Loan processing costs, information asymmetries and the speed of technology adoption

Economic Modelling 27 (2010) 358–367

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Economic Modelling j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / e c m o d

Loan processing costs, information asymmetries and the speed of technology adoption☆ Salvatore Capasso a,b,⁎, George Mavrotas c a b c

University of Naples “Parthenope”, Italy CSEF, University of Naples, Italy Global Development Network, India

a r t i c l e

i n f o

Article history: Accepted 24 September 2009 JEL classification: E44 O16 Keywords: Credit market Information asymmetries Loan processing costs Financial development Economic growth

a b s t r a c t This paper presents a simple model in which credit-constrained firms might delay the adoption of new and more productive technologies because of the external financing costs involved. Applying for a loan can be a very costly process by itself. Where there is a high degree of credit rationing, these costs might even deter entrepreneurs from applying for a loan altogether. We argue that the efficiency of the banking system, by affecting such costs, can influence the profitability of new technologies and entrepreneur's investment decisions, which, in turn, influence new technology implementation, capital accumulation and growth. © 2009 Elsevier B.V. All rights reserved.

1. Introduction In recent years, the impact of credit market frictions on real resource allocation has been addressed in a number of studies. The basic thesis of these analyses is that the cost of accessing external financial resources can affect a firm's investment decisions, and as a consequence, its capital accumulation and growth. As has been widely discussed in an extensive literature (Townsend, 1979; Stiglitz and Weiss, 1981; Williamson, 1986, 1987, among others), information asymmetries between borrowers and lenders, can significantly alter the optimal financial contract. Indeed, problems arising from moral hazard or adverse selection can increase the cost of external financial resources and, in this way, affect a firm's investment opportunities. On this theoretical basis, it is possible to explain the links between financial development and real economic growth by assuming that capital accumulation can influence one or more of the components involved in the financial contract. Decreasing monitoring or screening costs, a change in signalling costs or a change in the level of information asymmetry are all possible channels through which financial variables can affect real resource allocation and growth

☆ We wish to thank an anonymous referee for the very helpful comments. ⁎ Corresponding author. Department of Economics, University of Naples “Parthenope”, via Medina 40, 80133 Naples, Italy. Tel.: + 39 081 5474735; fax: + 39 081 5475750. E-mail address: [email protected] (S. Capasso). 0264-9993/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.econmod.2009.09.015

(Bernanke and Gertler, 1989; Bencivenga and Smith, 1991, 1993; Bose and Cothren, 1996, 1997; Boyd and Smith, 1998; Blackburn et al., 2005, among others). Typically, where there are information asymmetries the financial contract will involve higher interest rates, a lower level of funds transferred, or additional contractual elements that would otherwise be absent, such as the probability of rationing or screening. As a result of such distortions, some investment projects become more profitable than others, even when they offer a lower expected real rate of return. In this framework, the Modigliani–Miller theorem on the irrelevance of the financial structure is invalidated, and the nature of the financial system can, in fact, affect a firm's investment choice. Such information costs depend on, among other factors, the nature of the underlying investment projects which need to be financed. For instance, empirical evidence indicates that under information asymmetries the costs of external finance arising from credit market distortions are higher for high-tech firms and for high-tech–highgrowth projects. By analysing the investment decisions of a large cross-section of Italian manufacturing firms, Guiso (1998) finds that where there are credit constraints and information asymmetries, financial intermediaries are more likely to allocate a larger share of funds to traditional, low-risk–low-return projects than to high-risk– high-return projects. There may be various explanations for this finding. High-tech projects often involve innovative ideas as well as specialised know-how and private information, which make these projects less likely to be understood by outsiders (i.e. these projects

S. Capasso, G. Mavrotas / Economic Modelling 27 (2010) 358–367

entail a higher level of information costs). Moreover, the need to defend the market rent generated by a new design or technology will certainly involve a lower incentive for firms to disclose precious information to financial intermediaries or others. This behaviour may exacerbate the degree of information asymmetries and worsen credit market frictions. Finally, high-tech projects require, in general, a lot of R&D which, by its very nature, attracts more costly financial resources than other kinds of investment.1 The idea that high-tech projects might attract fewer and more costly financial resources is not new in the literature. Some decades ago, Arrow (1962) argued that where there are moral hazard problems, innovative investment might be more difficult to be financed. More recently, following the same line of reasoning, Stiglitz (1993) has argued that the level of credit rationing may be higher for firms that are more intensively engaged in innovative investments. Taking account of all these considerations, one can reasonably argue that the high financial costs might explain why entrepreneurs appear to be so resistant to undertaking high-tech investments when other investment opportunities are also available. Indeed, despite a relatively high level of internal profitability, high-tech projects, which usually also exploit the most innovative technologies, might involve very high financial costs in comparison to more traditional projects. In fact, well established empirical evidence shows that the implementation of new technologies is a slow and viscous process. In other words, the diffusion of technology appears not to be a linear and predictable phenomenon (Mansfield, 1989; Griliches, 1992; Pennings and Harianto, 1992; Jaffee et al., 1993; Karshenas and Stoneman, 1993; Jaffee and Trajtenberg, 1996, among others). The introduction of new technologies is not always followed by their direct and immediate adoption. It often takes a long time for firms to introduce new equipment or implement new production processes. Thus, the time path of technology usage seems to follow an S-curve: an initial slow implementation, followed by a rapid diffusion and a final lower rate of adoption. Many theoretical works have attempted to explain, in a variety of ways, these peculiar patterns (Stoneman, 1987; Metcalfe, 1988; Karshenas and Stoneman, 1995). Some of these works hinge on the assumption of exogenous frictions in technology diffusion (epidemic models); others focus on more firm-specific features in decision-making (probit models).2 Despite the attention of the literature, the issue is far from being resolved, and reasonable doubts remain as to whether the rationale behind this (very counterintuitive) phenomenon has been fully explained. In an attempt to extend the existing arguments on this subject, we provide a new explanation for the evidence regarding non-monotonicity and piecemeal implementation of new technologies. Focusing on the idea that a firm's investment decisions crucially depend on financial costs, we develop a model in which the slow adoption of new technologies is explained by frictions in the financial market caused by information asymmetries. A new available technology with a higher rate of return might not initially be implemented by the (skilled) entrepreneurs who could most benefit from it, simply because those entrepreneurs have alternative investment opportunities with lower financial costs. On the other hand, high-tech and innovative projects might initially attract mainly unskilled and venturous entrepreneurs with no alternatives. As a result of these financial frictions, the economy's growth path, which depends on the technology choices made by firms, will exhibit non-monotonic behaviour. In general, we also intend to give account of the two-way causal relationship between financial development and economic growth. By focusing on the efficiency of the banking system and on the costs of accessing credit, we develop a framework which is able to explain

1 The reason is that R&D usually involves larger initial outlays and higher probabilities of default and cannot be used as collateral. 2 For an excellent literature survey on this issue see Geroski (2000).

