed a solution procedure to solve a monetary search model when the buyers do not have full bargaining power in random match of agents. The proposed procedure extends the solution procedure developed in Molico (2006) for the case where the buyers have full bargaining power.
The procedure is first used to replicate Molico’s (2006) results for the case where the stock of money is constant. Next, the procedure is used to investigate the economic effect of changes in the buyers’ bargaining power. The results of simulations show that when there is decrease in the buyers’ bargaining power there is an increase in the Gini coefficient and thus an increase in the wealth inequality. I also find that when buyers’ bargaining power is very high, a reduction in their bargaining power improves the average welfare in the economy while the opposite result is true when their bargaining power is already low.
Chapter 2
The Effects of Monetary Growth on
Welfare and Wealth Inequality in a
Money Search Model without
Exogenous Breakdown
2.1
Introduction
The goal of this essay is to assess the effect of money growth on aggregate welfare and wealth inequality by extending the model in Molico (1997, 2006) to account for settings in which the buyers do not necessarily have all the bargaining power. Molico (1997, 2006) develops a search model of money with divisible goods and money and non-degenerated distribution of money holding where agents can have unbounded money holdings. The model uses a bargaining protocol where agents cannot meet other potential trading partners during the bargaining process. Following Trejos and Wright (1995), I call it bargaining without exogenous breakdown. Using numerical simulation, Molico (1997, 2006) finds that lump-sum creation of money has a beneficial effect on welfare. However, the finding is based on the assumption that in a random matching pair the buyer makes a take-it-or-leave offer to the seller. This assumption implies that the buyer takes all the surplus resulting from a trade and leaves the seller with zero surplus.
Deviatov and Wallace (2001) criticize this way of dividing the gain of the trade as being non-optimal. The authors suggest that the result of the beneficial effect of money creation in Molico (1997) may be driven by the extreme trade surplus sharing procedure. Deviatov and Wallace (2001) use an economic environment similar to that in Molico (1997) but in which money is indivisible and agents can hold only up to two units of money. It develops a trading protocol and lump-sum money trading along with some agent-individual ex-post rationality constraints to avoid a non-optimal sharing of trade surplus following random meetings. The model is solved using an analytical procedure and the results show that expansionary monetary policy is beneficial.
In a subsequent paper, Deviatov (2006) extends the model in Deviatov and Wallace (2001) by allowing the use of a lottery trade in a random meeting between buyers and sellers in order to alleviate the assumption of the indivisibility of money while keeping the upper bound of an agent’s money holding to two units of money. The author uses numerical computation to solve the model, and the results show that money creation is beneficial when sellers’ participation constraints are binding which is equivalent to buyers making take-it-or- leave-it offers in the model developed by Molico (2006). Deviatov (2006) then concludes that the trade surplus sharing rule adopted by Molico (1997) is in some sense optimal contrary to the conjecture in Deviatov and Wallace (2001). The conflicting results show the need to check the robustness of the findings in Molico (1997, 2006) by examining a case where the buyers do not have all the bargaining power.
The papers by Molico (1997, 2006), Deviatov and Wallace (2001), and Deviatov (2006) contribute to a strand of literature that argues that expansionary monetary policy can be welfare enhancing as a counterpoint to Friedman (1969) who argues in favor of a contrac- tionary monetary policy. According to Friedman (1969), since the cost of producing money is almost zero, then the optimal policy would be to set the rate of return on money equal to the rate of return of other assets. This is achieved by setting the nominal interest rate to zero which requires an inflation rate that is the negative of the real interest rate. So, the optimal
monetary policy would be a deflationary one. This monetary policy can be implemented through a lump-sum tax on the stock of agents’ money holdings.
