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2022 | OriginalPaper | Buchkapitel

7. Additional Endogenous Growth Models

verfasst von : Alfonso Novales, Esther Fernández, Jesús Ruiz

Erschienen in: Economic Growth

Verlag: Springer Berlin Heidelberg

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Abstract

We present in this chapter some additional mechanisms by which endogenous growth arises. We start with an economy in which technological progress shows up in the form of the number of varieties of products. After that, a model of technological diffusion between two countries, a leader in innovation and a follower, that adopts the innovations developed in the leading country. We then present a model economy with creative destruction à la Schumpeter in which growth is driven endogenously by attempts to improve the quality of existing goods through innovation. We close with an important model, that of a two-sector economy in which human and physical capital accumulate over time, and where time devoted to education plays an important role.

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Fußnoten
1
Other models consider endogenous growth in economies with a variety of consumer products. Since the treatment is relatively similar to that of models with a variety of products, we do not include those models here.
 
2
Previous studies by Spence [16], Dixit and Stiglitz [6] and Ethier [7] all consider the benefits of a variety of products. Spence [16] and Dixit and Stiglitz [6] considered a utility function defined on the set of consumption commodities as arguments, while Ethier [7] used a setup similar to that in Romer [12, 13]. The latter included a variety of productive inputs in the context of technological change and economic growth.
 
3
This is a simplifying assumption, equivalent to having a set of identical firms producing the final good.
 
4
The number of varieties will grow at a rate \(\gamma _{N_{t}}\) from its initial value N 0. This means that we have to consider N t to be a continuous variable taking values on the positive real line. As an exception, we will use the N-notation for the number of intermediate goods in an economy with constant population L.
 
5
Alternatively, x jt could be considered as the flow of services provided by a vector of durable intermediate goods subject to some depreciation. But we would then have to keep track of the quantities of each of intermediate good available at the firm each time period, which complicates the analysis significantly.
 
6
See Chap. 6 in Barro and Sala-i-Martin [3] for a model with random duration of monopoly rights.
 
7
If V t < η, then no resources would be devoted to further invention, and the number of goods would remain constant over time. On the other hand, if V t > η, then investing in R&D provides positive profits, so an infinite amount of resources would be devoted to that activity, and we could not possibly have an equilibrium.
 
8
The constant labour supply of L, together with the assumption of a unit endowment of labour for each consumer which is supplied inelastically, imply that we are dealing with a constant population. Per-capita and aggregate consumption then grow at the same rate, since \(\tilde {C}_{t}=L\tilde {c}_{t}\), and the same is true for per-capita and aggregate output, since \(\tilde {Y} _{t}=L\tilde {y}_{t}\).
 
9
We denote by asterisks the solution to the benevolent planner’s problem.
 
10
This is clearly a restrictive assumption, which neglects any consideration regarding the appropriate treatment of uncertainty under risk aversion.
 
11
A similar ratio was used as auxiliary variable when solving the discrete-time AK model.
 
12
Using the equality: \(1+\frac {z}{1+\gamma }=\frac {1+A^{\prime }(1+\alpha )\alpha ^{\frac {\alpha }{1-\alpha }}}{1+\gamma }\).
 
13
The cost of imitation, υ 2,t, could be allowed to exceed from η 2 even when N 2,t−1 < N 1,t−1 to capture a situation in which the yet uncopied goods from country 1 are hard to adapt for use in country 2. The proposed function does not allow for that possibility.
 
14
Interest rates may be different in both countries due to the fact that we do not consider international borrowing and lending.
 
15
It is also straightforward to adapt the program to compute impulse responses to a shock in the follower country. The reader will see that the leader country does not react to such a shock, and that responses in the follower country are as expected.
 
16
Our presentation follows Howitt and Aghion [9], in discrete time and in a stochastic setup.
 
17
Note that the natural definition involves the ratio of physical capital by the level of productivity, both variables taken at the beginning of the period.
 
18
Using the steady state version of (7.42), we see that any value of λ such that: \(\lambda \left ( Mn\right ) ^{b}<1\) in the definition of the probability of research success is admissible.
 
19
Remember that any variable x can be represented by \(e^{\ln x}\).
 
20
When, as in this model, an analytical proof does not exist, this supposed eigenvalue structure needs to be explored numerically, and may hold only in some region of the parameter space.
 
21
Since we assume a constant number of consumers, we can equivalently use per-capita or aggregate consumption.
 
22
That is, the socially available stock of human capital.
 
23
Without loss of generality, we will assume a production function in per capita terms.
 
24
We do not consider the experience in the job position as a way to obtain qualified work.
 
25
We assume that the consumer pays taxes on capital and labour rents obtained from the sector producing the final consumption good.
 
26
Alternatively, we could consider a random ratio of government expenditures to output ξ t = ξ + ε 3t, \(\varepsilon _{3t}\sim N(0,\sigma _{3}^{2})\), with \(\sigma _{3}^{2}\) small enough so that ξ t would fluctuate inside the (0,1) interval with probability one. That would introduce an additional source of randomness that could be interpreted as a possible error in controlling the level of government expenditures. We would then need at least a time varying tax rate so that the government budget constraint balances every period.
 
27
This latter condition is needed only when we do not assume constant returns to scale in either sector. In fact, that is the only case we will consider when solving the model. Then, the condition does not need to be imposed, since it will hold in equilibrium.
 
28
The ratios between growing variables will be the same in both economies, but not the levels of those variables.
 
29
Which is known as leapfrogging.
 
30
Beyond satisfying the restrictions imposed by rationality of expectations, if that is a maintained assumption.
 
31
That is, the version of the model without any random shock.
 
32
That is, the roots of the characteristic equation of the transition matrix Ψ of the vector AR(1) model in (7.111) are inside the unit circle.
 
33
Notice that the conditional expectations of highly nonlinear functions that initially appear in the optimality equations transform into conditional expectations of single variables when a linear or log-linear approximation is performed.
 
34
This is because under determinacy of equilibria, there is a one-to-one mapping between expectations errors and structural innovations to the model, while under indeterminacy, some expectations errors are left free.
 
35
Specify a process, a t = α 1 ε 1t + α 2 ε 2t + ξ t, with \(\xi \sim N(0,\sigma _{\xi }^{2}), {\textit {Cov}}(\xi _{t},\varepsilon _{1t})={\textit {Cov}}(\xi _{t},\varepsilon _{1t})=0\). From \({\textit {Cov}}(a_{t},\varepsilon _{1t})=\alpha _{1}\sigma _{\varepsilon 1}^{2}, {\textit {Cov}}(a_{t},\varepsilon _{2t})=\alpha _{2}\sigma _{\varepsilon 2}^{2}, {\textit {Var}}(a_{t})=\alpha _{1}^{2}\sigma _{\varepsilon 1}^{2}+\alpha _{2}^{2}\sigma _{\varepsilon 2}^{2}+\sigma _{\xi }^{2}\), we can use given values for \(\rho _{a,\varepsilon _{1}}, \rho _{a,\varepsilon 2}, {\textit {Var}}(a_{t})\) to choose the values of \(\alpha _{1}, \alpha _{2}, \sigma _{\xi }^{2}\).
 
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Metadaten
Titel
Additional Endogenous Growth Models
verfasst von
Alfonso Novales
Esther Fernández
Jesús Ruiz
Copyright-Jahr
2022
Verlag
Springer Berlin Heidelberg
DOI
https://doi.org/10.1007/978-3-662-63982-5_7