10 Horizontal Innovations and Model of Expanding Varieties (Romer, 1990)
10.1 Introduction
The next two chapters are devoted to models of endogenous technological change. We have seen in both Solow and Ramsey model that technical progress, left unexplained for now, is the main driver of economic growth once an economy as reached the steady state. Furthermore, as we have seen in the last previous chapters, the AK structure is particularly useful to provide sustained growth. We need therefore to understand how economic decisions lead to sustained growth and thus to open the “black box”. Two types of technological change is considered: horizontal and vertical innovations, which can be summarized as follows:
- horizontal innovations explain technological changes as appearing from the creation of new products
- vertical innovations look at technological changes as the result of improving quality of existing good (or decreasing cost of production).
While certainly not orthogonal, we discuss vertical innovations in the next chapter. The present chapter is devoted to horizontal innovations as introduced by Romer (1990).
The main ideas behind Romer (1990) model are:
technology is an important factor in production
technological progress is market outcome, i.e., it is endogenously generated
technology is a good with special features:
- technology is not rival, i.e., if it is created it can be used at zero cost any time after by everyone
- technology is at least partly excludable, i.e., one can restrict the access of others to its technology, to some extent (thus s/he can earn returns)
10.2 The Model
This is a multi-sector model of R&D-based endogenous growth
The sectors are:
- R&D sector that produces “blueprints” of new varieties/types of capital goods \(\dot{A}\). The R&D production uses \(L_{A}\) amount of total labour \(L\). The existing set of varieties \(A\) increases the productivity of the \(\mathrm{R} \& \mathrm{D}\) sector, i.e., positive knowledge externalities operate in the production of new blueprints, creating increasing returns in this sector
\[ \dot{A}=B A L_{A}, \quad B>0, \]
where \(B\) is the efficiency of “blueprint” creation. - An example could be the creation of wireless telephone while using the knowledge of transmitting information via radio waves and voice encoding.
Capital variety producing sector that uses the blueprints and produces intermediate capital goods for the final goods production. It is characterized by monopolistic competition. There is free-entry in the market of new blueprints. Entrepreneurs compete for patent that provides them with infinite-horizon property rights on a new blueprint. The acquisition of a patent allows an entrepreneur to employ exclusively the new blueprint and produce a distinct capital good thereafter. The production of capital goods/varieties requires investment in terms of the (foregone) final good. Romer motivates the price-setting assumption by entry (fixed) costs. Each and every capital good producing firm first buys (invests) the blueprint of capital good. It then enters to capital goods market and stays there forever. In capital goods market the firm has to have strictly positive profit streams in order to recover the entry cost. To have positive profits, it has to be a price setter. Moreover, in free entry equilibrium the investment cost equals to the value derived in the market, i.e., the present value of discounted profit streams.
Final good production sector where producers employ \(L_{Y}\) amount of total labor \(L\) and varieties ( defined as a set) of capital goods \(x(i), i \in[0, A]\) in the production, that is:
\[ Y=L_{Y}^{1-\alpha} \int_{0}^{A} x(i)^{\alpha} d i \]
These firms are fully competitive in input and output markets.
The final good is the numeraire and may be either consumed or invested.
- On the consumption-side, the representative household chooses its consumption and next period assets to maximize its lifetime utility \(U=\int_{0}^{\infty} u(C) e^{-\rho t} d t\), subject to the standard budget constraint.
We also assume that all capital varieties depreciate fully within one period and there is no growth in population, i.e., \(L\) is constant.
The following sections develop and solve the model.
This is the benchmark model. Two others versions show the robustness of the benchmark Romer (1990) model: the “lab-equipment” model where the final output is used to create new varieties and the “labor-for-intermediates” where labor is not embodied in the final good but is the unique input to produce the intermediate good. For sake of brevity, we redirect the reader toward Gancia and Zilibotti (2005).
10.3 Behaviors and Market equilibrium
10.3.1 Final goods production
The final good producers maximize their profits taking the price of their inputs, labour \((w)\) and capital goods/varieties \(\left(p_{x}(i), \forall i\right)\) as given. The problem of the representative final good producer is
\[ \max _{\{x(i)\}_{i \in[0, A]}, L_{Y}} L_{Y}^{1-\alpha} \int_{0}^{A} x(i)^{\alpha} d i-\int_{0}^{A} p_{x(i)} x(i) d i-w L_{Y} \]
The FOC are
\[ \begin{aligned} & \quad F_{L}=\frac{\partial Y}{\partial L_{Y}}=(1-\alpha) \frac{Y}{L_{Y}}=w, \\ & \quad F_{x(i)}=\frac{\partial Y}{\partial x(i)}=\alpha L_{Y}^{1-\alpha} x(i)^{\alpha-1}=p_{x}(i) ; \forall i, \end{aligned} \tag{10.1}\]
where the first expression describes the demand for labour and the second describes the demand for a capital good.
