3.3. Endogenous Technological Change
3.3. Endogenous Technological Change The graph below shows R&D intensities across OECD economies. Obviously there is a large variation in R&D intensity, even within the group of OECD countries. Two important questions are: [1] Does this variation explain differences in economic growth? This answer has been given at the beginning using growth regressions with R&D intensity as one explanatory variable. [2] How can we explain this variation in R&D intensities? Some determinants of R&D intensities might be: [i] availability of human capital; [ii] enforcement of intellectual property rights; [iii] sectoral structure of the economy; [iv] market structure affecting firm's profits; [v] public policies such as R&D subsidies.
Source : OECD Science, Technology and Industry Scoreboard
3.3.1. The Romer (1990) model: A Sketch of the Model Introduction The neoclassical growth model relies on exogenous technological progress as the engine of long-run growth. Romer (1990) was the first who formulated an explicit growth model with technical progress resulting from deliberate actions taken by private agents who respond to market incentives. Romer's analysis is based on three premises.
2
Romer_Model.nb
THREE PREMISES [1] Economic growth is driven by technological progress as well as capital accumulation; [2] Technological progress results from deliberate actions taken by private agents who respond to market incentives; [3] Technological knowledge is a non-rivalrous input (modelled as positive knowledge spill-overs).
FUNDAMENTAL MICROECONOMIC PROBLEM From a theoretical perspective, the problem was to explain technological progress endogenously resulting form a general equilibrium model with a consistent microeconomic structure. Earlier attempts to explain technical progress endogenously modelled technical progress as a by-product of capital accumulation (Arrow, 1962). The basic problem has been stated very precisely by Aghion and Howitt (1998, p. 23): "..., if A is to be endogenized, then the decisions that make A grow must be rewarded, just as K and L must be rewarded. But because F exhibits constant returns in K and L when A is held constant, it must exhibit increasing returns in three 'factors' K, L and A. Euler's theorem tells us that with increasing returns not all factors can be paid their marginal products. Thus something other than the usual Walrasian theory of competitive equilibrium, in which all factors are paid their marginal products, must be found to underlie the neoclassical model." To illustrate this argument consider the general production function Y = FHA, K, LL. This production function is assumed to exhibit constant returns to scale in the private factors, i.e. l Y = FHA, l K, l LL. If all factors should be rewarded according to their marginal products, then total output would be excessively exhausted: Y < A FA H.L + K FK H.L + L FL H.L.
Y = FHA, K, LL l Y = FHA, l K, l LL
Y < A FA H.L + K FK H.L + L FL H.L
A sketch of the model We consider a simplified model version in that there is only one type of labor. Romer (1990) distinguishes between unskilled labor and skilled labor (human capital). However, this distinction is not essential for the derived results (it merely relabels the relevant scale variable, see below). The production side comprises three sectors: a final output sector (competitive), an intermediate goods sector (monopolistic competition) and a research sector (competitive).
Firms FINAL OUTPUT SECTOR Firms in the final output sector produce a homogenous final output good that can be used for consumption or as an input for differentiated capital goods (i.e. investment in physical capital). The market for the final output good is perfectly competitive. The final output technology, assumed to exhibit CRS, can be expressed as Y = LY 1-a ‡ xHiLa „ i A
0
with 0 < a < 1
(1)
Romer_Model.nb
3
Y = LY 1-a A xa
(2)
Let an index of overall capital be defined as K := A x. This is motivated by the fact that it takes one unit of output not consumed ("raw capital") to produce one unit of x. Using this definition, the production function can be expressed as follows Y = LY 1-a A1-a K a
(3)
Y = HA LY L1-a K a
(4)
The output technology (4) represents a usual Cobb-Douglas technology with labor-augmenting technical progress. The magic thing is this: Even if one would hold K = A x constant (e.g. due to the fact that A x is produced from some constant resource R as Solow, 2000, p. 149, argues), then an increase in A (the number of varieties of capital goods) increases labor productivity. The technical progress should be interpreted as reflecting the Smith-Ethier effect, i.e. an increase in the division of labor rises labor productivity. This formalizes the basic idea of increasing productivity due to division of labor leading to specialization. x2 70 A 60 50 40
B
30 α=0.5
20
α=1
10 C 10
20
30
40
50
60
70
x1
The preceding graph illustrates this reasoning. Consider the technology Y = xa1 + xa2 with 0 < a b 1. The total amount of inputs is x1 + x2 . Compare point A (or C) to B. The total amount of inputs at A is x1 + x2 = 0 + 64 = 64 and at B it is x1 + x2 = 32 + 32 = 64. However, despite this fact, output produced with the input combination at A is smaller than output produced with input combination B, provided that a < 1.