359

how the prevailing financial contract affects entrepreneurial choices upon the available set of technologies and, hence, the volume and the nature of investment and growth. In turn, capital accumulation and growth can influence the optimal financial contract by affecting financial relevant variables such as the loan size and the probability of credit rationing and, potentially, the level of efficiency of the banking system. Our model is based on the assumption that firms have two alternative investment projects available to them. They can either undertake a low-investment–low-return project, in a traditional sector, which does not require external finance; or a high-investment–highreturn project, a typically high-tech investment, which requires, instead, a high initial outlay.3 The latter project can only be partially selffinanced and compels entrepreneurs to access the credit market. Information asymmetries and imperfections in the credit market, however, make access to loans very costly and deter investment. In order to convincingly pinpoint the rationale behind this entrepreneurial behaviour, we focus on specific frictions in the credit market which increase information costs and which have, until now, been disregarded by the literature on finance and growth, namely queuing costs and loan application costs. Indeed, empirical evidence seems to suggest that the process of applying for a loan can be very costly in itself for entrepreneurs. These costs arise from the time spent queuing, the production of papers and documentation and the consequent loss of alternative opportunities. Where there is a high degree of credit rationing, i.e. when entrepreneurs face a high probability of not obtaining credit, these costs might even deter entrepreneurs from applying for a loan altogether. By using Italian data from the Survey of Households Income and Wealth (SHIW), Guiso et al. (2004) are able to determine the number of borrowers (and the volume of the loans) that would have liked to demand credit over a one-year period, but they have not done so. They are called “discouraged” borrowers. According to Guiso et al., this latent demand for credit is relevant and it greatly contributes to the actual credit rationing to the extent that it is employed in determining a new measure of financial development. In a less recent empirical study, Cavalluzzo et al. (1999), using data from the 1993 National Survey of Small Business Finances in the US, show that in a three-year period, almost 50% of the firms they interviewed reported that they did not apply for a loan because they believed that they would not be able to obtain it. This interesting finding implicitly lends weight to the argument that applying for a loan is a costly process; most importantly, the large proportion of firms not accessing the credit market underlines the relevance of those costs to the decision-making process. The rest of this paper is structured as follows. In Section 2, we describe the economy in terms of technologies, agents' endowments and investment opportunities. In Section 3, we outline the main features of the credit market. The design of the optimal financial contract is determined and discussed in Section 4. Project choice is discussed in Section 5 and the description of the capital accumulation process is the subject of Section 6. Some concluding remarks are contained in Section 7. 2. The economy We consider an economy populated by an infinite sequence of identical two-period lived overlapping generations, in which time is discrete and indexed by 1, 2…,∞. Population is divided into a unit mass of countable entrepreneurs (or borrowers) and a number of n N 1 households (or workers). Young agents are endowed with one unit of labour which is inelastically supplied in the market in return for the current wage rate, wt. All agents are risk neutral and concerned only

3 One can think of this choice as a choice between the adoption of a completely new technology or the restoration and renovation of an existing one.

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S. Capasso, G. Mavrotas / Economic Modelling 27 (2010) 358–367

with old age consumption.4 Young entrepreneurs are also endowed with two alternative technologies to produce capital: a highinvestment–high-return technology (h-h technology) and a lowinvestment–low-return technology (l-l technology). The expected return on each technology depends on an entrepreneur's type. At birth entrepreneurs are divided into skilled (fraction 0.5 of the entrepreneur population) and unskilled (fraction 0.5 of the entrepreneur population). An entrepreneur's type is private information and unobservable by any other agent. The h-h technology requires an initial investment of q units of output at time t in order to deliver Q N q units of capital at time t + 1 with probability ps (pu) under a skilled (unskilled) entrepreneur's management. With probability 1 − ps (1 − pu) the project fails and delivers 0 unit of capital, where ps N pu. We further assume that wt b q implying that entrepreneurs can only partially finance the h-h project with their wage. The l-l technology requires an initial investment of b units of output at time t in order to deliver, with certainty at time t + 1, Bs units of capital to a skilled entrepreneur and Bu units of capital to an unskilled entrepreneur. This technology does not require external funding, since the initial outlay is low enough for the entrepreneur to completely self-finance the project, wt N b. Once one project is underway, an entrepreneur cannot switch to the other; i.e. investment is irreversible. One can think of the h-h project as a project of investment in a high-tech sector. This entails the adoption of riskier technologies, which comes with a high rate of internal return, but which also requires high initial capital outlay. A typical example is investment in R&D. This is usually very risky and very costly; however, it yields a high rate of return. Conversely, the l-l projects resemble the investment projects in low-tech sectors, or mature sectors. These involve the adoption of technologies which require relatively low initial capital outlay and yield a correspondingly low rate of return. All entrepreneurs produce output in the second period of their lives by employing capital and labour according to a standard Romertype production function: yt

+ 1

α 1
α α + 1 k t + 1 Lt + 1 ;

= ΘKt

ð1Þ

where y is individual firm's output, Θ is total factor productivity, k stands for individual firm's capital, K represents the aggregate level of capital and L denotes labour. Increasing returns to scale at aggregate level mainly ensures that the price of capital remains constant through time (see Eq. (3) below). This simplifies by a great extent the modelling process, since a changing price of capital would introduce problems of inconsistency in the decision process. Indeed, as we will see further, the price of capital enters the key Eqs. (7) and (8). By assuming constant returns to scale at the individual level, one reduces the difficulties of pinpointing the optimal loan size and makes it simpler the comparison between risky and safe projects. Increasing returns at the individual level would complicate the framework without adding much to the issue we try to investigate. We assume that the market for output is perfectly competitive and factors are paid their marginal productivity. As a result, by virtue of Eq. (1), the competitive wage rate can be expressed as wt + 1 = α
1 α
α α αΘKtα+ 1 k1
α t + 1 Lt + 1 and the interest rate as ρt + 1 = ð1
αÞΘKt + 1 kt + 1 Lt + 1 . Recalling that only old entrepreneurs produce output, the number of firms will be constant over time and equal to unity. The supply of labour, conversely, comes in each period from young workers, whose number is fixed to n, and from all young entrepreneurs, whose number 4 As in the prevailing literature (see Bencivenga and Smith, 1993; Bose and Cothren, 1996, 1997; Blackburn and Hung, 1998; Boyd and Smith, 1998 among others), we introduce the assumption of risk neutrality in order to simplify matters and to concentrate on the issue at stake. A deviation from risk neutrality would complicate matters without adding much to the results.

