國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
chapter instead analyzes the optimality of the Friedman rule via the implementation of inflation targeting.
The remainder of this chapter is organized as follows. Section 2.2 presents the structure of the one-sector cash-in-advance endogenous growth model in which the intermediate goods market is imperfectly competitive, the production of intermediate goods is subject to production externalities, and the monetary authority implements inflation targeting as a monetary policy rule.
Section 2.3 discusses the local dynamics of the economy’s balanced growth equilibrium and Section 2.4 examines the growth effect of adjusting the inflation rate and income tax rate.
Section 2.5 deals with whether fiscal and monetary authorities can choose a package of the income tax and inflation targeting to remedy the distortions stemming from market imperfections and production externalities. Finally, the main findings of our analysis are summarized in Section 2.6.
2.2. The model
The economy we deal with is composed of four sectors: a production sector, a household sector, a government, and a central bank. In what follows, we in turn describe the behavior of each of these sectors.
2.2.1. Firms
Following the approaches of Benhabib and Farmer (1994) and Farmer and Guo (1994), the production side of the economy consists of two markets: the perfectly competitive final goods market and the monopolistically competitive intermediate goods market. There is a continuum of intermediate goods y , j j∈[0, 1], which are used by a single representative firm to produce a
‧ 國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
final good y. The production of the final good takes the Dixit and Stiglitz (1977) technology that displays constant returns to scale:
) 1 /(
1 1
0
1 θ
θ −
−
=
∫
y djy j . (2.1)
To guarantee the existence of an equilibrium, the parameter θ is restricted at θ∈[0,1).9 As we will show later, θ measures the degree of monopoly power of the intermediate good firms.
Let P and p denote the prices of the final good and the jth intermediate good, j respectively. The maximization problem of the representative final good firm can be expressed as:
−
∫
y− dj −∫
p y djP
Max j j j
yj
1 0 ) 1 /(
1 1
0
1 θ
θ . (2.2a)
The first-order condition for the problem leads to the following constant-elasticity inverse demand function for the jth intermediate good:
P
y p y
j j
θ
= . (2.2b)
It is clear from equation (2.2b) that the demand function for the jth intermediate good has a constant price elasticity 1/θ . When θ =0, intermediate goods are perfect substitutes in the production of the final good, implying that the intermediate goods sector is perfectly competitive.
However, if 0<θ <1, intermediate good firms face a downward-sloping demand curve that can be exploited to manipulate prices. Accordingly, the parameter θ can be taken to reflect the degree of monopoly power of the intermediate good firms.
Given the fact that the final good market is perfectly competitive, profits of final good producers should be zero due to the free entry and exit assumption. Based on equations (2.2a)
9 See Blanchard and Kiyotaki (1987) for a detailed description.
‧ 國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
and (2.2b), the zero-profit condition for the final good sector implies that the following expression holds:
) 1 1 /(
0 / ) 1
( θ θ
θ
θ − −
−
−
=
∫
p djP j . (2.3)
The technology for producing the jth intermediate good can be described by a Cobb-Douglas production function characterized by both capital and labor externalities:10
β α
j j j
j A k n k n
y = ( , ) ; α,β∈(0,1), (2.4)
where k and j n denote the capital and labor inputs of the jth firm, and j A represents the j index of knowledge available to the jth intermediate good firm. The average economy-wide capital stock k and labor input n represent factors of production externalities (including capital and labor externalities) that are treated as given by the individual firm, due to a “learning by doing” mechanism proposed by Romer (1986).
In line with Benhabib and Farmer (1994) and Farmer and Guo (1994), the index of knowledge available to the jth intermediate good firm takes the following form:
η αn k A n k
Aj( , )= 1− ,
where the degrees of capital and labor externalities are 1−α and η, respectively.
Intermediate good firms hire capital and labor from perfectly competitive markets. Let Πj denote the profits of the jth firm in terms of the final good, and r and w denote the capital rental rate and the wage rate in terms of the final good, respectively. The optimization problem of the jth firm can be written as:
j j j j n j
k y wn rk
P Max p
j j
−
−
⋅
=
, Π , (2.5)
10 Guo and Lansing (1999, p. 972) provide a solid justification for the Cobb-Douglas production function: “The Cobb-Douglas specification implies that the income shares attributable to capital and labor inputs are constant over time, a feature that reasonably approximates the US economy.”
‧ 國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
subject to the inverse demand function stated in equation (2.2b) and the production function reported in (2.4). The first-order conditions for the physical capital and labor inputs are:
Pk r y p
j j
j =
− ) 1
( θ
α , (2.6a)
Pn w y p
j j
j =
− ) 1
( θ
β . (2.6b)
Equations (2.6a) and (2.6b) indicate that the jth firm hires both inputs until the marginal product of each input equals its market price. Moreover, both equations reveal that the demand for each input is inversely related to the monopoly power index θ.
Inserting equations (2.6a) and (2.6b) into (2.5), it is quite easy to deduce that the real profit of the jth intermediate good producer is given by:
[1 ( )(1 )] 0
≥
− = +
= −
Π P
y p P
y
pj j j j
j
ν θ
β
α , (2.6c)
where ν =1−(α+β)(1−θ). Equation (2.6c) indicates that the real profit of the jth intermediate good is positively related to the degree of monopoly power.