Friedman’s proposal has been found to be optimal in most monetary models, for instance in a money-in-the-utility function model by Benhabib and Bull (1983) and in a cash-in- advance model by Grandmont and Younes (1973). In addition, Lagos (2010) shows that Friedman’s proposal can also be implemented in a search theoretical model of money. In contrast, a few models find that the Friedman proposal in not the best monetary policy. For example, Levine (1991), and Kehoe et al. (1992) develop heterogeneous agents models in which expansionary monetary policies are optimal policies. Kehoe et al. (1992) is an extended version of Levine (1991). Both papers present an endowment economy with infinitely-lived agents who, subject to preference shocks, randomly alternate between being buyers or sellers. They find that an expansionary monetary policy through a lump-sum transfer to agents is welfare improving. This is due to the fact that an equal transfer in effect redistributes real balances from the relatively rich to the relatively poor agents which in turn leads to an increase in the number of trades. However, as pointed out by Kehoe et al. (1992), the model leads to an equilibrium where the money holdings degenerate in equilibrium, in particular at the end of each period sellers hold all the money stock.
In this paper I study the economic effect of an expansionary monetary policy in a search model where buyers do not necessarily have full bargaining power. In the model, agents are not allowed to meet other potential partners while trading. The expansionary monetary policy is implemented through a lump-sum transfer to each agent in each period after the closing of the goods market. Due to the complexity of the model, the steady state results were obtained through computer simulations. The solution procedure used in this paper is an adaptation of the procedure developed in Chapter 1.
The pattern of the impact of money creation on the average welfare depends on the bargaining power of the buyer. When the bargaining power of the buyer is high, low rates of money supply increase welfare and high rates reduce welfare. However, when the bargaining
power of the buyers is low, a faster rate of money creation can increase the average welfare even at moderate rates. These results show money creation can be welfare improving even when buyers do not have full bargaining power.
The results also reveal that when buyers have low bargaining power, a faster rate of monetary expansion always leads to a decline in wealth inequality. However, when buyers’ bargaining power is high, at low rates an increase in the growth rate of money supply reduces wealth inequality but at high rates, an increase in the rate of monetary expansion increases wealth inequality.
The rest of this paper is organized as follows. Section 2.2 describes the environment. Section 2.3 deals with the value function. In Section 2.4, I formulate the Nash bargaining problem. Section 2.5 defines the steady state equilibrium. In Section 2.6, I present the computational procedure. Section 2.7 is devoted to the presentation and the discussion of the results, and Section 2.8 concludes the paper.
2.2
Economic environment
The economic environment in this chapter is the same as the one described in Chapter 1 with one major difference: the stock of money is no longer constant. As in Chapter 1, I assume that in a random match with a single coincidence of wants, the partners in a current bargaining pair cannot meet other potential partners during the period of delay between two successive offer proposals. Following Trejos and Wright (1995), I call this type of bargaining a bargaining without exogenous breakdown.
The government changes the money supply by making lump-sum transfer τMtto individ-
uals, where Mt is the total per capita money holdings (or equivalently the aggregate money
holdings since the total measure of agent is one) at the beginning of period t. Therefore, if Mt+1 denotes the per capita money holding at the beginning of period t + 1, then
Mt+1 =
Z ∞
0
(mt+ τMt) dFt(mt) = (1 + τ)Mt (2.1)
where Ft(mt) is the distribution of money holdings mt at the end of the period t and before
the government lump-sum transfer to agents. We also have that for all t:
Mt=
Z ∞
0
(mt)dFt(mt). (2.2)
The timing is the following. At the beginning of period t, the goods market opens and individuals trade using the money holdings carried from the previous period. After the closing of the goods market, the government proceeds with the monetary transfer to all the individuals.
In this chapter I focus on the time invariant state equilibrium. When the growth rate of the money supply is different from one (τ 6= 1) then the aggregate money stock will either grow or shrink continually over time, making it difficult to obtain stationary equilibrium values for nominal variables such as the individual money holdings, (mt), or the quantity
of money exchanged during trading, (dt). Therefore to obtain a time invariant stationary
equilibrium I normalize nominal variables of period t by Mt which is the total money stock
at the beginning of the period. So, if yt denotes a nominal variable and ˜yt denotes its
corresponding normalized version then ˜yt is defined as:
˜ yt≡
yt
Mt
(2.3)
Now let m∗t be the money holding of an agent at the closing of the goods market in period t and before the lump-sum transfer. Let ˜mt+1 denotes the quantity of money that agent will
carry to the beginning of the next period t + 1. We have that:
Dividing both sides of equation (2.4) by Mt+1 we get: ˜ mt+1= m∗t Mt+1 + τMt Mt+1 . (2.5)
Now, using equation 2.1 we get:
˜ mt+1 =
˜ m∗t + τ
1 + τ . (2.6)
The actual value of ˜m∗t will depend in part on whether in a period t the agent had the opportunity to meet a partner or not, and whether the agent was a buyer or a seller in case the agent had the opportunity to trade.