is 1. Since in equilibrium, each firm employs an equal amount of capital and labour, per firm capital equals the aggregate capital, Kt + 1 = kt + 1, while per firm labour supply is Lt + 1 = 1 + n. Using these formulae, the equilibrium wage rate and the equilibrium interest rate can be expressed as follows: wt ρt

+ 1

+ 1

α
1

= αΘð1 + nÞ

kt

+ 1;

ð2Þ α

= ρ = ð1
αÞΘð1 + nÞ :

ð3Þ

Entrepreneurs wishing to run the h-h technology can obtain financial resources by applying for a bank loan. As in Diamond and Dybvig (1983), we assume that banks are institutions consisting of a coalition of agents born at t. Banks obtain financial resources by offering competitive deposits at a competitive rate to young households, and employ these resources either by offering loans to entrepreneurs or by investing in a technology which can only be implemented on a large scale and is, therefore, inaccessible to single agents. The latter delivers 1 unit of capital at time t + 1 per unit of output invested at time t. We assume that this technology is always dominated by both the h-h and l-l technology and is, therefore, used by banks only residually for the employment of resources not lent to entrepreneurs. More formally we assume that: i

i

pQ B N N 1; where i = s; u q b

ð4Þ

Finally, the banking system is competitive, and therefore the banks' profit is zero. Applying for a bank loan is costly. An entrepreneur who enters a bank to ask for a loan will need to queue, fill in forms, provide all the necessary documentation, etc. These are obstacles for entrepreneurs, and certainly represent a cost in terms of time, energy and missed opportunities. From this perspective, it is reasonable to maintain that these costs reflect the efficiency of the banking system. The more efficient the banking system, the lower will be the queuing time, the level of bureaucracy to contend with, the amount of paperwork required etc., in order to obtain a loan. These costs might not directly affect the efficiency of the project to be financed, but they will certainly affect the utility of the entrepreneur seeking a loan. If these costs are very high, they might even deter the entrepreneur from applying for the loan at all. It is a reasonable assumption that these inefficiency costs, which we will refer to as queuing costs, closely resemble in their final effects the shoe leather costs or menu costs involved in a high level of inflation. Entrepreneurs seeking a loan have to make frequent visits to the bank, where they have to queue and waste time. They have to contend with long and complex bureaucratic procedures; they have to revise their production plans, and they may even have to bribe one or more bank officials. As to inflation costs, we assume that queuing costs translate into fewer resources available to the entrepreneur. Specifically, we assume that a fixed proportion of the entrepreneur's wage, 1 − γ, is wasted in queuing, where 0 b γ b 1. One can retain that such cost depends on the efficiency of the banking system, and in this sense it is endogenous to the extent that the efficiency of the banking system is endogenous and it increases with capital accumulation. We do not explicitly model the nature of such cost in terms of capital, yet in Section 5 we discuss the implication of a change in this cost on the optimal financial contract and on agents' choices.5 Given the above, at the end of the loan application process, the entrepreneur is left with γwt resources available for investment. If 5 Of course, one could also assume a differentiated queuing cost among entrepreneurs. If, on the one hand, this would make the dynamic process in the credit market richer, on the other, it would not alter the basic mechanism through which such costs affect agents' investment decisions.

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refused a loan, the entrepreneur, who cannot run the safe project, has only one option: to invest these resources in an inefficient storing technology.6 We assume that a skilled entrepreneur is able to transform 1 unit of time t output into βs b 1 U of time t + 1 capital. Unskilled entrepreneurs have less efficient storing technology, and are only able to transform 1 unit of time t output into βu units of time t + 1 output. In order to simplify matters further we set βu = 0.7 3. The credit market The credit market works in the following way. At time t young entrepreneurs supply labour in the market, obtain the wage and decide whether to employ h-h or l-l technology. The latter can be completely self-financed and does not require credit (recall that wt N b). Therefore, an entrepreneur will only require external finance, and apply for a bank loan, if she decides to employ h-h technology. An entrepreneur's type is private information, which cannot be accessed either by banks or by any other agent in the economy. The presence of information asymmetries generates agency costs in the credit market, which agents try to minimise by designing optimal and incentive-compatible financial contracts. An entrepreneur who wishes to undertake the h-h project requires a total of q units of output to invest. Therefore, the minimum amount of funds an entrepreneur needs to borrow is q − γwt, given that each one can self-finance the project by a maximum of γwt units of output (the fraction 1 − γ of an entrepreneur's resources are lost in applying for the loan). On the assumption that capital production technology delivers the highest return per unit of output invested, each entrepreneur optimally invests the maximum amount of available funds (wt) in the project and requires a loan of q − γwt units of output. The interest rate banks charge on this loan to a skilled entrepreneur is easily determined by the bank's zero profit constraint: s

Rt

+ 1

=

ðq
γwt Þρt ps

+ 1

;

ð5Þ

ensures that by lending out to borrowers, banks can expect the same return as they obtain by investing in the large-scale technology (Eqs. (5) and (6)) and, given that banks' profit constraints must be binding, this is also the competitive deposit rate banks can offer. 4. The optimal financial contract In order to discriminate between borrowers, banks design different financial contracts for each type of borrower in such a way that any entrepreneur seeking a loan will automatically select the contract designed for her own type over other contracts. This being the case, her choice of contract will reveal the borrower's identity. With this in mind, we now turn to contract design. In our simple economy, the financial contract must specify three items: loan size, interest rate and the probability of credit being granted. We now compute each one of these in turn. The size of the loan is easily determined by recalling that the expected return on the project dominates the large-scale technology, which is the bank's alternative investment opportunity. On this assumption, banks will transfer to entrepreneurs as much as entrepreneurs demand and entrepreneurs will invest as much as they can in the project. Given that the maximum amount of self-finance is wt — of which 1 − γ units are wasted in the queuing process, and that investment in hh technology requires q units of output, each entrepreneur (of whatever type) requires a loan of q − γwt units of output. The interest rate is straightforwardly determined by the bank's zero profit constraint — Eqs. (5) and (6). As we have already stressed, since skilled entrepreneurs pay a lower interest rate, unskilled entrepreneurs have an incentive not to reveal their type and to pretend to be skilled. What banks can do to deter unskilled entrepreneurs from selecting the contract designed for skilled entrepreneurs is to credit ration the latter. As a result, the financial contract will also entail a probability of credit rationing which is determined by the following maximisation problem: s

psRst + 1,

which requires that the expected repayment on the loan, equals the alternative cost of investment in the large-scale technology, (q − γwt), in output terms. Similarly, the interest rate charged to an unskilled borrower is: u

