11  Vertical innovations and Model of Quality Ladders (Grossman-Helpman ,1991; Aghion and Howitt 1992)

11.1 Introduction

In this chapter, we present another version of technological change modelling, i.e. vertical innovations. Instead of seeing new varieties as the result of innovations (e.g. development of DVD products), vertical innovations explains the increase in quality of existing goods (e.g. from DVD to Blu-Ray). This has led to the so-called Schumpeterian Growth literature as the augmented-quality goods replace existing goods.

11.2 The Model

  • A multi-sector model of R&D-based endogenous growth that is driven by “creative destruction.”
  • There are two factors of production: a fixed amount of labour and fixed number, \(N\), of capital good types. Within each variety \(j\), capital goods differ in their quality.
  • Qualities are of distance \(q>1\) of each other. The best quality within every sector \(j\) is \(q^{\kappa_{j}}\), where \(\kappa_{j} \in \mathbb{N} \cup\{0\}\). The initial quality is normalized to one \(\left(\left.\kappa_{j}\right|_{\kappa=0}=1\right)\).
  • The R&D sector produces “blueprints” for improved quality capital goods of each known variety. The input to the \(\mathrm{R} \& \mathrm{D}\) production is investment in units of the final output. For the variety \(j\) of quality \(\kappa_{j}\), the \(\mathrm{R} \& \mathrm{D}\) expenditures are \(Z_{j \kappa_{j}}\). The output of the \(\mathrm{R} \& \mathrm{D}\) production is uncertain. The \(\mathrm{R} \& \mathrm{D}\) expenditures result in the new variety \(\left(\kappa_{j}+1\right)\) with probability \(p_{j \kappa_{j}}\). The technology of \(\mathrm{R} \& \mathrm{D}\) production is linear in \(\mathrm{R} \& \mathrm{D}\) investment,

\[ p_{j \kappa_{j}}=\phi\left(\kappa_{j}\right) Z_{j \kappa_{j}} \tag{11.1}\]

As quality improves, new discoveries become more expensive in terms of the required investment of resources, i.e., \(\phi^{\prime}\left(\kappa_{j}\right)<0\) gives diminishing returns to R&D input. As the probability depends only on the current quality level, it suggests that innovation occurs like a Poisson process.

Note

Linearity implies absence of congestion. Innovation in each sector is “jumpy” (takes place in a discreet manner), however the existence of many sectors and the Law of Large Numbers ensures a smooth outcome at the aggregate level.

  • The discovery of a better quality capital good of a particular variety provides an entrepreneur with monopoly rights over the use of the “blueprint.” He produces the distinct capital variety with a linear technology that transforms one unit of final output into one unit of capital good.
  • There is free entry into the capital goods industry.
  • The final good sector operates under perfect competition. It combines labour, \(L\), with qualityadjusted input \(\tilde{X}_{j}\) of every variety \(j\) of the existent (fixed) set of capital varieties, \(j \in\{1, \ldots, N\}\), i.e.,

\[ Y=A L^{1-\alpha} \sum_{j=1}^{N} \tilde{X}_{j}^{\alpha} \]

There is additive separability in all varieties of capital and all of them are used in the final good production due to standard neoclassical function assumptions (Inada conditions). - Different qualities of capital goods of a variety \(j\) are perfect substitutes of each other. Hence, if \(\kappa_{j}\) is the best quality known, the total input employed in the final good sector of the \(j\) th capital good variety is:

\[ \tilde{X}_{j}=\sum_{k=0}^{\kappa_{j}} q^{k} x_{j k} \] - It is assumed here that only the highest quality capital good survives of each variety of capital goods, hence

\[ \tilde{X}_{j}=q^{\kappa_{j}} x_{j \kappa_{j}} \]

This “guess” regarding the properties of the equilibrium path is to be verified below, by examining the conditions that support it.

Note: The survival of the best quality only across sectors means that the model features technological obsolescence. This results in that the decision to conduct research in order to invent a better quality capital good is based on two forces. First, the discovery is to be overtaken by another researcher in the future (this decreases incentives for research). Second, the discovery of an improved quality capital good implies that there will be a transfer of the monopoly rent from the previous best-quality discovery owner (this increases the incentives for research). - From the consumption-side, the representative HH chooses its consumption and assets to maximize its intertemporal utility: \(\int_{0}^{\infty} u(C) e^{-\rho t} d t\) subject to the standard budget constraint.

  • All capital goods depreciate fully within one period.
  • No population growth.