We can now use equation (2.6) to write the law of motion of an agent’s money holding in the stationary equilibrium as:
˜ m0 = ˜ m− ˜d+τ
1+τ with probability ασ (agent was a buyer)
˜ m+ ˜d+τ
1+τ with probability ασ (agent was a seller) ˜
m+τ
1+τ with probability 1 − 2ασ (agent was not involved in bargaining)
(2.7)
where ˜m0 is the agent’s beginning of next period normalized money holding, α is the prob- ability of meeting someone in given period, and σ is the probability of single coincidence of wants in a random match.
2.3
Value function
Let V ( ˜m) be the steady state value function of an agent who enters a new period with a normalized amount of money ˜m. Let d(˜b, ˜s) be the quantity of money given by the buyer, with normalized money holding ˜b, to the seller, with normalized money holding ˜s, in exchange for the production of the quantity q(˜b, ˜s) in equilibrium. Also, let F (˜y) be the measure of agents with normalized money holdings ˜m such that ˜m <= ˜y in equilibrium. Then the
agent’s steady state value function V ( ˜m) must satisfy the following Bellman equation: V ( ˜m) = 1 1 + r ασ Z ∞ 0 U (q( ˜m, ˜ms)) + V ˜m − ˜d( ˜m, ˜ms) + τ 1 + τ − V ˜m + τ 1 + τ dF ( ˜ms) + ασ Z ∞ 0 −C(q( ˜mb, ˜m)) + V ˜m + ˜d( ˜mb, ˜m) + τ 1 + τ − V ˜m + τ 1 + τ dF ( ˜mb) + V ˜m + τ 1 + τ . (2.8)
In Equation (2.8) the first term of the expression inside the curly braces is the expected increase in the value function of the agent when he is a buyer and the second term is the expected increase in his value function when is he is a seller. The third term is just the agent’s continuation value.
2.4
Generalized Nash bargaining
I use the Generalized Nash bargaining procedure to determine the terms of trade (q, ˜d). Fol- lowing Trejos and Wright (1995), since by assumption agents cannot meet potential partners during a bargaining process, I will set their threat points to zero. Therefore, the terms of trade (q, ˜d) between a buyer with money holding ˜mb and a seller with money holding ˜ms
solve the following problem:
max q,d U (q) + V ˜mb− ˜d + τ 1 + τ θ −C(q) + V ˜ms+ ˜d + τ 1 + τ (1−θ) (2.9) subject to 0 ≤ q ≤ 1 (2.10) 0 ≤ ˜d ≤ ˜mb (2.11) U (q) + V ˜mb− ˜d + τ 1 + τ ≥ V ˜mb+ τ 1 + τ (2.12) − C(q) + V ˜ms+ ˜d + τ 1 + τ ≥ V ˜ms+ τ 1 + τ (2.13) ˜ ms+ ˜d + τ 1 + τ ≤ ˜m (2.14)
where ˜m is an upper bound of money holding such that the mass of agents with money holdings greater than ˜m is negligible and θ is the buyer’s bargaining power.
Equation (2.10) expresses the constraint imposed on the quantity of good exchanged in order to get a non-negative disutility for the seller based on the specific functional form used in this chapter. Equation(2.11) takes into account the buyer’s budget constraint. Equations (2.12) and (2.13) are respectively the participation constraints of the buyer and the seller. Equation (2.14) imposes a constraint on the money holding of a seller to ensure that seller’s money holding does not exceed the upper bound of the domain [0, ˜m].
2.5
Equilibrium definition
An equilibrium in this economy consists of a distribution of money holdings F ( ˜m), terms of trade q( ˜mb, ˜ms) and ˜d( ˜mb, ˜ms), a value function V ( ˜m) such that given the constant growth
rate τ the following requirements are meet:
1. F is an invariant distribution given ˜d;
2. V satisfies the Bellman equation (2.8) given (q, ˜d) and F ;
3. (q, ˜d) solves the generalized Nash bargaining problem equations 2.9–2.14 given V .
