We devise a simple way of incorporating the financial sector into a growth model that is useful pedagogically. Financial innovation raises the efficiency of financial intermediation, which facilitates capital accumulation. The model may be extended to include real R&D as a symbiotic source of endogenous growth.Financial innovations, economic growth

Yuan K. Chou

English

Research Papers in Economics

  1
ISSN  0819-2642 
  ISBN 0 7340 2544 0 
 
 
 
 
 
THE UNIVERSITY OF MELBOURNE 
 
DEPARTMENT OF ECONOMICS 
 
 
RESEARCH PAPER NUMBER 888 
 
 
NOVEMBER 2003 
 
 
A PEDAGOGICAL TOOL FOR ILLUSTRATING  
THE REAL IMPACT OF THE FINANCIAL SECTOR 
 
by 
 
Yuan K. Chou 
 
 
 
 
   
Department of Economics 
The University of Melbourne  
Melbourne      Victoria     3010 
Australia. 
   2
A Pedagogical Tool for Illustrating the Real 
Impact of the Financial Sector 
 
Yuan K. Chou
1 
 
Department of Economics, University of Melbourne 
 
Abstract 
We devise a simple way of incorporating the financial sector into a growth model that is useful 
pedagogically. Financial innovation raises the efficiency of financial intermediation, which 
facilitates capital accumulation. The model may be extended to include real R&D as a 
symbiotic source of endogenous growth. 
 
JEL codes: G20, O41 
JEL keywords: Financial innovations, economic growth 
 
 
1. Introduction 
 
Students in introductory and intermediate-level undergraduate economics courses are often 
puzzled about the role of the financial sector in the functioning and performance of the real 
economy. While they are exposed to constant media coverage of financial firms and their 
activities, discussions about the financial sector are often cursory or non-existent in most 
standard macroeconomic texts. On the topic of economic growth, real R&D activities that 
produce technological innovations are often touted as the primary engine of growth in 
advanced economies. Since financial firms attract many of the brightest college and business 
school graduates, the financial sector may appear to affect growth adversely by draining 
precious talent and human capital away from the science and engineering fields. 
In this paper, we construct an extension of the classic Ramsey-Cass-Koopmans model 
to show in a succinct fashion why the financial sector matters for the real economy. The 
financial sector in our model comprises financial innovators and financial intermediaries. 
Financial innovations, designed by those talented college and MBA graduates, increase the 
efficiency of financial intermediation by improving the match between the needs of savers and 
borrowers, which in turn facilitates capital accumulation and increases future output. As our 
model of finance and growth differs from previous models by focusing on innovative financial 
activities, we begin with a short discussion on financial innovations.  
Financial innovations may be motivated by the need to hedge some new economic risk, 
or by new regulation, a change in fiscal and monetary policies, demand for intertemporal or 
spatial wealth transfers, the need to lower transactions costs, or the desire to reduce agency 
costs due to asymmetric information. Financial innovation may involve inventing a brand new 
class of products, the modification of existing products, or the combination of the 
characteristics of several existing products (“spectrum filling”), which moves the financial 
system closer to the Arrow-Debreu ideal where all transactors can ensure themselves delivery 
of goods and services in all states of nature. 
                                                 
1 Address: Department of Economics, University of Melbourne, VIC 3010, Australia. Phone: +61 3 8344 5287 
Fax: +61 3 8344 6899 Email: ychou@unimelb.edu.au. The author would like to thank Martin Chin for his 
research assistance.   3
    Financial  innovations  differ  from  real  technological innovations in that they are not 
protected by patents. However, financial innovators gain first-mover advantages such as 
acquiring reputation and expertise in pricing and exploiting the properties of the product. 
[Tufano (1989), Herrera and Schroth (2001)]. Financial innovations may be classified 
according to their risk-transferring, liquidity-enhancing, or equity-generating abilities. 
Financial innovations may aid real innovative activities (such as venture capital), assist 
corporate expansion (bonds), facilitate profit-taking and the spreading of risk (mutual funds, 
CDs, derivatives, hedge funds), and help refinance obligations or mobilize assets (swaps, 
mergers, futures).  
 
2. Modelling the Financial Sector: The Key Equations 
 
2.1 Financial Innovators 
 
    Financial  innovators  produce  new  financial products and services using labor (and the 
embodied human capital) that is diverted from the production of the final consumption good.  
The stock of financial products (that is, old financial innovations) is denoted as τ. 
    The development of the financial sector is characterized by an ever-expanding variety of 
financial products. The existing stock of financial products affects the production of new 
financial ideas according to 
()
λ φ
τ τ τ =  Fu L ,     (1) 
where τ denotes the quantity of financial innovations per unit time, uτ is the fraction of the 
labor force employed by the financial sector, L is the aggregate stock of labor which grows at 
rate  n,  F is a productivity parameter,  (0,1) λ∈  is an elasticity parameter, and  (0,1) φ ∈  
measures the extent of positive spillovers from existing financial products (the “spectrum 
filling effect” discussed previously). The production function for financial innovations exhibits 
diminishing marginal returns with respect to labor: as more and more individuals are engaged 
in designing new financial products, the probability of inefficient replication rises. 
 