Rt

+ 1

=

ðq
γwt Þρt pu

+ 1

:

ð6Þ

Given that Rst + 1 N Rut + 1 , unskilled entrepreneurs have an incentive to pretend to be skilled in order to pay a lower rate of interest. As with standard adverse selection problems, banks can solve the information problem by discriminating between borrowers. In this framework, as is commonly assumed, discrimination can be achieved through credit rationing. Since different borrowers incur different costs when they are refused credit and, in particular, skilled borrowers incur lower costs (βs N βu = 0), banks can ensure that a contract designed for good borrowers is unattractive to unskilled borrowers by introducing a probability of credit rationing. Let us now turn to the analysis of credit supply. Financial resources in the market are supplied by young households, who transfer to banks the remuneration for their work, the wage rate, in exchange for a competitive deposit rate. The deposit rate is determined by the return on the large-scale investment technology. Indeed, arbitrage

6 We also assume that entrepreneurs only learn the outcome of their requests for loans at the close of the credit market. This means that once refused a loan, an entrepreneur cannot deposit the resources in a bank. 7 One needs to assume a low enough return on the storing technology of unskilled entrepreneurs, βs N βu, for the single crossing property to be satisfied and lenders being able to separate borrowers. We assume βs N βu = 0 just to simplify matters.

361

Max πt p ðQρt πt

s

+ 1


Rt

+ 1Þ

s

+ ð1
πt Þγβ wt

ð7Þ

s.t. u

p ðQ ρt πt ≥0;

u

+ 1


Rt

u + 1 Þ≥πt p ðQ ρt + 1

s


Rt

+ 1Þ

ð8Þ ð9Þ

where πt is the probability of credit being granted. Banks maximise a skilled entrepreneur's expected utility subject to the incentive compatibility constraint, Eq. (8), and non-negativity constraint, Eq. (9). Eq. (7) represents a skilled entrepreneur's expected utility. With probability πt, the loan application is successful, and the entrepreneur will run the h-h technology which delivers a net profit of ps ðQ ρt + 1
Rst + 1 Þ. With probability 1 − πt the loan is not granted, and the entrepreneur can only invest her residual wage in the storing technology, which will yield γβswt. In order to discriminate between borrowers, the contract must be incentive-compatible, which implies that, given the probability of being credit rationed, an unskilled entrepreneur must be at least indifferent to revealing her own type or pretending to be a skilled entrepreneur, Eq. (8). It is easy to demonstrate that because of Eq. (4), the risky project yields a higher expected return than the storing technology, ps ðQ ρt + 1
Rst + 1 Þ N γβs wt ρt + 1 for every b ≤ wt ≤ q. Therefore, we can state that an entrepreneur's expected utility is a strictly increasing function of πt.8 As a corollary, maximisation implies that πt is set at the

8 Indeed, since βs b 1, Eq. (4) implies that (psQ − q)/q N βs and therefore, psQ − q + γwt N psQ − q N βsq N γβswt for all wt b q.

362

S. Capasso, G. Mavrotas / Economic Modelling 27 (2010) 358–367

maximum and the incentive compatibility constraint is binding. Hence, by virtue of Eqs. (5) and (6), we can write: πt =

ps ½pu Q
ðq
γwt Þ : pu ½ps Q
ðq
γwt Þ

ð10Þ

The latter establishes that the probability of not being credit rationed is a function of the wage rate, πt(wt). It is tedious, but easy, to show that this is an increasing, ∂πt(wt)/∂wt N 0, and concave, ∂2πt(wt)/ (∂wt)2 b 0, function of wt.9 Since the wage rate, in turn, is a linear increasing function of capital (see Eq. (2)), the probability that a skilled borrower will obtain a loan is ultimately an increasing and concave function of capital. In other words, as capital accumulates, the prevailing financial contract will entail a higher probability of granting a loan (πt increases) which translates into a lower level of credit rationing. Interest rate, loan size and the probability of credit rationing, as determined above, allow us to rewrite the skilled entrepreneur's expected utility when applying for a loan to finance the h-h project as follows:  ps pu Q
ðq
γwt Þ
s s ˆ p Q
ðq
γwt Þ
γβ wt ρt Uðw t ; γÞ = u s p p Q
ðq
γwt Þ s

+ γβ wt ρt

+ 1

ð11Þ

+ 1:

s

+ 1

+ β ðwt
bÞρt

+ 1:

ð12Þ

5. Project choice We now have enough information to study the entrepreneur's optimal project choice. By comparing expected utility in the two alternatives, it is possible to determine when a skilled entrepreneur will find it ex ante optimal to apply for a bank loan in order to adopt the risky high-return technology, or, alternatively, will find it optimal to self-finance the safe low-return project. Clearly, since these payoffs, Eqs. (11) and (12), are a function of the wage rate, the optimal choice will depend on, among other factors, the wage rate prevailing in the economy. In more general terms, a skilled entrepreneur will find ˆ t ; γÞN Uðw ˜ t Þ, while selfit optimal to apply for a bank loan only if Uðw ˆ t ; γÞ b Uðw ˜ t Þ. Skilled entrepreneurs will financing will prevail if Uðw ˆ t ; γÞ = have no preference for either of the two alternatives if Uðw ˜ Uðwt Þ. Let us define:
ˆ t ; γÞ
Uðw ˜ t Þ = fπt ps Q ρt F ≡ Uðw + γβ wt
B
β ðwt
bÞgρt s

9

s

s

s

+ 1 + 1:

A formal proof is given in Appendix A.1.