In this chapter I focus on a monetary equilibrium, that is, an equilibrium in which there exists at least one matching trade where the buyer and the seller exchange strictly positive quantities of good q and money ˜d.
2.6
Computational procedure
To solve the model, I use a modified version of the solution procedure presented in Chapter 1 to account for the change in money stock from one period to another. For example, in step 4 of the numerical procedure described in Chapter 1, I use the following new update equation
for the value function from iteration k to the next iteration k + 1: Vk+1( ˜m) = 1 1 + r ασ Z ∞ 0 U (q( ˜m, ˜ms)) + Vk ˜m − ˜d( ˜m, ˜ms) + τ 1 + τ − Vk ˜m + τ 1 + τ dF ( ˜ms) + ασ Z ∞ 0 −C(q( ˜mb, ˜m)) + Vk ˜m + ˜d( ˜mb, ˜m) + τ 1 + τ − Vk ˜m + τ 1 + τ dF ( ˜mb) + Vk ˜m + τ 1 + τ . (2.15)
In addition, I use Equations 2.9–2.14 to determine the terms of trade and Equation 2.6 to update the money holdings of the agents.
Table 2.1: Replication of effects of monetary expansion published in Molico (2006), page 719
Money growth rate (%) 0 0.5 1 2 3 5 10 20
Standard deviation of money (%) 36.753 39.213 43.114 49.049 52.421 56.860 61.416 62.737 Coefficient variation (prices) 0.396 0.246 0.239 0.254 0.271 0.292 0.324 0.352 Average quantity (good) 0.848 0.851 0.848 0.842 0.835 0.823 0.795 0.752 Real balance(M/P) 32.895 12.924 8.719 5.567 4.203 2.949 1.867 1.250 Annual velocity money 0.026 0.066 0.097 0.151 0.199 0.281 0.432 0.617 Welfare 13.664 13.854 13.855 13.827 13.787 13.698 13.482 13.120
Notes: α = 1, σ = 0.25, r = 0.01 and M = 100.
Table 2.2: Molico’s original results of the effects of monetary expansion
µ 0% 0.5% 1% 2% 3% 5% 10% 20% Std. ˆm(%) 42.3 40.6 44.2 49.4 53.2 57.0 61.8 62.0 M/E(p) 31.8 12.85 8.60 5.47 4.15 2.93 1.83 1.18 Coef. Var. p 0.46 0.25 0.24 0.25 0.27 0.30 0.33 0.36 E(q) 0.843 0.851 0.848 0.840 0.830 0.821 0.794 0.756 W 13.57 13.86 13.87 13.84 13.79 13.68 13.42 12.95 Annual vel 0.026 0.066 0.098 0.153 0.198 0.280 0.433 0.638 x = 0.25, r = 0.01.
Source: Molico (2006), page 719.
I tested the modified solution procedure on the Molico’s model where buyers have all the bargaining power. The results, presented in Table 2.1 are in general in line with Molico’s
results shown in Table 2.2, where µ is the growth rate of money, x is the probability of a single coincidence of wants, and r is the discount rate.
2.7
Simulation results
In this section I present the simulation results of the economic effects of changes in the growth rate of money using the parameters values from Molico (2006). To account for the non-monotonic effects of bargaining power on the outcome of the simulations I run two sets of simulations: one set for the case where the buyers’ bargaining power is high (θ = 0.8) and another set for the case where the buyers’ bargaining power is low (θ = 0.4). Tables 2.3 and 2.4 summarize the results of the different simulations.
In equilibrium, an increase in the rate of money growth always raises the average real price and decreases the dispersion of prices when the bargaining power of the buyer is low. However, when the bargaining power of the buyer is high, low rates of money creation decrease the price dispersion while higher rates increase the price dispersion.
Also, in general, a faster growth rate of money supply leads to a decline in the average quantity of good consumption exchanged in equilibrium. However, when buyers have low bargaining power, low growth rates of money will have limited impact on the quantity of consumption good exchanged.
