Policy and Intellectual Property RightsPolicy and Intellectual Property Rights
Lemma 2: The equilibrium labor allocations are given by
4.2. A search-based monetary growth model
4.2.1.2. Households' optimization in the market 1
From standard optimization, the optimality condition for consumption in the market 2 is 1 .
t
t w
A
x = (4.3)
Equation (4.3) implies that all households consume the same amount of general goods x in the t market 2 regardless of their holdings of capital and money. This useful property results from the quasi-linear utility function, which is a standard simplifying assumption in this branch of model to eliminate any dispersion in money holdings that arises from trades in the market 1.52 The standard intertemporal optimality conditions for the accumulation of capital and money are respectively
Equations (4.3) to (4.5) imply that all households enter the market 1 in the next period with the same holdings of capital and money. In addition, the familiar envelope conditions are
1 ,
4.2.1.2. Households' optimization in the market 1
In the market 1, a household either becomes (a) a buyer, (b) a seller or (c) a nontrader. The
52 See for example, Rocheteau and Wright (2005) and Aruoba et al. (2011) for a useful discussion.
53 Here we can also use other vintages of trading friction as shown in Berentsen et al. (2007). That is, in the beginning of the first market, a household receives a preference shock that determinies whether he consumes or produces. With probability σ a household can produce but cannot consume, while with probability 1−σ the household can consume but cannot produce. This environment may exhibit a smaller welfare effect than the environment of Lagos and Wright. Therefore, it may be fruituful for studies to futher revisit the growth and welfare effects of monetary policy using a search-based monetary growth model. This question was brought to our attention
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policy would have no effects on economic growth and social welfare. This taste-and-technology-shock specification shows a matching technology that buyers meet with sellers and is a standard feature of the Lagos-Wright model. As a result of this taste-and-technology shock, the value of entering the market 1 is
, ) , ( ) 2 1 ( ) , ( )
, ( )
,
(mt kt Vb mt kt Vs mt kt W mt kt
V =σ +σ + − σ (4.8)
where Vb(.) and Vs(.) are the values of being a buyer and a seller respectively.
To analyze Vb(.) and Vs(.), we consider the following functional forms for the buyers'
preference and the sellers' production technology. In the market 1, each buyer's utility lnqtb is increasing and concave in the consumption of special goods. Each seller produces special goods
s
q by combining her capital t k and effort t e subject to the following Cobb-Douglas t production function.
1 α α η,
t t t s
t z k e
q = − (4.9)
where z denotes aggregate technology. To achieve endogenous growth, we will follow Romer t (1986) to assume that capital has a positive externality effect on aggregate technology such that
t kt
z = , where k is the aggregate holding of capital in the economy.t 54 The parameter )
1 , 0
∈(
α determines capital share. To ensure constant returns to scale, we will impose α
η=1− for labor share; however, it would be useful for us to first present the analysis with η in order to isolate the effects of capital and labor shares.
by Professor Yiting Li, to whom we are most grateful.
54 It is useful to note that kt=kt in equilibrium.
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Rewriting equation (4.9), we can express the utility cost of production as
. seller are respectively56
, obtain the following envelope condition for k . t
55 As a result of these different money holdings at the end of the market 1, households supply different amounts of labor in the market 2 that eliminate any dispersion in money holdings.
56 Adding a disutility parameter to the supply of effort in the market 1 would not change our qualitative and quantitative results. Therefore, we follow Aruoba et al. (2011) and Waller (2011) to normalize this parameter to unity.
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competitive equilibrium with price taking as in Aruoba et al. (2011) and Waller (2011).57 Under price taking, once buyers and sellers are matched, they both act as price takers. Given the pricept
subject to the budget constraint
t
In the market 1, buyers spend all their money,58 so that the money constraint implies that
t t b
t m p
q = /~ . (4.17)
As for sellers' maximization problem in the market 1, it is given by
57 In addition to the competitive equilibrium with price taking, Aruoba et al. (2011) and Waller (2011) also consider bargaining between buyers and sellers to determine the terms of trade, which is also a common approach in the literature. In the present study, we only consider the competitive equilibrium with price taking because of economic growth. In the case of generalized Nash bargaining as in Aruoba et al. (2011) or proportional bargaining as in Waller (2011), the bargaining condition is incompatible with balanced growth because the buyers' utility, which determines their surplus, is increasing overtime due to economic growth whereas the sellers' disutility of effort is stationary on a balanced growth path. In Appendix A, we demonstrate this problem under proportional bargaining and show that only a special case in which buyers gain all surplus is consistent with balanced growth. The same result can also be shown for the case of generalized Nash bargaining.
58 See Appendix B for a proof. Intuitively, due to the opportunity cost of holding money and the possibility of not being a buyer in the marke 1, households do not carry a sufficient amount of money to the market 1. Therefore, if a household turns out to be a buyer in the market 1, it would be optimal to spend all the money on qtb.
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Sellers' optimal supplies of special goods can be obtained from the following condition.
t
where the second equality of (4.19) makes use of (4.7) and (4.10).
Using (4.17) and (4.19), we can obtain ∂qtb/∂mt =1/p~t , ∂ /∂ t =1 (4.19) into (4.13) and (4.14), we can derive the following conditions.
t
Intuitively, (4.20) states that the marginal benefit of holding money is the sum of the marginal utility from being able to consume special goods with probability σ (i.e., the household
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exert less effort (recall that e2<0) in producing special goods in the market 1 with probability σ (i.e., the household becomes a seller in the market 1).59