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The remainder of this chapter is organized as follows. In Section 2.2, we describe the basic model structure. In Section 2.3, we investigate the growth e¤ects of capital taxation. In Section 2.4, we calibrate the model to provide a quantitative analysis of capital taxation. Finally, some concluding remarks are discussed in Section 2.5.
2.2 The model
The model that we consider is an extension of the seminal workhorse R&D-based growth model from Romer (1990).3 In the Romer model, R&D investment creates new varieties of intermediate goods. We extend the model by introducing endoge-nous labor supply and distortionary income taxes. In what follows, we describe the model structure in turn.
2.2.1 Household
The economy is inhabited by a representative household. Population is stationary and normalized to unity. The household has one unit of time that can be allocated between leisure and production. The representative household’s lifetime utility is given as:4
U = Z 1
0
e t[ln C + (1 L)1
1 ] dt, (1)
3In the case of extending the model into a scale-invariant semi-endogenous growth model as in Jones (1995), the long-run growth e¤ect of capital taxation simply becomes a level e¤ect. In other words, instead of increasing (decreasing) the growth rate of technology, capital taxation increases (decreases) the level of technology in the long run.
4For notational simplicity, we drop the time subscript.
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where the parameter > 0 is the household’s subjective discount rate and the parameter > 0 determines the disutility of labor supply. 0 determines the Frisch elasticity of labor supply. The utility is increasing in consumption C and decreasing in labor supply L 2 (0; 1).
Two points regarding the utility function in eq. (1) should be noted. First, to make our analysis tractable, the household is speci…ed to have a quasi-linear utility function. In the quantitative analysis in Section 2.4, we will consider a more general utility function in order to examine the robustness of our results. Second, as pointed out by Hansen (1985) and Rogerson (1988), the linearity in work hours in the utility function can be justi…ed as capturing indivisible labor.
The representative household maximizes its lifetime utility subject to the bud-get constraint:
K + _a = (r_ A+ _V =V )a + (1 K)rKK + (1 L)wL C Z. (2) The variable K denotes the stock of physical capital. The variable a (= V A) denotes the value of equity shares of monopolistic …rms, in which A is the number of monopolistic …rms and V is the market value of an invented variety, w is the wage rate. rAis the rate of dividends, _V =V is the rate of gain or loss of an invented variety, and rK is the capital rental rate.5 The policy instrument Z is a lump-sum tax.6 The other policy instruments f L; Kg < 1 are respectively the labor and capital income tax rates.7
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The rates of return on the two assets, physical capital and equity shares, must follow a no-arbitrage condition at any time:
rA+ _V =V = (1 K)rK (3a)
We denote the common net return on both assets as r, i.e., r rA+ _V =V = (1 K)rK.
By solving the household’s optimization problem, we can easily derive the typ-ical Keynes-Ramsey rules:
C_
C = (1 K)rK , (3b)
and also the optimality condition for labor supply, which is in the form of a hori-zontal labor supply curve given the quasi-linear utility function in eq. (1):
(1 L) = (1 L)w. (4)
2.2.2 Final goods
There is a single …nal good Y , which is produced by combining labor and a con-tinuum of intermediate goods, according to the following aggregator:
Y = L1Y Z A
0
xidi, (5)
where LY is the labor input in …nal goods production, xi for i 2 [0; A] is the intermediate good of type i, and A is the number of varieties of intermediate goods. The …nal good is treated as the numeraire, and hence in what follows its price is normalized to unity. We assume that the …nal goods sector is perfectly
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competitive. Pro…t maximization of the …nal goods …rms yields the following conditional demand functions for labor input and intermediate goods:
LY = (1 )Y =w, (6)
xi = LY ( =pi)11 , (7)
where pi is the price of xi relative to …nal goods.
2.2.3 Intermediate goods
Each intermediate good is produced by a monopolist who owns a perpetually protected patent for that good. Following Romer (1990), capital is the factor input for producing intermediate goods, and the technology is simply a linear one-to-one function. That is, the production function is expressed as xi = ki, where ki is the capital input used by intermediate …rm i. Accordingly, the pro…t of intermediate goods …rm i is:
i = pixi rKki. (8)
Pro…t maximization subject to the conditional demand function for intermediate goods …rm i yields the following markup-pricing rule:
pi = rK
> rK. (9)
Equation (9) implies that the level of price is the same across intermediate goods
…rms. Based on eq. (7) and the production function xi = ki, we have a symmetric
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2.2.4 R&D
In the R&D sector, the familiar no-arbitrage condition for the value of a variety V is:
rV = + _V. (11)
Equation (11) states that, for each variety, the rate of return on an invention must be equal to the sum of the monopolistic pro…t and capital gain (or loss) . As in Romer (1990), labor is the factor input of R&D. The innovation function of new varieties is given by:
A = AL_ A, (12)
where > 0 is the R&D productivity parameter and LA denotes R&D labor.
Given free entry into the R&D sector, the zero-pro…t condition of R&D is
AV = wL_ A , AV = w. (13)
2.2.5 Government
The government collects taxes, including capital income tax, labor income tax, and lump-sum tax, to …nance its public spending. At any instant of time, the government budget constraint can be expressed as:
KrKK + LwL + Z = G. (14)
The variable G denotes government spending, which is assumed to be a …xed proportion 2 (0; 1) of …nal output such that
G = Y. (15)
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2.2.6 Aggregation
Since the intermediate …rms are symmetric, the total amount of capital is K = Aki = Ak. Given xi = ki, xi = x, ki = k, and K = Ak, the …nal output production function in eq. (5) can then be expressed as:
Y = A1 K L1Y : (16)
After some calculations using eqs (2), (6), (7), (11)-(14), and (16), we can derive the resource constraint in this economy:
K = Y_ C G: (17)
2.2.7 Decentralized equilibrium and the balanced-growth path
The decentralized equilibrium is a sequence of allocations fC; K ; A; Y ; L; LY; LA; x ; Gg1t=0, prices fw; r; rK; pi; Vg1t=0, and policies f K; L; Zg, such that at any instant of time:
a. households maximize lifetime utility (1) taking prices and policies as given;
b. competitive …nal goods …rms choose fx; LYg to maximize pro…t taking prices as given;
c. monopolistic intermediate …rms i 2 [0; A] choose fki; pig to maximize pro…t
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g. the government budget constraint is balanced, i.e., KrKK + LwL + Z = G.
The balanced growth path is characterized by a set of constant growth rates of all economic variables. Let denote the growth rate of technology and a “~”
over the variable denote its steady-state value. Along the balanced growth path, we have
K_ K =
Y_ Y =
C_ C = w_
w = A_
A = ~, _L = _LY = _LA= _V = 0. (17a)