When buyers have low bargaining power, a faster rate of monetary expansion always leads to a decline in wealth inequality. However, when buyer’s bargaining power is high, at low rates an increase in the money supply growth rate reduces inequality but a high rates, an increase in the rate of monetary expansion increases wealth inequality.
In addition, the average real money balance is always decreasing in the money growth rate while the annual velocity of money is always increasing in the money growth rate.
The pattern of the impact of money creation on the average welfare depends on the bargaining power of the buyer. When the bargaining power of the buyer is low, low rates of
Table 2.3: Effect of monetary expansion when the buyer’s bargaining power is 0.80
Money growth rate(%) 0 2 5 10 20
Standard deviation (money) 0.509 0.495 0.564 0.610 0.620 Gini coefficient 0.291 0.276 0.313 0.336 0.342
Average price 0.068 0.183 0.340 0.536 0.801
Coefficient variation (prices) 0.544 0.255 0.293 0.329 0.351 Average quantity (good) 0.846 0.842 0.823 0.796 0.752 Real balance(M/P) 14.635 5.451 2.945 1.864 1.249 Annual velocity money 0.057 0.155 0.281 0.433 0.618
Welfare 13.730 13.826 13.698 13.482 13.120
Notes: α = 1, σ = 0.25, r = 0.01 and M = 1.
money supply increase welfare and high rates reduce welfare while at high rates, the results are reversed. However, when the bargaining power of the buyers is low, a faster rate of money creation can increase the average welfare even at moderate rates.
To get an insight into these results, we need to take into account two effects of a lump-sum transfer policy in addition to the two effects of the bargaining power mentioned in Chapter 1. First, a lump-sum transfer has a redistributive effect by working as a subsidy for the poor (agents with money holding less than the average money holding) and as a tax for the rich (agents with money holding more than the average money holding). Second, the lump-sum transfer, by inducing inflation, has a negative real balance effect which reduces the average real money balance of agents.
The intuition behind the effect of money creation on wealth inequality is the following. On the one hand, the redistributive effect of a lump-sum money transfer reduces wealth inequality, but on the other hand, the real balance effect of money creation can increase inequality. This negative effect comes from the fact that the real balance effect increases the opportunity cost of holding money, therefore poor agents in particular will have to transfer a more significant portion of their wealth to rich agents to induce them to produce as the growth rate of money increases. This leads to an increase in both the average real price and
Table 2.4: Effect of monetary expansion when the buyer’s bargaining power is 0.40
Money growth rate(%) 0 2 5 10 20
Standard deviation (money) 0.818 0.806 0.774 0.716 0.654 Gini coefficient 0.425 0.424 0.412 0.388 0.358
Average price 0.579 0.643 0.700 0.774 0.901
Coefficient variation (prices) 0.836 0.725 0.639 0.522 0.421 Average quantity (good) 0.751 0.754 0.751 0.744 0.726 Real balance(M/P) 1.726 1.556 1.428 1.292 1.109 Annual velocity money 0.452 0.500 0.541 0.592 0.675
Welfare 12.777 12.863 12.907 12.918 12.833
Notes: α = 1, σ = 0.25, r = 0.01 and M = 1.
wealth inequality.
In addition, as mentioned in Chapter 1 the seller’s bargaining power has a negative ef- fect on wealth inequality. Therefore in this environment, the net effect of money creation on wealth inequality is ambiguous. When the seller’s bargaining power is low, money cre- ation can reduce inequality only when the money growth rate is low. However, when the seller’s bargaining power is high, poor buyers are in such a disadvantaged position that the redistributive effect of a money transfer policy can give them enough endogenous bargain- ing power and lead to a reduction in wealth inequality as they get more money transfer. The effects of money creation on the distribution of wealth are illustrated in Figure 2.1 and Figure 2.2.
The effect of money creation on price dispersion seems closely related to that on wealth inequality. In general, an increase in wealth inequality leads to an increase in price volatil- ity. Since the higher the dispersion of money holdings, the higher the occurrence of large differences in money holdings between buyers and sellers and thus the more dispersed the distribution of prices. The distribution of prices is reported in Figure 2.3.
Since the real-balance effect reduces the incentive to hold money, there is a lower demand for money which will lead to smaller quantity of good exchanged. However, this negative
effect can be dominated by the redistributive effect by benefiting the poor and giving them