ðq
γwt Þ
γβ wt



Proposition 1. If s

β≤

The latter is a non-linear function of the wage rate and capital: as capital accumulates, and the wage rate increases, the expected utility of skilled entrepreneurs in applying for a bank loan will change nonmonotonically. It is also worth stressing that, ceteris paribus, an increase in the bank's loan processing costs — a decrease in γ —, will reduce the level of expected utility. Instead of applying for a bank loan in order to implement the h-h project, skilled entrepreneurs can opt for the l-l technology. The latter, as outlined, requires a minimum initial investment of b units of output, which can be completely covered by the entrepreneur's wage, wt N b, and it yields Bs units of capital after one period. Output that is left over after this investment, wt −b, is stored and will be transformed into βs(wt −b) units of next period capital. Therefore, the l-l project yields to the entrepreneur the expected utility: ˜ t Þ = Bs ρt Uðw

This is a non-monotonic function of the wage rate, whose behaviour depends on the parameter values. In fact the wage rate has two opposite effects on the skilled entrepreneur's expected utility. On the one hand, an increase in the wage rate directly increases the volume of funds that can be stored and, as a result, causes the relative cost of applying for a bank loan to increase. Therefore, by implication, the cost of the h-h project increases. On the other hand, an increase in the wage rate, by decreasing the probability of credit rationing and also the amount of self-finance, lowers the queuing cost and makes the h-h project more profitable. The relative magnitudes of these two effects, which will determine the entrepreneur's optimal project choice, depend on the values of the parameters and the level of the wage rate. The behaviour of Eq. (13) largely depends on the parameter values. However, it is sufficient to assume (we provide a formal proof in Appendix A.2) a small enough rate of return on the storing technology, βs, to ensure that the difference in expected payoff, F(wt;γ), is a well behaved function, concave and monotonically increasing in wt. We summarise these results in the following:

ð13Þ


s  ∂π γπðqÞ + p Q
qð1
γÞ ∂w jW =q 1
½1
πðqÞγ +

j

∂π ∂w W = q

;

then F ′ ðwt ; γÞN 0 and F ″ ðwt ; γÞ b 0 ∀wt ∈½b; q. In order to have a meaningful framework, in which one project's expected return is not always dominated by the other's, one also needs to make sure that, at least for some values in the feasible set of wt, the difference in expected utility F(wt;γ) is such that entrepreneurs find it optimal to choose the h-h technology over the l-l — i.e. F(wt;γ) N 0 — and the l-l over the h-h project — i.e. F(wt;γ) b 0. In more rigorous terms we can state: Proposition 2. If limþ Fðwt ; γÞ b 0 b = N πðbÞ b

wt →b

Bs
γβs b p Q
ðq
γbÞ
γβs b s

ð14Þ

and lim Fðwt ; γÞ N 0 b = N

wt →q−

Bs + βs ðq
bÞ
γβs q b πðqÞ; ps Q
qð1
γÞ
γβs q

ð15Þ

then there is one and only one w = wc ∈½b; q such that Fðwc ; γÞ = 0. A formal proof of Proposition 2 is contained in Appendix A.3. The latter ensures that the F(wt;γ) function, for feasible values of the wage rate, can intersect the horizontal axis at most once. Therefore, if this proposition is satisfied, there is a critical value of the wage rate, wc, such that for this wage skilled entrepreneurs will be indifferent between loan financing the h-h project and self-financing the safe l-l project, i.e. F(wc;γ) = 0. As a corollary, by recalling that F(wt;γ) is monotonically increasing in [b,q], this proposition implies that ∀ wt b wc, the difference in expected payoff is negative, F(wt;γ) b 0, — ˆ ˜ i.e. Uðw t ; γÞ b Uðwt Þ — and is, in fact, positive F(wt;γ) N 0 — i.e. ˜ ˆ Uðw ; γÞ N Uðw Þ−∀w N wc . In other words, Proposition 2 ensures that t

t

t

for a wage level lower than the critical value, it is optimal for a skilled entrepreneur to self-finance the safe low-return l-l technology. As wage increases, the h-h project becomes more and more profitable. However, only when the wage rate is high enough, wt N wc, does it turn out to be optimal for skilled entrepreneurs to adopt the high-return risky technology. Skilled entrepreneurs' technology choice depends on the relative costs of the projects. These costs in turn depend on the degree of asymmetric information in the economy. From this perspective, Proposition 2 requires that the minimum level of information asymmetries — corresponding to the highest level of probability of a

S. Capasso, G. Mavrotas / Economic Modelling 27 (2010) 358–367

loan's being granted, π(q) — is low enough to make the h-h project more profitable than the l-l project. Similar logic, but different arguments, allow us to state that Proposition 2 requires that the maximum level of information cost — corresponding to the lowest level of probability of a loan's being granted, π(b) — is high enough to make the l-l project more profitable. Moreover, the information cost depends on the wage level in the economy, π(wt), and, in turn, on the level of capital — see Eq. (2). We assume that the economy starts off with an initial level of capital, k0, such that w(k0) = w0 b w(kc) = wc. Given the above, this is equivalent to assuming that the economy starts off with an equilibrium in which skilled entrepreneurs find it optimal not to ask for a bank loan to finance the h-h project. Even if available, the financial costs involved are so high that it is not optimal for skilled entrepreneurs to adopt the new technology, so they prefer to selffinance the safe l-l project, instead (we recall that F(•) b 0). As capital accumulates and the wage rate increases, the F(•) increases and, when wt reaches the critical value, wc, the economy switches to a new equilibrium in which skilled entrepreneurs find it optimal to apply for a bank loan in order to implement the h-h technology (in this case F(•) N 0). We anticipate, however, that the migration to the new equilibrium might not occur if the economy reaches the steady state before the wage rate reaches the critical value. We will analyse in detail such possibilities, and the resulting impact on the growth rate, in the following section. It is useful to provide a formal definition of credit market equilibrium. Definition 1. At each point in time t and ∀ wt ≥ wc, the equilibrium in the credit market will be characterised by a financial contract in which skilled entrepreneurs obtain a bank loan of size q − wt with probability πt (Eq. (10)) and an interest rate Rst + 1 (Eq. (3)). At each point in time t and ∀ wt b wc, skilled entrepreneurs will self-finance and not access the credit market. It is worth noting that the critical value of the wage rate, wc, which defines the boundary for financial regime switching, depends on the level of loan processing costs, γ. More specifically, it can straightforwardly be shown that an increase in this cost — a decrease in γ — by shifting the F(wt;γ) downwards increases the value of wc, and, therefore, delays the transition of the system to a new financial regime in which skilled entrepreneurs adopt the h-h technology. Indeed, an increase in the expected cost of applying for a loan acts as a deterrent to entrepreneurs from approaching a bank. Following this line of argument, it is possible to infer that a low enough γ might deter entrepreneurs from applying for credit for all values of wt. We depict in Fig. 1, two possible F(wt;γ) curves corresponding to two different sets of loan application costs, γ1 and γ2. Let us consider how unskilled borrowers are likely to behave. The costs incurred in an unskilled entrepreneur's alternative to loan finance are very low. Indeed, such entrepreneurs do not have access to a storing technology, (βu = 0). Moreover, unskilled entrepreneurs are never credit rationed. They are neither credit rationed in a discriminating equilibrium, with information asymmetries, nor when agents are fully informed (i.e. when skilled entrepreneurs do not access the credit market).10 Therefore, applying for a bank loan and running the risky project always yields to an unskilled entrepreneur a return which is independent of financial frictions. In order to simplify matters, we assume that this expected return is higher than the return on the safe technology: u

u

p Q
q + b N B :

ð16Þ

10 This is equivalent to arguing that unskilled entrepreneurs always get their first best and, hence, that their choice does not depend on the prevailing financial contract.