CAPITAL GOODS SECTOR The capital goods sector produces differentiated capital goods. As a technical (and legal) prerequisite for production, capital goods producer must at first purchase a blueprint (patent). The production function of the final output producers implies that the market for capital goods is monopolistically competitive. It is assumed that capital goods can be produced from final output "one by one". Consequently, aggregate capital obeys the following law of motion (d = 0) ° ° K = A x + A x° = Y - C
(5)
4
Romer_Model.nb
COMMENTS [1] Along the BGP x° = 0. [2] Notice that capital goods here are differentiated capital goods and not intermediate goods which are put together and finally make up the final output good [as in Grossman and Helpman (1991)]. Put differently, capital goods here are modelled as a stock and not as a flow (Solow, 2000, p. 147 uses the flow interpretation; however, Jones, 1995, p. 781, uses the phrase "total stock of producer durables"). Capital goods producers rent the producer durables to the FO sector. Hence, x is a stock here. The profit function of the typical capital goods producer is p = @ pHxL - rD x.
RESEARCH SECTOR Firms in the research sector conduct R&D, i.e. they search for new and economically valuable ideas (alternatively, the model can be interpreted in the sense that R&D is done within the capital goods firms). R&D firms produce blueprints for the production of new types of capital goods (designs). The market for designs is perfectly competitive. The research technology is of the following form ° A = h A LA
with h > 0
(6) °
°
d ln A d ln A Notice that there is a double knife-edge restriction implicit in this formulation: [1] ÅÅÅÅÅÅÅÅ ÅÅÅÅÅ = 1 and [2] ÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅ = 1. The first is d ln A d ln L A needed for sustained growth to be feasible. The second is required for a consistent microeconomic structure, i.e. a perfectly competitive market requires CRS in the single private input factor L.
Households As in the standard neoclassical model, households derive utility from consumption only and supply inelastically one unit of labor each period of time (labor supply is exogenous). Maximization of the present value of the infinite utility stream subject to a dynamic budget constraint leads to the familiar Keynes-Ramsey rule (KRR) of optimal consumption r-r ° C = C ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ s
with s, r > 0
(7)
where r is the net interest rate.
The balanced growth rate We now sketch the strategy how to solve for the balanced growth rate. To this end consider the production technologies Y = HA LY L1-a K a
(8)
° A = h A LA
(9)
The Y -technology indicates that, along the BGP, this model is equivalent to a neoclassical growth model with labor-aug` ` ` ` menting technical progress. Therefore, the following relations must hold along the BGP: g := Y = A = K = C. How can the balanced growth rate be determined? The growth rate of A may be expressed as ` A = h LA
(10)
`* This expression shows that L A must be constant along a BGP. The balanced growth rate is A = h L*A . To determine the constant allocation L*A it is necessary to analyze the market solution more deeply (see below). Here we simply report the resulting long-run growth rate, which is given by
Romer_Model.nb
5
ahL- r gM = ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ a+s
(11)
IMPLICATIONS [1] The long-run growth rate is completely determined by technology and preference parameters. Growth is endogenous in two senses: First, the growth rate results from the optimizing decisions of private agents (households and entrepreneurs). Second, the presence of h indicates that an increase in the productivity in R&D (equivalent to a reduction in R&D costs) stimulates growth. Hence, an R&D subsidy can affect the long-run growth rate, i.e. policy is effective. [2] The above displayed growth rate applies to the market solution. Since there are a number of "market failures" (positive spill-over effects and imperfect competition, which are essential for sustained endogenous growth to be feasible), the market solution does not coincide with the socially optimal solution. The socially optimal long-run growth rate is given by h L-r ÅÅÅÅÅÅ , where gS > gM . gS = ÅÅÅÅÅÅÅÅ s [3] The long-run growth rate exhibits a scale effect. The larger the economy (measured by L), the faster it grows. The economic intuition behind the scale effect is straightforward: The larger the amount of L, the higher is the amount of labor allocated to R&D and the higher is the long-run growth rate (g = h L A ). [4] The growth condition is a h L - r > 0. This implies that the economy must be sufficiently large in order to exhibit sustained growth (there is a threshold effect). Notice that in the original Romer (1990) formulation the relevant scale variable is human capital H rather than labor L.