11.3 Market equilibrium

11.3.1 Final goods production

The final good producers maximize their profits taking the price of their inputs, labour \((w)\) and capital goods \(\left(P_{j} \kappa_{j}, \forall j\right)\) as given. Standard conditions imply that:

\[ \begin{aligned} F_{L} & =\frac{\partial Y}{\partial L}=(1-\alpha) \frac{Y}{L}=w \\ F_{X_{j \kappa_{j}}} & =\frac{\partial Y}{\partial x_{j \kappa_{j}}}=A \alpha L^{1-\alpha} x_{j \kappa_{j}}^{\alpha-1} q^{\alpha \kappa_{j}}=P_{j} \kappa_{j} ; \forall j \end{aligned} \]

11.3.2 Capital goods production

Each capital good producer, within every period maximizes its profits, \(\pi_{j \kappa_{j}}\), by selecting its price, \(P_{j} \kappa_{j}\), and quantity of production, \(x_{j \kappa_{j}}\). For every unit of capital that it produces, it needs to invest one unit of final good that it “borrows” from HH at the current output price, i.e.,

\[ \pi_{j \kappa_{j}}=P_{j} \kappa_{j} x_{j \kappa_{j}}-x_{j \kappa_{j}} \]

The monopolist takes as given the price of the output it uses in its production and the demand that its good is facing from the final good producers, i.e., its problem is

\[ \begin{aligned} & \max _{P_{j} \kappa_{j}, X_{j \kappa_{j}}} \pi_{j \kappa_{j}} \\ & \text { s.t. } \\ & P_{j} \kappa_{j}=A \alpha L^{1-\alpha} x_{j \kappa_{j}}^{\alpha-1} q^{\alpha \kappa_{j}} \end{aligned} \]

The FOCs of this optimal program imply

\[ \begin{aligned} x_{j \kappa_{j}} & =L A^{\frac{1}{1-\alpha}} Q^{\frac{2}{1-\alpha}} q^{\frac{\alpha}{1-\alpha} \kappa_{j}} \\ P_{j \kappa_{j}} & =\frac{1}{\alpha} \end{aligned} \]

:::{.callout-note} It was assumed that only the best available quality is available within each type of capital good. Suppose instead that this was not the case and the second-best quality is also available. The marginal product of any two consecutive qualities differs by their quality difference, i.e., by a factor of \(q\), which means that the price differential supported by the market equilibrium between the first and second highest quality is: \(\frac{P_{j \kappa_{j}}}{P_{j \kappa_{j}-1}}=q\). Therefore, the monopoly producer of the second highest quality can at most charge \(P_{j \kappa_{j}-1}=\frac{1}{\alpha q}\). If \(\alpha q>1\), then the second best quality producer cannot cover with such price its marginal cost of production and is driven out of the market. Alternatively, if \(\alpha q<1\), the result that only the leading technology survives within each variety can still be an equilibrium outcome, where the leader follows limit pricing. In such case it charges \(P_{j} \kappa_{j}=q-\varepsilon\) \((\varepsilon \rightarrow+0)\) and the next quality producer then could charge \(1-\frac{\varepsilon}{q}<1\). :::

With the leading technology only, then the final output production in equilibrium is described by

\[ \begin{aligned} Y & =A L^{1-\alpha} \sum_{j=1}^{N} q^{\alpha \kappa_{j}} x_{j \kappa_{j}}^{\alpha} \\ & =L A^{\frac{1}{1-\alpha}} \alpha^{\frac{2 \alpha}{1-\alpha}} \sum_{j=1}^{N} q^{\frac{\alpha}{1-\alpha} \kappa_{j}} \end{aligned} \]

11.3.3 the \(R\& D\) Sector

Given the equilibrium outcome of the capital goods, labour and final output market, the next step is to examine the decision of the entrepreneurs to conduct \(\mathrm{R} \& \mathrm{D}\) for the discovery of the \(\kappa_{j+1}\) quality of every type of capital good. Access to the market is free, therefore, every entrepreneur should be in the limit equating his cost in investing in \(\mathrm{R} \& \mathrm{D}\) with his expected profit.