363

Fig. 1. Difference in expected payoff and loan processing costs.

The latter implies that unskilled entrepreneurs strictly prefer to apply for bank loans, whatever the wage rate level. As a corollary of Eq. (16), for low wage rate levels, when skilled entrepreneurs do not access the credit market and prefer to self-finance the l-l project, only unskilled entrepreneurs will run the risky projects.11 6. Capital accumulation Since the wage rate is a linear positive function of capital, wt(kt), as determined by Eq. (2), the prevailing financial regime is ultimately a function of the accumulated stock of capital. As we have already argued, given a sufficiently low initial capital stock, k0, such that w(k0) b w(k c), the economy will display well defined dynamics. At initial stages of development, when the level of capital accumulated is low, k0 b kt b kc, the degree of information asymmetry in the market, and the financial costs, are so high that skilled entrepreneurs prefer to self-finance low-return projects rather than to embark on risky projects which require costly external finance. For these values of capital, the h-h technology is relatively unprofitable and will remain idle, even if ready for adoption. In these circumstances, banks will only be approached by unskilled entrepreneurs, who, by assumption, always find it optimal to go for adventurous enterprises. In consequence, the average interest rate will be higher, as witnessed by empirical evidence, and the average capital productivity low. As capital accumulates, the h-h project becomes more and more profitable. However, only when capital crosses the threshold level for regime switching, i.e. kt ≥ kc, is the financial cost associated with the loan application sufficiently low to ensure that the expected return on the investment in the h-h technology will be higher than the return on the self-financed l-l project. For these levels of capital, skilled entrepreneurs find it optimal to apply for a bank loan in order to avail themselves of the risky high-return technology. Since the h-h is more productive than the l-l technology, and involves a higher expected return, the regime switching will clearly have a positive long term impact on the rate of growth. However, contrary to all intuition, we find that in this process the economy might initially experience a 11 This is only a simplifying assumption. We could modify the framework to take into account a more multifarious environment in which only a fraction of unskilled entrepreneurs opt for the h-h project. Yet, this would not add significant insights. For instance, we could simply assume different returns on the l-l technology for different groups of unskilled entrepreneurs.

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drop in the aggregate level of capital. Indeed, the transition of the economy from one regime to the other, as well as the impact of regime switching on growth, depends on the parameter values. In order to understand the nature of these dynamics, we need to analyse in detail the behaviour of all possible capital accumulation paths. As outlined above, by affecting the level of credit rationing, πt, and the volume of funds that can be transferred and stored, economic growth can affect the prevailing financial contract and the optimal investment allocation. In turn, a change in the investment allocation, by influencing the average capital productivity will, inevitably, affect the process of capital accumulation and growth. Indeed, as we show below in detail, the economy moves along two different capital accumulation paths, depending on the nature of the choices made by entrepreneurs with regard to resource allocation. At a low level of capital accumulation, k0 b kt b kc, when skilled entrepreneurs prefer to self-finance their projects by implementing the l-l technology, the total amount of capital available in each period is given by the capital produced by unskilled entrepreneurs operating the h-h technology, 0.5puQ, capital produced by skilled entrepreneurs running the l-l technology, 0.5[Bs + βs(wt − b)], and capital produced by banks by means of the large-scale technology, [nwt − (q − γwt)0.5]. Bearing this in mind, and assuming that expected and actual values of capital coincide i.e. E[kt + 1] = kt + 1 because of the law of large numbers, we can express the capital accumulation path as follows: k˜ ≡ kt

+ 1

 s u s  = 0:5 B + p Q
q
β b

ð17Þ


  s α
1 + n + γ + β 0:5 αΘð1 + nÞ kt ; where we have substituted for wt = αΘ(1 + n)α − 1kt. As the economy reaches the critical level of capital, kc, and skilled entrepreneurs find it optimal to apply for bank loans in order to avail themselves of the h-h technology, the aggregate level of capital will follow a different dynamics. In each period, capital is now given by capital produced by skilled entrepreneurs who produce 0.5πtpsQ units of capital if they obtain credit and 0.5(1 − πt)βsγwt if they are credit rationed; capital produced by unskilled borrowers, 0.5puQ, and capital produced by banks which invest in the large-scale technology what is left over from their lending activities, [nwt − πt(q − γwt)0.5 − (q − γwt)0.5]. Hence, in this case, capital accumulation is described by the following: kˆ ≡ kt

+ 1

 s  s s = 0:5ðp Q
qÞ + 0:5πðkt Þ p Q
½q
γwðkt Þ
β γwðkt Þ


  s + 0:5γ 1 + β + n wðkt Þ:

ð18Þ

It can easily be proved that the latter is an increasing and concave function.12 The following definition may therefore be proposed: Definition 2. Given an initial capital stock k0: w(k0) b wc(kc), a competitive equilibrium for this economy is a sequence of {wt, πt, ρt + 1}t∞= 0, that • maximise agents' expected utility (equilibrium in the credit market), • satisfy Eq. (2), Eq. (3), and • satisfy Eq. (17) ∀ kt: w(kt) b wc(kc) and Eq. (18) ∀ kt: w(kt) ≥wc(kc).13 The dynamics of the economy is determined by the relative position of the above capital accumulation paths, Eqs. (17) and (18), and by the F(•) function which regulates the technology choices made by entrepreneurs, Eq. (13). 12

Formal proof is given in the Appendix A.4. Of course, the existence of equilibrium is ensured by the nature of the capital accumulation path which we discuss in Appendix A.4. 13