References Romer, P. M., Endogenous technological change, Journal of Political Economy, 1990, 98, S71-S101.
3.3.2. The Romer (1990) model: The Microeconomic Structure Introduction We now consider the microeconomic structure of the model. The model is further simplified in that there is no physical capital. This has two consequences: [1] Differentiated capital goods (interpreted as a stock) are now intermediate goods (a flow); [2] there is no (predetermined) state variable in this model and hence no transitional dynamics. The basic reference is Gancia and Zilibotti (2005).
Households The economy is populated by infinitely lived agents who derive utility from consumption and supply inelasticaly labor. Population is constant and equal to L. The optimal consumption plan follows the familiar Keynes-Ramsey rule: r-r ° C = C ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ s There is no physical capital and savings are used to finance innovation.
(12)
6
Romer_Model.nb
Firms On the production side there are two sectors: [1] A competitive sector producing a homogenous final output good and [2] an imperfectly competitive sector producing differentiated intermediate goods (R&D is conducted within the intermediate good firms, hence there is no separate R&D sector). The final goods sector employs labor and a set of intermediate goods as inputs according to the following technology a Y = L1-a ‡ xHiL di Y A
(13)
0
where xHiL is the quantity of intermediate good i, A is the measure of intermediate goods available at each point in time, LY is labor allocated to Y -production, and 0 < a < 1. This production function is due to Ethier (1982). The intermediate goods sector consists of monopolistically competitive firms, each producing a different variety i. The production of one unit of xHiL requires one unit of Y . In addition, each intermediate goods producer is subject to a sunk cost to design a new intermediate input variety (i.e. R&D is conducted within the intermediate goods firm). New designs are produced without uncertainty. The innovating firm can patent the design (intellectual property rights are enforced) and thereby acquire a perpetual monopoly power over the production of the corresponding input. R&D uses only labor as private input. However, innovations generate intertemporal knowledge spill-overs. The production function for additional designs reads ° A = h A LA
(14)
where h > 0 and L A is labor devoted to R&D. Note that the rate of technical change is a linear function of the amount of labor employed in R&D. The amount of labor required to produce a unit measure of new designs requires a labor input of 1 ê Hh AL . Finally, feasibility requires that LY + L A § L
(15)
Static Equilibrium Prices: w is the wage rate, pHiL the price of xHiL, and pY = 1 (i.e. Y is chosen as the numeraire). Equilibrium in the Y -sector. The representative firm in the competitive Y -sector takes all prices as given and chooses production and technology so as to maximize profit, given by a pY = L1-a ‡ xHiL di - w LY - ‡ Y A
0
A
pHiL xHiL di
(16)
0
The first-order conditions yield the following (inverse) factor demands: xHiLa-1 pHiL = a L1-a Y
" i œ @0, AD
a w = H1 - aL L-a Y ‡ xHiL di
(17)
A
(18)
0
Equilibrium in the intermediate goods sector. A firm owning a patent sets its production level so as to maximize the profit subject to the demand function (17). The profit of the firm producing xHiL is
Romer_Model.nb
7
pHiL = pHiL xHiL - xHiL
(19)
The optimal price set by the monopolist and the associated quantity are (see appendix below) pHiL = p = 1 ê a
(20)
xHiL = x = a2êH1-aL LY
(21)
Hence, the maximum profit for an intermediate producer is 1-a pHiL = p = H p - 1L x = ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ a2êH1-aL LY a
(22)
Substitution of (21) into (18) yields the equilibrium wage rate w = H1 - aL a2 aêH1-aL A
(23)
Appendix: The problem of the intermediate goods producer In[3]:=
α Ly1−α xα−1 , xD êê Simplify
Solve@p
−1+α Ly−1+α p y z j 99x → i == z j α { k 1
Out[3]=
−1+α −1+α i Ly−1+α p y i Ly−1+α p y z z z z j j equ1 = DAp j −j , pE z z j j α α { { k k 1
In[7]:=
I Ly α
−1+α
Out[7]=
In[8]:= Out[8]=
M H−1 + p αL p H−1 + αL
p
1
0 êê FullSimplify
1 −1+α
0
Solve@equ1, pD 99p →
1 == α