The cost of \(\mathrm{R} \& \mathrm{D}\) is the investment in terms of current output, \(Z_{j \kappa_{j}}\). The probability of a successful innovation of the quality \(\kappa_{j}+1\) is \(p_{j \kappa_{j}}\) while the expected value from it is \(V_{j \kappa_{j}+1}\). More rigorously, \(V_{j \kappa_{j}+1}\) is the expected present value of the profit flows of the producer of \(\kappa_{j}+1\), which within every period is \(\pi_{j \kappa_{j}+1}=\frac{1-\alpha}{\alpha} x_{j \kappa_{j}+1}\), until its position is overtaken by the discovery of the next quality of the same type of capital good. The latter depends on the probability of the discovery of the next higher quality, \(p_{j \kappa_{j}+1}\). Therefore, the successful innovator of the \(\kappa_{j}+1\) quality has \(V_{j \kappa_{j}+1}\) that satisfies in equilibrium:

\[ \begin{aligned} r V_{j \kappa_{j}+1} & =\pi_{j \kappa_{j}+1}-p_{j \kappa_{j}+1} V_{j \kappa_{j}+1} \\ & \Longrightarrow V_{j \kappa_{j}+1}=\frac{\pi_{j \kappa_{j}+1}}{r+p_{j \kappa_{j+1}}} \end{aligned} \tag{11.2}\]

The last condition is essentially an arbitrage condition. The entrepreneur should be indifferent between lending \(V_{j \kappa_{j}+1}\) units of output and earning the market interest rate, i.e. \(r V_{j \kappa_{j}+1}\) and holding the firm that provides him with a profit flow, \(\pi_{j \kappa_{j}+1}\), and there is a probability that at the end of the period he loses the value of its firm because it is overtaken by a new higher quality product (i.e., its capital loss will be \(-V_{j \kappa_{j}+1}\) ).

Therefore, given that there is free entry into capital goods sector in equilibrium

\[ p_{j \kappa_{j}} V_{j \kappa_{j}+1}=Z_{j \kappa_{j}} \]

From this condition and the \(R \& D\) production function (11.1) together with (11.2) it follows that

\[ p_{j \kappa_{j}+1}=\phi\left(\kappa_{j}\right) \pi_{j \kappa_{j}+1}-r \]

Therefore, the equilibrium probability of a new innovation is influenced by two forces

  1. A higher quality of a type of capital variety implies higher demand for this type of capital and thereby higher profits for the successful innovator.

  2. The marginal cost of discovering a higher quality capital variety increases.

When the positive effect dominates, newer sectors grow faster than older ones, i.e. there are increasing returns to scale. When the negative effect dominates there is convergence to à la Ramsey. When the two effects balance each other out (i.e. when there are CRS), there is balanced growth in all sectors.

11.4 Balanced growth path

All variables of the model need to grow at constant rates (BGP). A specification for \(\phi\left(\kappa_{j}\right)\) that ensures balanced growth is the following

\[ \phi\left(\kappa_{j}\right)=\frac{1}{\zeta} q^{-\left(\kappa_{j}+1\right) \frac{\alpha}{1-\alpha}} \]

This implies that the free-entry condition boils down to

\[ \frac{1}{\zeta} \frac{1-\alpha}{\alpha} L A^{\frac{1}{1-\alpha}} \alpha^{\frac{2}{1-\alpha}}=r+p_{j \kappa_{j}+1}=r+p \]

Note that this specification was chosen in order to eliminate the asymmetry across the different sectors of the economy, given that it is sufficient for a constant probability of new innovations taking place across all sectors. The equilibrium R&D investment is

\[ Z_{j \kappa_{j}}=\frac{p}{\frac{1}{\zeta} q^{-\left(\kappa_{j}+1\right) \frac{\alpha}{1-\alpha}}}=q^{\left(\kappa_{j}+1\right) \frac{\alpha}{1-\alpha}}\left(\frac{1-\alpha}{\alpha} L A^{\frac{1}{1-\alpha}} \alpha^{\frac{2}{1-\alpha}}-r \zeta\right) \]

Note that there are scale effects in this model, as larger sectors spend more on R&D.

Define the aggregate quality index as

\[ Q \equiv \sum_{j=1}^{N} q^{\kappa_{j} \frac{\alpha}{1-\alpha}} \]

Then, the aggregate output and quantity of capital goods are given by

\[ \begin{aligned} & Y=L A^{\frac{1}{1-\alpha}} \alpha^{\frac{2 \alpha}{1-\alpha}} Q \\ & X=\sum_{j=1}^{N} X_{j \kappa_{j}}=L A^{\frac{1}{1-\alpha}} \alpha^{\frac{2}{1-\alpha}} Q \end{aligned} \]

Aggregate R&D expenditures are

\[ Z=\sum_{j=1}^{N} Z_{j \kappa_{j}}=\left(\frac{1-\alpha}{\alpha} L A^{\frac{1}{1-\alpha}} \alpha^{\frac{2}{1-\alpha}}-r \zeta\right) q^{\frac{\alpha}{1-\alpha}} Q \]