10.3.2 Individual variety
Each capital variety producer \(i\), within every period maximizes her profits \(\pi_{x}(i)\), by selecting the price \(p_{x}(i)\) and the quantity of production \(x(i)\). For every unit of capital that it produces it needs to invest one unit of final good that it “borrows” from households at the current price of final output (which is set to one, i.e., the final good is the numeraire), i.e., \(\pi_{x}(i)=p_{x}(i) x(i)-x(i)\). The firm takes as given the price of the output it uses in the production and the demand that it’s good is facing from the final good producers. Since the firm does not have dynamic constraints it’s problem is1
\[ \begin{aligned} & \max _{p_{x}(i), x(i)}\left\{\pi_{x}(i)=p_{x}(i) x(i)-x(i)\right\} \\ & \text { s.t. } \\ & p_{x}(i)=\alpha L_{Y}^{1-\alpha} x(i)^{\alpha-1} \end{aligned} \]
The optimal rule(s) are derived by plugging the inverse demand function of capital good to the profit function and taking the derivative with respect \(x(i)\), i.e., solve the following problem
\[ \begin{aligned} \max_{x(i)} \left\{\alpha L_{Y}^{1-\alpha} x(i)^{\alpha}-x(i)\right\} \\ \Rightarrow 1=\alpha^{2} L_{Y}^{1-\alpha} x(i)^{\alpha-1} \Rightarrow x(i)=\alpha^{\frac{2}{1-\alpha}} L_{Y} \end{aligned} \tag{10.2}\]
From (10.1) and (10.2), we derive:
\[ \begin{aligned} & p_{x}(i)=\alpha L_{Y}^{1-\alpha} x^{\alpha-1}=\frac{1}{\alpha} \Rightarrow \\ & \pi_{x}(i)=\left[p_{x}-1\right] x=\frac{1-\alpha}{\alpha} \alpha^{\frac{2}{1-\alpha}} L_{Y}>0 \end{aligned} \tag{10.3}\]
Given the symmetry across the different varieties in the final good production (as (10.2) does not depend on index \(i\)), the equilibrium implies that all capital good varieties producers will make the same optimal price and quantities choices, i.e., \(p_{x}(i)=p_{x}\) and \(x(i)=x \forall i\).
Note that the imperfect competition in the market equilibrium implies that the price of capital good is a constant mark-up \(\left(\frac{1-\alpha}{\alpha}\right)\) above its marginal cost, and the quantity of supplied capital good will be lower than the one selected in a perfectly competitive market (first-best outcome). Note also that in equilibrium the final output is linear in technology since \(Y=Y=L_{Y}^{1-\alpha} \int_{0}^{A} x^{\alpha} di\) for all \(i\) which implies \(Y =L_{Y}^{1-\alpha} A x^{\alpha}=\alpha^{\frac{2 \alpha}{1-\alpha}} L_{Y} A\), and the economy can experience long-run growth driven by technology progress, that here takes the form of expanding variety of capital goods.
10.3.3 Firm entry into capital goods market
The potential capital good producer in order to establish its firm competes with other potential producers in bidding for a new blueprint, where the blueprint is produced in fully competitive market. It makes an up-front (prior to entry) payment for the blueprint. In free entry equilibrium
this payment (cost of entry) is equal to the value derived by the firm in the capital market,
\[ V_{x}(t)=\int_{t}^{\infty} \pi_{x}(\tau) e^{-\int_{t}^{\tau} r(s) d s} d \tau \tag{10.4}\]
where \(t\) is the entry date and \(r(s)\) is the instantaneous real interest rate that the representative household earns on its asset holdings.