Dynamic Equilibrium We guess and verify the existence of a balanced growth equilibrium such that C, Y and A grow at the same constant rate g, and LY and LA are constant. Note that this implies that along the BGP x and p are constant. Free entry into intermediate goods production implies that the present discounted value (PDV) of profits from innovation cannot exceed the entry costs, i.e. the costs of an additional design (w ê h A). By the KRR the interest rate is also constant along the BGP. Hence, the PDV of profits earned in the intermediate goods sector is p ê r. The free entry condition thus can be expressed as: w p ÅÅÅÅÅÅ § ÅÅÅÅÅÅÅÅÅÅÅ r hA
(24)
Using (22) and (23) this condition can be expressed as 1-a 2êH1-aL ÅÅÅÅ ÅÅÅÅÅÅ a LY H1 - aL a2 aêH1-aL a ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅ § ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ r h
(25)
8
Romer_Model.nb
The RHS is the marginal cost of innovation, which is independent of A due to the cancellation of two opposite effects: [1] Labor productivity, and hence equilibrium wages, grow linearly with A. [2] The productivity of researchers increases with A due to the intertemporal knowledge spill-over effect. Thus the unit cost of innovation is constant over time. Note that, without knowledge spill-overs, the cost of innovation would grow over time, and technical progress and growth would come to a halt, as in the neoclassical model. ° For A > 0 condition (25) must hold with equality. Using LY = L - LA and LA = g ê h (resulting from (14) and balanced growth) condition (25) can be expressed as r = aHh L - gL
(26)
This is the equilibrium condition on the production side, which says that the higher the interest rate that firms must pay to finance innovation expenditure, the lower is L A and hence growth. Finally, the KRR can be expressed as r= r+sg
(27)
which shows the usual relationship between r and the growth rate of consumption. Solving these two equilibrium conditions for g gives the interior solution haL- r g = ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ a+s
(28)
Provided that h a L - r > 0 , g is positive. Otherwise, L A = 0 and g = 0.
References Gancia and Zilibotti (2005), Horizontal Innovation in the Theory of Growth and Development, Handbook of Economic Growth.
Source : OECD Science, Technology and Industry Scoreboard
3.3.1. The Romer (1990) model: A Sketch of the Model Introduction The neoclassical growth model relies on exogenous technological progress as the engine of long-run growth. Romer (1990) was the first who formulated an explicit growth model with technical progress resulting from deliberate actions taken by private agents who respond to market incentives. Romer's analysis is based on three premises.
2
Romer_Model.nb
THREE PREMISES [1] Economic growth is driven by technological progress as well as capital accumulation; [2] Technological progress results from deliberate actions taken by private agents who respond to market incentives; [3] Technological knowledge is a non-rivalrous input (modelled as positive knowledge spill-overs).
FUNDAMENTAL MICROECONOMIC PROBLEM From a theoretical perspective, the problem was to explain technological progress endogenously resulting form a general equilibrium model with a consistent microeconomic structure. Earlier attempts to explain technical progress endogenously modelled technical progress as a by-product of capital accumulation (Arrow, 1962). The basic problem has been stated very precisely by Aghion and Howitt (1998, p. 23): "..., if A is to be endogenized, then the decisions that make A grow must be rewarded, just as K and L must be rewarded. But because F exhibits constant returns in K and L when A is held constant, it must exhibit increasing returns in three 'factors' K, L and A. Euler's theorem tells us that with increasing returns not all factors can be paid their marginal products. Thus something other than the usual Walrasian theory of competitive equilibrium, in which all factors are paid their marginal products, must be found to underlie the neoclassical model." To illustrate this argument consider the general production function Y = FHA, K, LL. This production function is assumed to exhibit constant returns to scale in the private factors, i.e. l Y = FHA, l K, l LL. If all factors should be rewarded according to their marginal products, then total output would be excessively exhausted: Y < A FA H.L + K FK H.L + L FL H.L.