Aggregate output, capital and \(\mathrm{R} \& \mathrm{D}\) expenditures are proportional to the quality index \(Q\), which is itself function of time. Therefore, in steady-state

\[ g_{Y}=g_{Z}=g_{X}=g_{Q} \equiv g \]

The resource constraint of this economy is

\[ Y=C+X+Z \]

implying that in the steady-state it is also true that

\[ g_{C}=g \]

On average in the economy in all different capital goods “industries”, at every point in time, an innovation of a higher quality capital good takes place with probability \(p\). Therefore, for a large \(N\) the Law of Large Numbers provides with the growth rate

\[ g^{*} \approx E\left(\frac{\Delta Q}{Q}\right)=\frac{\sum_{j=1}^{N} p\left(q^{\left(\kappa_{j}+1\right) \frac{\alpha}{1-\alpha}}-q^{\kappa_{j} \frac{\alpha}{1-\alpha}}\right)}{\sum_{j=1}^{N} q^{\kappa_{j} \frac{\alpha}{1-\alpha}}}=p\left(q^{\frac{\alpha}{1-\alpha}}-1\right) \]

This growth rate is implied from the production-side of the economy. It is negatively related to \(r\) since \(p=\frac{1}{\zeta} \frac{1-\alpha}{\alpha} L A^{\frac{1}{1-\alpha}} \alpha^{\frac{2}{1-\alpha}}-r\). On the other hand, the standard optimization condition for the representative household gives a positive relation between the growth rate of the economy and the real return on assets, \(g=\frac{1}{\theta}(r-\rho)\). Therefore, the equilibrium interest rate and growth rate are

\[ \begin{aligned} r & =\theta p\left(q^{\frac{\alpha}{1-\alpha}}-1\right)+\rho \\ & =\frac{\theta\left(q^{\frac{\alpha}{1-\alpha}}-1\right) \frac{1}{\zeta} \frac{1-\alpha}{\alpha} L A^{\frac{1}{1-\alpha}} \alpha^{\frac{2}{1-\alpha}}+\rho}{1+\theta\left(q^{\frac{\alpha}{1-\alpha}}-1\right)} \\ g & =\frac{1}{\theta}\left(r^{*}-\rho\right) \end{aligned} \]

TVC is satisfied for parameters that ensure \(r>g\).

11.5 Further comments

The decentralized equilibrium in this model does not achieve first best allocations due to the following distortions

  1. Monopoly pricing in the capital goods sector,

  2. Obsolescence of the lower quality capital goods.

The first distortion implies that there is low investment and growth in the economy. The second implies that when a new innovation is made, then the successful innovator takes over the profits made by the previous leader in the particular type of capital variety. This “rat race” effect boosts the incentive to conduct \(R \& D\) for the purpose of product innovation. At the same time though, for the same potential innovator there is positive probability that himself will be overtaken by the next discovery, which reduces the incentives for R&D. Because of discounting, as current profits matter more than future ones, the outcome is that in equilibrium there is more than optimal \(R \& D\) investment, which would tend to increase growth. The net effect of the two distortions is ambiguous.

The decentralized growth rate will be equal to the social planner’s one when relative prices are corrected and the successful innovators compensate their immediate predecessors for the loss of their monopoly rents. In the case that all innovations are conducted by the leaders in each sector, i.e. when there is no need for compensation, equilibrium \(R \& D\) would be too low. If instead relative prices are corrected, but no compensation is offered, then \(R \& D\) would be too high.

Contrast with Romer (1990) The horizontal expansion in the capital good varieties model may be better suited for large-scale inventions (e.g. the ones implying the establishment of a new industry). The vertical expansion of every variety of capital goods is better suited to account for the smaller and gradual improvements in the quality of the capital goods.

On the one hand, both of these endogenous growth models share key features and thereby predictions. In both models, the production-side equilibrium implies that the interest rate has a negative growth effect as it reduces the present discounted value of the expected profits from innovations. Also, both models “suffer” from scale-effects. The engine of long-run growth is the R&D production that is conducted given market-based incentives. Noteworthy, the organizational capital/institutional level of the economy, as captured by the term \(A\) in the aggregate production function, has not only a level effect (as in Solow), but a permanent growth effect.

On the other hand, there are important differences in their specifications. Romer’s model lacks the feature of creative destruction as it assumes that the different varieties are not direct substitutes or complements. As a result, in the decentralized equilibrium the R&D effort cannot be too high.