From (10.4), it follows that
\[ \begin{aligned} \dot{V}_{x}(t) & =-\pi_{x}(t) e^{-\int_{t}^{t} r(s) d s}+\int_{t}^{\infty} \pi_{x}(\tau) \frac{\partial}{\partial t} e^{-\int_{t}^{\tau} r(s) d s} d \tau \\ & =-\pi_{x}(t)+\int_{t}^{\infty} \pi_{x}(\tau)\left\{\frac{\partial}{\partial t}\left[-\int_{t}^{\tau} r(s) d s\right] \frac{\partial}{\partial\left[-\int_{t}^{\tau} r(s) d s\right]} e^{-\int_{t}^{\tau} r(s) d s}\right\} d \tau \\ & =-\pi_{x}(t)+r(t) V_{x}(t) \end{aligned} \]
which is standard Hamilton-Jacobi-Bellman equation. It can be rewritten as
\[ r V_{x}=\pi_{x}+\dot{V}_{x} \]
If \(V_{x}(t)\) is constant in equilibrium over time, then
\[ V_{x}(t)=\frac{\pi_{x}(t)}{r(t)} \]
This condition implies that at every point in time, the instantaneous excess of revenue over marginal cost must be just sufficient to cover the interest rate cost on the initial investment on a new blueprint. Another way of thinking this is that a household lends \(V_{x}\) to the entrepreneur for him to “buy” a blueprint and establish a firm and then receives in every period the “dividends” that equal to the per period profits.
Under the free-entry the value generated by the entry of a firm \(V_{x}(t) \dot{A}\) is equal to the cost of generating the blueprint \(w L_{A}\),
\[ V_{x}(t) \dot{A}=w L_{A} \tag{10.5}\]
10.3.4 The \(R \& D\) sector
Any blueprint is owned by a capital good producer which has a value \(V_{x}(t)\). Thus, the price/value of a “blueprint” is \(V_{x}(t)\) and the problem of a “blueprint” producer is
\[ \max_{L_{A}} V_{x}(t) \underbrace{B A L_{A}}_{=\dot{A}(t)}-w L_{A} \]
Assuming fully competitive market with free-entry and zero-profit conditions, we get from ( 10.5):
\[ w(t)=V_{x}(t) B A(t) \]
10.3.5 Labour market
The labour market equilibrium needs to guarantee that labor is freely mobile between the final good and R&D sectors, i.e. the value of the marginal product of labour is equated in these two sectors. Therefore, the wage rate needs to be equal in both sectors:
\[ w(t)=V_{x}(t) B A(t)=(1-\alpha) \frac{Y(t)}{L_{Y}}=\alpha^{\frac{2 \alpha}{1-\alpha}}(1-\alpha) A(t) \tag{10.6}\]
Thus,
\[ V_{x}(t)=\frac{\alpha^{\frac{2 \alpha}{1-\alpha}}(1-\alpha)}{B} \equiv V_{x} \]
Therefore, indeed \(V_{x}=\frac{\pi_{x}}{r}\).
From (10.3) and (10.7) it also follows that:
\[ \begin{aligned} w & =V_{x}(t) B A(t)=\frac{\pi_{x}}{r} B A(t) \\ & =\frac{1-\alpha}{\alpha} \alpha^{\frac{2}{1-\alpha}} L_{Y} \frac{1}{r} B A(t) \frac{Y(t)}{Y(t)} \\ & =\alpha(1-\alpha) \frac{1}{r} B Y(t) . \end{aligned} \]
Plugging back this expression into (10.7), we get: \[ L_{Y}=\frac{r}{\alpha B},\quad L_A=L-L_Y \tag{10.7}\]
10.4 The household side
From the standard household intertemporal maximization problem, it follows that its consumption over time follows the path
\[ \frac{\dot{C}(t)}{C(t)}=\frac{1}{\theta}(r(t)-\rho) \tag{10.8}\]
The standard transversality condition ensures that the value of the asset holdings of the households is equal to zero in the limit, i.e., the growth of the assets does not exceed the real interest rate.
10.5 Balanced growth path
All variables of the model need to grow at constant rates (BGP).
For \(g_{A} \equiv \frac{\dot{A}(t)}{A(t)}=BL_A\), to be constant in equilibrium, it must be that the allocation of labour in the research and final good sector are constant over time, i.e., \(\dot{L}_{Y}=0\). From (10.8), it follows that the interest rate should be constant on BGP and \(g_{C} \equiv \frac{\dot{C}(t)}{C(t)}=\frac{1}{\theta}(r-\rho)\). Therefore, from (10.7) it follows that \(\dot{L}_{Y}=0\).
From the FOCs of the intermediate capital good producers, it follows that \(x\) is time invariant, implying that aggregate capital stock available in the economy at every point in time \(K=A(t) x\) grows at rate \(g_{K}=g_{A}\). The economy grows due to capital-deepening which is entirely driven by the expansion of capital varieties.
The production function of final goods \(Y(t)=\alpha^{\frac{2 \alpha}{1-\alpha}} L_{Y} A(t)\) implies that along the BGP \(g_{Y}=g_{A}\).