Y = FHA, K, LL l Y = FHA, l K, l LL
Y < A FA H.L + K FK H.L + L FL H.L
A sketch of the model We consider a simplified model version in that there is only one type of labor. Romer (1990) distinguishes between unskilled labor and skilled labor (human capital). However, this distinction is not essential for the derived results (it merely relabels the relevant scale variable, see below). The production side comprises three sectors: a final output sector (competitive), an intermediate goods sector (monopolistic competition) and a research sector (competitive).
Firms FINAL OUTPUT SECTOR Firms in the final output sector produce a homogenous final output good that can be used for consumption or as an input for differentiated capital goods (i.e. investment in physical capital). The market for the final output good is perfectly competitive. The final output technology, assumed to exhibit CRS, can be expressed as Y = LY 1-a ‡ xHiLa „ i A
0
with 0 < a < 1
(1)
Romer_Model.nb
3
Y = LY 1-a A xa
(2)
Let an index of overall capital be defined as K := A x. This is motivated by the fact that it takes one unit of output not consumed ("raw capital") to produce one unit of x. Using this definition, the production function can be expressed as follows Y = LY 1-a A1-a K a
(3)
Y = HA LY L1-a K a
(4)
The output technology (4) represents a usual Cobb-Douglas technology with labor-augmenting technical progress. The magic thing is this: Even if one would hold K = A x constant (e.g. due to the fact that A x is produced from some constant resource R as Solow, 2000, p. 149, argues), then an increase in A (the number of varieties of capital goods) increases labor productivity. The technical progress should be interpreted as reflecting the Smith-Ethier effect, i.e. an increase in the division of labor rises labor productivity. This formalizes the basic idea of increasing productivity due to division of labor leading to specialization. x2 70 A 60 50 40
B
30 α=0.5
20
α=1
10 C 10
20
30
40
50
60
70
x1
The preceding graph illustrates this reasoning. Consider the technology Y = xa1 + xa2 with 0 < a b 1. The total amount of inputs is x1 + x2 . Compare point A (or C) to B. The total amount of inputs at A is x1 + x2 = 0 + 64 = 64 and at B it is x1 + x2 = 32 + 32 = 64. However, despite this fact, output produced with the input combination at A is smaller than output produced with input combination B, provided that a < 1.
CAPITAL GOODS SECTOR The capital goods sector produces differentiated capital goods. As a technical (and legal) prerequisite for production, capital goods producer must at first purchase a blueprint (patent). The production function of the final output producers implies that the market for capital goods is monopolistically competitive. It is assumed that capital goods can be produced from final output "one by one". Consequently, aggregate capital obeys the following law of motion (d = 0) ° ° K = A x + A x° = Y - C
(5)
4
Romer_Model.nb
COMMENTS [1] Along the BGP x° = 0. [2] Notice that capital goods here are differentiated capital goods and not intermediate goods which are put together and finally make up the final output good [as in Grossman and Helpman (1991)]. Put differently, capital goods here are modelled as a stock and not as a flow (Solow, 2000, p. 147 uses the flow interpretation; however, Jones, 1995, p. 781, uses the phrase "total stock of producer durables"). Capital goods producers rent the producer durables to the FO sector. Hence, x is a stock here. The profit function of the typical capital goods producer is p = @ pHxL - rD x.
RESEARCH SECTOR Firms in the research sector conduct R&D, i.e. they search for new and economically valuable ideas (alternatively, the model can be interpreted in the sense that R&D is done within the capital goods firms). R&D firms produce blueprints for the production of new types of capital goods (designs). The market for designs is perfectly competitive. The research technology is of the following form ° A = h A LA
with h > 0
(6) °
°
d ln A d ln A Notice that there is a double knife-edge restriction implicit in this formulation: [1] ÅÅÅÅÅÅÅÅ ÅÅÅÅÅ = 1 and [2] ÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅ = 1. The first is d ln A d ln L A needed for sustained growth to be feasible. The second is required for a consistent microeconomic structure, i.e. a perfectly competitive market requires CRS in the single private input factor L.
Households As in the standard neoclassical model, households derive utility from consumption only and supply inelastically one unit of labor each period of time (labor supply is exogenous). Maximization of the present value of the infinite utility stream subject to a dynamic budget constraint leads to the familiar Keynes-Ramsey rule (KRR) of optimal consumption r-r ° C = C ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ s
with s, r > 0
(7)
where r is the net interest rate.