As we have already argued, the F(•) function determines, given bank queuing cost, the threshold level of capital, k c, at which skilled entrepreneurs show no preference between the risky technology and the self-financed safe project — i.e. F[w(k c);γ] = 0. For levels of capital higher than kc, the h-h risky project, which requires external finance, will dominate the safe l-l project. In this case, capital follows the dynamic path as determined by Eq. (18). The opposite occurs for levels of capital lower than kc. For these levels of capital, the safe technology dominates, and the dynamics is determined by Eq. (17). Therefore, the dynamics of the economy is determined by the relative position of the two paths and by the level of capital for technology regime switching. From this perspective, assuming a low enough initial level of capital, k0 b kc, one can argue that the economy will initially display the dynamics shown in Eq. (17), k̃. The latter is an upward straight line with a positive intercept. This ensures that, at this initial stage of development, the economy displays a positive rate of growth. In order to simplify the exposition, we assume that the slope of Eq. (17) is less than unity, i.e. [n + (γ + βs)0.5]αΘ(1 + n)α − 1 b 114. Indeed, if this is the case, the system will involve a steady state level of capital: ss

k =

  0:5 Bs + pu Q
q
βs b 1
½n + ðγ + βs Þ0:5αΘð1 + nÞα
1

:

ð19Þ

As capital accumulates, the h-h project becomes more and more profitable. However, only when capital reaches the threshold level, kc, does technology regime switching take place. At this point the economic system will jump onto a new capital accumulation path, Eq. (18). Clearly, the economic system might experience a sudden boost in the level of capital only if at kc the new capital accumulation path is above that in Eq. (17) — see Fig. 2A. The economy will, instead, face an initial decrease in the stock of capital if at kc the capital accumulation path in Eq. (18) is below that in Eq. (17) — see Fig. 2B. The drop (or increase) in the level of capital, and output, during the phase of technological transition depends on the relative return on the alternative technologies. In particular, it is easy to see that if the return on the skilled entrepreneurs' safe project is particularly low — specifically if pu(Bs − βsb) b ps(puQ − q)15 — then the capital accumulation path in Eq. (18) will always lie above that in Eq. (17) and the economy will always experience a boost in the level of capital during the transition from one technological system to the other. At the same time, after the discontinuous change in the level of capital, the economy will grow at a higher rate of growth. Indeed, it is easy to show that by Proposition 1 the capital accumulation path in Eq. (18) always has a higher slope than that of the capital accumulation path in Eq. (17). This implies that as the economy reaches the critical level of capital at which the h-h project becomes more profitable, entrepreneurs will apply for bank credit to adopt the risky technology and the rate of growth will increase. We can summarise the above results as follows. Given an initial sufficiently low level of capital, the economy will display a positive rate of growth (bold segment of Fig. 2A and B). At this initial stage of development, the information costs are particularly high and entrepreneurs find it optimal not to apply for bank loans in order to adopt high-return riskier technologies, even though these are readily available on the market. They will, instead, run the safe technology which does not require external finance. As capital accumulates, information costs decrease, and the financial costs involved in the adoption of the risky technology decrease accordingly. Such technology, however, will only be implemented when the economy has 14 This is not a necessary assumption. If the slope is greater than one, the system displays positive rates of growth, and more articulated dynamics. This, however, does not add significant insights to the analysis. 15 This occurs if for kt → 0+ the intercept of Eq. (17) is lower than the intercept of Eq. (18).

S. Capasso, G. Mavrotas / Economic Modelling 27 (2010) 358–367

365

Fig. 3. A growth trap.

increase in γ —, given the wage rate and the level of capital accumulated in the economy, will make it more profitable for skilled entrepreneurs to apply for a loan in order to run the h-h technology. Graphically, an increase in γ results in an upward shift of the F(wt;γ) curve and reduces the critical level of capital, and the wage rate, above which it is optimal to finance the risky project technology. 7. Concluding remarks

Fig. 2. Growth with technology transition.

reached a sufficiently high level of capital accumulation. At this time, the economy will undergo a technology regime change which will positively (Fig. 2A) or negatively (Fig. 2B) affect the level of capital, depending on the return of the technologies. The technology change, however, has a positive impact on the rate of capital accumulation and the economy will display a higher rate of growth (bold segment of Fig. 2A and B). Despite all this, the technology transition might not occur at all. If the economy reaches the steady state before the level of capital is high enough to make the h-h project profitable, that is if kc N kss, entrepreneurs will never find it optimal to adopt the high-return risky technology, and the economy will be trapped in a low growth, technologically underdeveloped state (see Fig. 3). Only if the critical level of capital is sufficiently low, kc b kss, will the economy be able to migrate onto a higher capital accumulation path. Crucially, given that the relative profitability of risky technology depends, inter alia, on bank loan processing costs, 1 − γ, the critical level of capital, and the occurrence of technological adjustment will also depend on these. A decrease in bank queuing costs — i.e. an

The efficiency of the banking system can have a profound impact on real resource and investment allocation, not only directly, by reducing the amount of resources channelled into the credit market, but also indirectly, by affecting entrepreneurs' investment decisions. In a credit market distorted by information asymmetries, financial intermediaries often react by credit rationing firms: loan requests are frequently rejected or only partially satisfied. On this basis, one can reasonably argue that if the probability of credit rationing is very high, and applying for a loan is costly, entrepreneurs might ex ante find it optimal not to seek external financial resources and to stay out of the financial market. This occurs when other investment opportunities, which do not require access to credit, are also available. In line with the relevant empirical evidence, we maintain that high-tech investments are usually the kind of investments which are riskier and require larger initial investment outlay. For this reason, such investments, which are typically also the means by which new technologies are introduced, require a greater degree of external finance and are more likely to be credit rationed. Conversely, one can argue that investments in more mature traditional sectors, which do not involve innovative technologies, are less costly and can very often be self-financed. Based on these premises, we have developed a model of information asymmetries in the credit market in which the high costs of processing bank loan applications might obstruct investments in high-tech projects and favour, instead, low-return, self-financed investments in mature technologies. The result, as intuition would indicate, is that these kinds of costs have a negative impact on average capital productivity and also on the rate of economic growth. In specific circumstances, the combination of these costs and the dynamics of capital accumulation can be such that the economy