Households’ total assets are \(b(t)=V(t)A(t)\equiv V_xA(t) \Rightarrow \dot{b}(t)=V_x\dot{A}(t)\). From households’ budget constraint, the law of motion of assets is defined by
\[ \dot{b}(t)=r(t) b(t)+w(t) L-C(t) \]
Noting that \(rb(t)=rV_xA(t)=\pi_x A(t)= \frac{(1-\alpha)Y(t)}{\alpha}\) and \(w(t)L=\frac{(1-\alpha)Y}{L_Y}\), we have:
\[ \begin{aligned} \dot{b}(t)\equiv V_x\dot{A}(t)= & rV_xA(t)+\frac{(1-\alpha)Y}{L_Y}L-C(t) \\ = & \frac{(1-\alpha)Y(t)}{\alpha}+\frac{(1-\alpha)Y}{L_Y}L-C \\ = & \left(\frac{1}{\alpha}+\frac{L}{L_Y} \right)(1-\alpha)Y(t)-C(t) \\ \Rightarrow g_A= & \left(\frac{1}{\alpha}+\frac{L}{L_Y} \right)(1-\alpha)\frac{Y(t)}{V_x A(t)}-\frac{C(t)}{V_x A(t)} \end{aligned} \]
Thus, \(g_{C}=g_{A}=g_{Y}=g_{K} \equiv g\).
Note that in equilibrium, there is a positive relation between growth and the real interest rate, as implied by the Euler equation (10.8), but a negative one implied by the / production-side, as
\[ g_{A}=B\left(L-L_{Y}\right)=B\left(L-\frac{r}{\alpha B}\right) \tag{10.9}\]
The latter is due to the fact that higher interest rate reduces the present discounted value of any capital variety firm. By doing so, it reduces the market incentives to direct (labour) resources away from the final good production into the production of new assets/savings instruments. Along the unique BGP, these two forces are equated implying a unique real interest rate and labour allocation. These two can be derived by equating the (10.8) and (10.9)
\[ \begin{gathered} \frac{1}{\theta}(r-\rho)=B\left(L-\frac{r}{\alpha B}\right) \Rightarrow \\ r=\frac{\alpha}{\alpha+\theta}(\theta B L+\rho) \\ \Rightarrow L_{Y}=\frac{\theta B L+\rho}{(\alpha+\theta) B} \end{gathered} \] and therefore
\[ g=\frac{r-\rho}{\theta}=\frac{\alpha B L-\rho}{\alpha+\theta} \]
The condition for positive long-run growth sets a minimum bound on the scale of the economy: \(L>\frac{\rho}{B \theta}\). Note that the transversality condition is always satisfied under this condition, i.e., \(r>g_{A}\).
10.6 Comparative statics
- Increase in \(B\) and \(L\) increase \(g\), by reducing \(L_{Y}\)
- Increase in \(\theta\) or \(\rho\) decreases \(g\), by increasing \(L_{Y}\)
- Increase in \(\alpha\) also increases \(g\), by reducing \(L_{Y}\)
10.7 Kaldor stylized facts and the first models of endogenous growth
- \(Y / L=A x^{\alpha}\) increases at a rate \(g\)
- \(K / L=A \frac{x}{L}\) also increases at a rate \(g\)
- \(Y / K\) is constant
- The real interest rate \(r\) is constant
- The wage rate \(w=(1-\alpha) \frac{Y}{L_{Y}}=(1-\alpha)\left(\frac{x}{L_{Y}}\right)^{\alpha} A\) increases at rate \(g\)
- Growth rates across countries differ in the long-run due to technology and preference parameters. Long-run growth takes place due to the endogenous expansion of the capital-varieties. Short-episodes of fast growth are due to changes in the underlying parameters and/or the scale of the economy.
The model predicts that there is no convergence in terms of GDP per capita.
10.8 Social Planner’s problem
In the decentralized market equilibrium there are two sources of inefficiency that drive the equilibrium growth outcome away from the first-best:
Monopoly rights/patents
Positive knowledge externalities in the production of new blueprints of capital goods
The social planner faces the following optimal control problem, where the state of the economy is summarized by \(A\)
\[ \begin{aligned} & \max _{x(i), L_{Y}, C} \int_{0}^{\infty} \frac{C^{1-\theta}-1}{1-\theta} e^{-\rho t} d t \\ & \text { s.t. } \\ & L_{Y}^{1-\alpha} \int_{0}^{A} x(i)^{\alpha} d i=C+\int_{0}^{A} x(i) d i \\ & \dot{A}=B\left(L-L_{Y}\right) A \\ & A(0)>0 \text { given. } \end{aligned} \]
where the second equation is the resource constraint. The third term in resource constraint is the available capital stock. In this model the capital depreciates in a period; thus, the investment equals to the capital stock.