The balanced growth rate We now sketch the strategy how to solve for the balanced growth rate. To this end consider the production technologies Y = HA LY L1-a K a
(8)
° A = h A LA
(9)
The Y -technology indicates that, along the BGP, this model is equivalent to a neoclassical growth model with labor-aug` ` ` ` menting technical progress. Therefore, the following relations must hold along the BGP: g := Y = A = K = C. How can the balanced growth rate be determined? The growth rate of A may be expressed as ` A = h LA
(10)
`* This expression shows that L A must be constant along a BGP. The balanced growth rate is A = h L*A . To determine the constant allocation L*A it is necessary to analyze the market solution more deeply (see below). Here we simply report the resulting long-run growth rate, which is given by
Romer_Model.nb
5
ahL- r gM = ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ a+s
(11)
IMPLICATIONS [1] The long-run growth rate is completely determined by technology and preference parameters. Growth is endogenous in two senses: First, the growth rate results from the optimizing decisions of private agents (households and entrepreneurs). Second, the presence of h indicates that an increase in the productivity in R&D (equivalent to a reduction in R&D costs) stimulates growth. Hence, an R&D subsidy can affect the long-run growth rate, i.e. policy is effective. [2] The above displayed growth rate applies to the market solution. Since there are a number of "market failures" (positive spill-over effects and imperfect competition, which are essential for sustained endogenous growth to be feasible), the market solution does not coincide with the socially optimal solution. The socially optimal long-run growth rate is given by h L-r ÅÅÅÅÅÅ , where gS > gM . gS = ÅÅÅÅÅÅÅÅ s [3] The long-run growth rate exhibits a scale effect. The larger the economy (measured by L), the faster it grows. The economic intuition behind the scale effect is straightforward: The larger the amount of L, the higher is the amount of labor allocated to R&D and the higher is the long-run growth rate (g = h L A ). [4] The growth condition is a h L - r > 0. This implies that the economy must be sufficiently large in order to exhibit sustained growth (there is a threshold effect). Notice that in the original Romer (1990) formulation the relevant scale variable is human capital H rather than labor L.
References Romer, P. M., Endogenous technological change, Journal of Political Economy, 1990, 98, S71-S101.
3.3.2. The Romer (1990) model: The Microeconomic Structure Introduction We now consider the microeconomic structure of the model. The model is further simplified in that there is no physical capital. This has two consequences: [1] Differentiated capital goods (interpreted as a stock) are now intermediate goods (a flow); [2] there is no (predetermined) state variable in this model and hence no transitional dynamics. The basic reference is Gancia and Zilibotti (2005).
Households The economy is populated by infinitely lived agents who derive utility from consumption and supply inelasticaly labor. Population is constant and equal to L. The optimal consumption plan follows the familiar Keynes-Ramsey rule: r-r ° C = C ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ s There is no physical capital and savings are used to finance innovation.