We confine the analysis to a symmetric equilibrium where all intermediate goods firms behave in the same manner, i.e., kj =k =k, nj =n=n and pj = p for all j∈[0, 1]. Then,
from equations (2.3) and (2.4) we have P= p and y=yj =Aknη+β. As the owners of all firms, with Πj =νpjyj/P, p=P, and yj =y, the households receive real profits Π(=
∫
01Πjdj)= νy from all intermediate goods firms in the form of dividends. Moreover, given P= p= pi,n
nj = , kj =k, the rent for capital and the wage for labor reported in (2.6a) and (2.6b) can be rewritten as:
β
θ η
α − +
= An
r (1 ) , (2.7a)
) 1
1
( − + −
=β θ Aknη β
w . (2.7b)
‧ 國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
2.2.2. Households
Consider an economy that is populated by a large number of identical and infinitely-lived households. For simplicity, population is normalized to unity. The representative household derives utility from consumption of final goods c and incurs disutility from working n . In line with Benhabib and Farmer (1994), the lifetime utility of the representative household can be written as:
∫
∞ + −
+
− −
= 0
1
1
ln n 1 e dt
c
U ρt
ε
ε . (2.8)
In equation (2.8), the absolute value of ε (≥0) stands for the inverse of the intertemporal elasticity of substitution in labor supply, and ρ(>0) denotes the constant rate of time preference.
As in the Lucas (1980) cash-in-advance model, the representative household faces the following liquidity constraint:
c
m≥ , (2.9)
where m(=M /P) is real money balances with M denoting nominal money balances.
Equation (2.9) states that real money balances are needed by the household in order to purchase final goods.
As the owners of all firms, the households receive real profits in the form of dividends. In addition to holding money and physical capital as assets, each household also has access to nominal government bonds B that pay the nominal interest rate R. Let b(=B/P) denote real government bonds. The household’s flow budget constraint can then be written as follows:
c m rk
wn b
R k m
b&+ &+ &=( −π) +(1−τ)( + +Π)−π − , (2.10)
‧ 國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
where τ represents the income tax rate and π(=P&/P) refers to the inflation rate. Moreover, a dot on the top of a variable indicates its rate of change with respect to time.
The household treats w, r , R, π, and Π as given and maximizes (2.8) subject to equations (2.9) and (2.10) by choosing a sequence {c,n,k,b,m}∞t=0. Then, the current-value Hamiltonian can be formulated as follows:
+
+
− −
= +
ε
ε
1 ln 1
n1
c
H λ1[(R−π)b+(1−τ)(wn+rk+Π)−c−πm]+λ2(m−c), (2.11) where λ is the shadow value of wealth 1 a(=b+m+k) in terms of utility and λ stands for 2 the Lagrange multiplier for the cash-in-advance constraint reported in equation (2.9).
The first-order conditions with respect to the indicated variables are:
:
c c−1 =λ1+λ2, (2.11a) :
n nε =λ1(1−τ)w, (2.11b) :
k λ&1−λ1ρ=−λ1(1−τ)r, (2.11c) :
b λ&1−λ1ρ=−λ1(R−π), (2.11d) :
m λ&1−λ1ρ=−(λ2−λ1π). (2.11e) By means of some simple manipulations to delete λ1 and λ2, we can derive the following necessary conditions for an optimum:
R R R
c c
− +
−
−
= 1
&
&
ρ
π , (2.12a)
) 1 (
) 1 (
R c w
n +
= −τ
ε , (2.12b)
π τ = −
− )r R 1
( . (2.12c)
Equation (2.12a) is the Keynes-Ramsey Rule, indicating how consumption will change over time.
Equation (2.12b) indicates that the marginal rate of substitution between consumption and leisure
‧ 國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
is equal to the after-tax real wage rate in terms of the effective price of consumption, which is the opportunity cost of holding an additional unit of money, (1+R). Equation (2.12c) represents an arbitrage condition between holdings of physical capital and government bonds.
2.2.3. Central bank and government
Let µ(=M&/M) denote the growth rate of the nominal money stock. Then, according to the definition of real money balances m=M/P, the evolution of m can then be expressed as:
=µ−π m
m&
. (2.13)
Moreover, the government finances its budget deficits, i.e., the difference between expenditures on interest payments and income tax, by issuing money and bonds. Thus, the government’s flow budget constraint can be written as:
µm+b&+πb=Rb−τ(wn+rk+Π). (2.14) Under a regime of targeting the inflation rate, the inflation rate is set to be a constant for all times. The central bank must adjust the money growth rate µ to whatever level is needed to maintain the target value of the inflation rate.
2.2.4. Decentralized competitive equilibrium
By including the government’s budget constraint (2.14), the household’s budget constraint (2.10), the rent for capital r=α(1−θ)Anη+β in equation (2.7a), the wage for labor
) 1
1
( − + −
=β θ Aknη β
w in equation (2.7b), and the profit Π (=
∫
01Πjdj)=νy together, the aggregate resource constraint of the economy is given by:c y
k&= − . (2.15)
The main equations of the symmetric equilibrium of the economy can then be described in summary fashion as follows:
‧
k . In the next section, we will study the dynamic property of the balanced growth equilibrium.