Let \(\lambda_{K}\) and \(\lambda_{A}\) denote the shadow prices of “capital” and “knowledge” respectively. The optimal rules imply the following conditions that govern equilibrium in every point in time
\[ \begin{aligned} & C^{-\theta}=\lambda_{K} \\ & x(i)=\alpha^{\frac{1}{1-\alpha}} L_{Y}, \forall i \\ & \lambda_{K}(1-\alpha)\left(\frac{x}{L_{Y}}\right)^{\alpha}=B \lambda_{A} \\ & \dot{\lambda}_{A}=\rho\lambda_{A} -\lambda_{K}\left[L_{Y}^{1-\alpha} x^{\alpha}-x\right]-\lambda_{A} B\left(L-L_{Y}\right) \end{aligned} \]
The first equation shows that the social planner would choose to produce to the point that the marginal product of each capital variety equates its marginal cost (one unit of output), hence \(x^{S P}>x\). The second equates the value of the marginal product of labour in the final good and \(\mathrm{R} \& \mathrm{D}\) production. It implies that over time, the rates of returns are equated, i.e., \(-\frac{\dot{\lambda}_{K}} {\lambda_{K}}=-\frac{\dot{\lambda}_{A}}{\lambda_{A}}\). From the optimal rule for consumption then follows that
\[ \frac{\dot{C}}{C}=-\frac{1}{\theta} \frac{\dot{\lambda}_{K}}{\lambda_{K}}=-\frac{1}{\theta} \frac{\dot{\lambda}_{A}}{\lambda_{A}} \] where \(\frac{\dot{\lambda}_{A}}{\lambda_{A}}\) can be derived from the system above:
\[ \begin{aligned} \frac{\dot{\lambda}_{A}}{\lambda_{A}} & =\rho-\frac{\lambda_{K}}{\lambda_{A}}\left[L_{Y}^{1-\alpha}\left(\alpha^{\frac{1}{1-\alpha}} L_{Y}\right)^{\alpha}-\alpha^{\frac{1}{1-\alpha}} L_{Y}\right]-B\left(L-L_{Y}\right) \\ & =\rho-\frac{\lambda_{K}}{\lambda_{A}}\left[L_{Y} \alpha^{\frac{\alpha}{1-\alpha}}-\alpha^{\frac{1}{1-\alpha}} L_{Y}\right]-B\left(L-L_{Y}\right) \\ & =\rho-B\left[(1-\alpha)\left(\frac{x}{L_{Y}}\right)^{\alpha}\right]^{-1} L_{Y} \alpha^{\frac{\alpha}{1-\alpha}}(1-\alpha)-B\left(L-L_{Y}\right) \\ & =\rho-B\left[\left(\frac{x}{L_{Y}}\right)^{\alpha}\right]^{-1} L_{Y} \alpha^{\frac{\alpha}{1-\alpha}}-B\left(L-L_{Y}\right) \\ & =\rho-B \alpha^{-\frac{\alpha}{1-\alpha}} L_{Y} \alpha^{\frac{\alpha}{1-\alpha}}-B\left(L-L_{Y}\right) \\ & =\rho-B L . \end{aligned} \]
10.8.1 Balanced growth path
From the blueprint production follows that \(L_{Y}\) is constant on BGP. Given this, the \(x\) is also constant. Therefore, from resource constraint follows that the macroeconomic aggregates grow at the same constant rate on balanced growth path. Denote that constant growth rate by \(g^{SP}\).
The socially optimal long-run growth is given by
\[ g^{S P}=\frac{B L-\rho}{\theta} . \]
Since \(\alpha \in(0,1)\) the \(g^{S P}>g\),i.e.,
\[ \frac{B L-\rho}{\theta}>\frac{\alpha B L-\rho}{\alpha+\theta} \]
Moreover, since \(g^{S P}>g\) from the “blueprint” production follows that \(L_{Y}^{S P}<L_{Y}\), i.e., the social planner would allocate more of the labour resources into the production of R&D, since the latter is the engine of growth. Therefore, the growth promoting policies increase the incentives to innovate by subsidies to the production of \(R \& D\) (e.g., subsidies to the employment of labour in R&D) that will make the firms internalize the knowledge externalities they generate by each new variety that they discover. The distortion of the imperfect competition may be alleviated by a subsidy to the purchases of the capital goods and/ or subsidies to the production of final output that would increase the demand of capital goods. These policies would finance those subsidies by lump-sum taxes on household.