(12)
6
Romer_Model.nb
Firms On the production side there are two sectors: [1] A competitive sector producing a homogenous final output good and [2] an imperfectly competitive sector producing differentiated intermediate goods (R&D is conducted within the intermediate good firms, hence there is no separate R&D sector). The final goods sector employs labor and a set of intermediate goods as inputs according to the following technology a Y = L1-a ‡ xHiL di Y A
(13)
0
where xHiL is the quantity of intermediate good i, A is the measure of intermediate goods available at each point in time, LY is labor allocated to Y -production, and 0 < a < 1. This production function is due to Ethier (1982). The intermediate goods sector consists of monopolistically competitive firms, each producing a different variety i. The production of one unit of xHiL requires one unit of Y . In addition, each intermediate goods producer is subject to a sunk cost to design a new intermediate input variety (i.e. R&D is conducted within the intermediate goods firm). New designs are produced without uncertainty. The innovating firm can patent the design (intellectual property rights are enforced) and thereby acquire a perpetual monopoly power over the production of the corresponding input. R&D uses only labor as private input. However, innovations generate intertemporal knowledge spill-overs. The production function for additional designs reads ° A = h A LA
(14)
where h > 0 and L A is labor devoted to R&D. Note that the rate of technical change is a linear function of the amount of labor employed in R&D. The amount of labor required to produce a unit measure of new designs requires a labor input of 1 ê Hh AL . Finally, feasibility requires that LY + L A § L
(15)
Static Equilibrium Prices: w is the wage rate, pHiL the price of xHiL, and pY = 1 (i.e. Y is chosen as the numeraire). Equilibrium in the Y -sector. The representative firm in the competitive Y -sector takes all prices as given and chooses production and technology so as to maximize profit, given by a pY = L1-a ‡ xHiL di - w LY - ‡ Y A
0
A
pHiL xHiL di
(16)
0
The first-order conditions yield the following (inverse) factor demands: xHiLa-1 pHiL = a L1-a Y
" i œ @0, AD
a w = H1 - aL L-a Y ‡ xHiL di
(17)
A
(18)
0
Equilibrium in the intermediate goods sector. A firm owning a patent sets its production level so as to maximize the profit subject to the demand function (17). The profit of the firm producing xHiL is
Romer_Model.nb
7
pHiL = pHiL xHiL - xHiL
(19)
The optimal price set by the monopolist and the associated quantity are (see appendix below) pHiL = p = 1 ê a
(20)
xHiL = x = a2êH1-aL LY
(21)
Hence, the maximum profit for an intermediate producer is 1-a pHiL = p = H p - 1L x = ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ a2êH1-aL LY a
(22)
Substitution of (21) into (18) yields the equilibrium wage rate w = H1 - aL a2 aêH1-aL A
(23)
Appendix: The problem of the intermediate goods producer In[3]:=
α Ly1−α xα−1 , xD êê Simplify
Solve@p
−1+α Ly−1+α p y z j 99x → i == z j α { k 1
Out[3]=
−1+α −1+α i Ly−1+α p y i Ly−1+α p y z z z z j j equ1 = DAp j −j , pE z z j j α α { { k k 1
In[7]:=
I Ly α
−1+α
Out[7]=
In[8]:= Out[8]=
M H−1 + p αL p H−1 + αL
p
1
0 êê FullSimplify
1 −1+α
0
Solve@equ1, pD 99p →
1 == α
Dynamic Equilibrium We guess and verify the existence of a balanced growth equilibrium such that C, Y and A grow at the same constant rate g, and LY and LA are constant. Note that this implies that along the BGP x and p are constant. Free entry into intermediate goods production implies that the present discounted value (PDV) of profits from innovation cannot exceed the entry costs, i.e. the costs of an additional design (w ê h A). By the KRR the interest rate is also constant along the BGP. Hence, the PDV of profits earned in the intermediate goods sector is p ê r. The free entry condition thus can be expressed as: w p ÅÅÅÅÅÅ § ÅÅÅÅÅÅÅÅÅÅÅ r hA
(24)
Using (22) and (23) this condition can be expressed as 1-a 2êH1-aL ÅÅÅÅ ÅÅÅÅÅÅ a LY H1 - aL a2 aêH1-aL a ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅ § ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ r h
(25)
8
Romer_Model.nb
The RHS is the marginal cost of innovation, which is independent of A due to the cancellation of two opposite effects: [1] Labor productivity, and hence equilibrium wages, grow linearly with A. [2] The productivity of researchers increases with A due to the intertemporal knowledge spill-over effect. Thus the unit cost of innovation is constant over time. Note that, without knowledge spill-overs, the cost of innovation would grow over time, and technical progress and growth would come to a halt, as in the neoclassical model. ° For A > 0 condition (25) must hold with equality. Using LY = L - LA and LA = g ê h (resulting from (14) and balanced growth) condition (25) can be expressed as r = aHh L - gL
(26)
This is the equilibrium condition on the production side, which says that the higher the interest rate that firms must pay to finance innovation expenditure, the lower is L A and hence growth. Finally, the KRR can be expressed as r= r+sg
(27)
which shows the usual relationship between r and the growth rate of consumption. Solving these two equilibrium conditions for g gives the interior solution haL- r g = ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ a+s
(28)
Provided that h a L - r > 0 , g is positive. Otherwise, L A = 0 and g = 0.
References Gancia and Zilibotti (2005), Horizontal Innovation in the Theory of Growth and Development, Handbook of Economic Growth.