366

S. Capasso, G. Mavrotas / Economic Modelling 27 (2010) 358–367

falls into a “technology trap”, in which even readily available new technologies will never be adopted (or will only be adopted by very adventurous high-risk firms with no alternatives) because of high friction and inefficiencies in the credit market, a situation that would seem to obtain in many developing countries. Our model offers an explanation for the irregular dynamics observed in the rate of new technology usage. Alternative investment opportunities, the financial endowments of individual firms and the structure of the credit market, as outlined above, exert a strong influence on the investment decisions that firms make and also on the types of projects undertaken. Readily available new technologies may not be adopted simply because they are rendered unprofitable by the very high financial costs they involve. As a consequence, it is possible to have economies in which firms do not implement new technologies, or where these are adopted only after a substantial delay. One can go further and argue that if competition is positively correlated to the level of inefficiency, then government policies aimed at reducing the degree of monopoly in the credit market and improving the efficiency of the financial sector should remove these costs and frictions and thereby achieve a greater expansionary impact on the economic system. Appendix A A.1

Recalling that because of A.1, ∂πt/∂wt N 0, the latter is always satisfied since by assumption psQ N q. Therefore, we can write 2

∂ F ″ ≡ F ðwt ; γÞ b 0 ∀wt ∈½b; q: ð∂wt Þ2 Given that F″(wt;γ) b 0, we can state that F′(wt;γ) N 0 ∀ wt ∈ [b,q] iff ∂F F′(q) N 0. Therefore, a necessary condition for ∂w = F ′ ðwt ; γÞN 0 is that t

j

∂πt ∂wt

∂F ∂wt

s

w=q

s

s

s

s

½p Q
qð1
γÞ
γβ q + πðqÞð1
β Þγ + γβ
β N 0

Rearranging the latter we find that a necessary condition to have = F ′ ðwt ; γÞ N 0 is

s

β≤

∂π γπðqÞ + ½ps Q
qð1
γÞ ∂w jw=q

1
½1
πðqÞγ +

j

∂π ∂w w = q

Since ∂ F 2 ≡F ″ ðwt ; γÞ b 0 ∀ wt ∈ [b,q], we can state that if the latter ð∂wt Þ is satisfied then F′(wt;γ) N 0 and F″(wt;γ) b 0 ∀ wt ∈ [b,q]. This proves Proposition 1. 2

A.3 By differentiating Eq. (10), we obtain s

s

The proof of this proposition simply follows from Proposition 1. Indeed because of the latter, the difference in the payoff function is a strictly increasing function of the wage rate in the definition set, i.e. F′(wt;γ) N 0 and F″(wt;γ) b 0 ∀ wt ∈ [b,q]. Therefore, if lim þFðwt ; γÞb0

u

∂πt p γQðp
p Þ = u s p ½p Q
ðq
γwt Þ2 ∂wt

wt →b

s

u

which is strictly positive, given that p N p . Analogously, by differentiating the latter we get, ∂2 πt ps 2γ2 Q ðps
pu Þ ∂π 2γ b 0: =
u s =
t s 2 p ½p Q
ðq
γwt Þ3 ∂wt ½p Q
ðq
γwt Þ ð∂wt Þ

and lim
Fðwt ; γÞN 0 then there must exist a value of the wage rate, wt →q

wc ∈ ]b,q[, for which the F(wc;γ) = 0. By virtue of Eq. (13), we can restate that if lim Fðwt ; γÞ b 0 b = N πðbÞb

wt →bþ

given that psQ N (q − γwt) ∀ wt ∈ [b,q].

and

A.2 lim Fðwt ; γÞ N 0 b = N

wt →q


By differentiating F(•;γ), we obtain ∂F = ∂wt

Bs
γβs b ps Q
ðq
γbÞ
γβs b



  ∂πt
s s s s s p Q
ðq
γwt Þ
γβ wt + πt 1
β γ + γβ
β ρt ∂wt

Bs + βs ðq
bÞ
γβs q b πðqÞ; ps Q
qð1
γÞ
γβs q

then there exists one and only one w =wc ∈ ]b,q[, such that F(wc;γ) = 0. + 1

A.4 and 2

∂ F = ð∂wt Þ2

(

)

 ∂ πt
s ∂π  s s p Q
ðq
γwt Þ
γβ wt + 2 t 1
β γ ρt ∂wt ð∂wt Þ2 2

+ 1

Using A.1 and rearranging the latter we can write
s  ∂2 F ∂π 2γ s p Q
ðq
γwt Þ
γβ wt b0b = N
t s ∂wt ½p Q
ðq
γwt Þ ð∂wt Þ2 ∂π s + 2 t ð1
β Þγ b 0; ∂wt which implies,  s  ∂πt p Q
ðq
γwt Þ
γβs wt s + 1
β 2γ
b 0 ∀wt ps Q
ðq
γwt Þ ∂wt

Since π(wt) is a continuous and differentiable function in the domain wt ∈ ]b,q[, the continuity theorem will ensure that Eq. (18), which is a polynomial involving the sum and products of continuous functions, will also be continuous in the domain. We can now prove that the capital accumulation path in Eq. (18) is an increasing and concave function. For this purpose, it will be sufficient to prove that Eq. (18) is increasing and concave in wt, given that the wage rate is a linear function of capital. Indeed, by differentiating Eq. (18) we obtain ∂kt + ∂wt

1

=

 ∂πðwt Þ
s s s p Q
ðq
γwt Þ
β γwt 0:5 + πðwt Þγð1
β Þ0:5 ∂wt
 s + γð1
β Þ0:5 + n N 0

tÞ given that ∂πðw N 0 ∀wt ∈½b; q and that [psQ − (q − γwt) − βsγwt] N 0 ∂wt and βs b 1 by assumption.

S. Capasso, G. Mavrotas / Economic Modelling 27 (2010) 358–367

Moreover, since as already verified in A.2 ∂πt γβ wt  + 2 ∂w ð1
βs Þγ b 0, we can write t s

∂2 πt ð∂wt Þ2

½ps Q
ðq
γwt Þ


 ∂2 kt + 1 ∂2 πðwt Þ
s s = p Q
ðq
γwt Þ
β γwt 0:5 2 ð∂wt Þ ð∂wt Þ2 ∂πðwt Þ  ∂πðwt Þ  s s + γ 1
β 0:5 + γ 1
β 0:5 b 0 ∂wt ∂wt Finally, by comparing the slope of the two capital paths we can state that the slope of Eq. (18) is higher than the slope of Eq. (17) ∀ wt ∈ [b,q]. Indeed,  ∂kˆ ∂k˜ ∂πðwt Þ
s s N that is p Q
ðq
γwt Þ
β γwt ∂wt ∂wt ∂wt  s s + πðwt Þγ 1
β > β ð1
γÞ: The latter is always satisfied if

∂F ∂wt

= F ′ ðwt ; γÞN 0 (see A.2).

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