2.2 Financial Intermediaries 
 
    Financial  intermediaries  are  responsible for intermediating funds between borrowers 
(producers of the final consumption good) and lenders (households). Unlike conventional 
growth models, not all household savings will automatically be transformed into funds that are 
utilizable by firms for investment in new plant and machinery. In particular, some risk-averse 
savers will continue to hold liquid but unproductive assets unless offered a sufficient variety of 
financial products. The efficiency of intermediation is measured by 
 /
κ ξτ ≡ L ,       ( 2 )  
where  (0,1) ξ ∈ , and  (0,1) κ ∈  is a parameter that measures the degree of rivalry in τ. ξ may be 
interpreted as the state of development or sophistication of the financial sector. 
        Why does the efficiency of intermediation diminish as L increases? As the labor force 
increases, so does the volume and complexity of funds that have to be intermediated. A larger 
population may exhibit more diverse preferences for the risk, maturity and other characteristics 
of financial instruments. For some financial products and services (such as branch banking), 
the increase in population creates congestion that can only be relieved by financial innovations 
(like phone and Internet banking). By restricting κ to lie strictly between 0 and 1, financial 
innovations in the aggregate are neither fully rivalrous (κ = 1) nor completely non-rivalrous (κ   4
= 0). Our model allows κ to be infinitely small but not zero. The capital accumulation process 
therefore takes the form: 
      ( ) ξ δ =− −  KY CK ,       ( 3 )  
where K denotes the stock of capital, C is the level of aggregate consumption, and δ is the rate 
at which capital depreciates. 
    The aggregate production function for the final good, Y, is of the Cobb-Douglas form: 
      ()
1 α α − = YA Ku L ,       ( 4 )  
where A denotes the exogenous level of technology, uY the fraction of the labor force employed 
by the final goods sector, and  (0,1) α ∈  is capital's share of income from final goods 
production. 
    In the steady state, ξ must be constant by definition (and bounded from above at one in a 
closed economy). Therefore, if the labor force grows at the constant rate n, then the rate of 
financial innovations in the steady state must equal κn. 
 
3. Microeconomics of the Model 
 
  Tufano (1989)'s empirical findings on the pricing behavior of financial innovators are 
consistent with the hypothesis of competitive innovation: investment banks that create new 
products do not charge higher prices in the brief period of monopoly before imitative products 
appear. The profit of a price-taking financial innovator, to be maximized by its choice of τ, is 
      τ ττ τ π τ = −  Pw u L ,       ( 5 )  
where  τ P  is the price of each financial innovation. The first order condition implies that 
     
(1 )/
1/
λ λ τ
τ λ τ
λ
− =  
w
P
F
.       ( 6 )  
This equation may be interpreted as the optimal pricing schedule for τ. 
        Downstream in the financial sector, financial intermediaries purchase innovations from 
financial innovators (which, in the real world, are probably sister divisions of the same 
financial firms) and use them in transforming savings into productive investment. As the focus 
of our model is on financial innovations, we model the financial intermediaries very simply. 
They are passive, price-taking entities engaged in perfect competition who derive their income 
from charging firms a higher interest rate,  K r , for renting capital than it pays out to households 
for their savings,  V r . The interest rate differential,  K r -  V r , may be thought of as the 
commission charged for intermediating funds. For simplicity, we assume that financial 
intermediation requires no labor input. 
    In each period, the financial intermediary ensures that interest revenues received from firms 
equal the costs of acquiring deposits from households and purchasing new financial products 
from financial innovators: 
      ττ = +  KV rK rK P .       ( 7 )  
We can show that this results in an arbitrage equation governing the evolution of  τ P : 
     
τ
τ τ
ξ
τ
= +
 
K
P V
r
PP
.       ( 8 )  
    In the final goods sector, firms maximize profits by choosing the optimal account of labor 
and capital at each point in time: 
     
,
max  − −
Y
YY K
uK
Yw u Lr K .     (9) 
The first order conditions require   5
      (1 ) /( ) α = − YY wY u L ,       ( 1 0 )  
      / α δ = − K rY K .       ( 1 1 )  
        Finally, to close the model, we examine the consumption decision of households. The 
representative household maximizes a discounted stream of lifetime utility subject to an 
intertemporal budget constraint where instantaneous utility is of the CRRA form: 
    
1
0 ,
1
max
1
θ
ρ
θ
− ∞ − −
− ∫
Y
t
Cu
C
ed t  
 
subject to 
     ττ τ π =+ + + − 
VY Y Vr Kw u L w u L C , 
     ξ =  KV , 
    1 τ =+ Y uu ,         ( 1 2 )  
where   V  represents the flow of households’ stock of assets (that is, savings). We assume that 
households are ultimate owners of all capital and shareholders of final goods firms and 
financial innovators. In equilibrium, wages are equal across all labor markets, that is  τ = Y ww . 
Substituting this and equation (7) into the constraint results in the Hamiltonian: 
    ()
1 1
1
θ
ρ λλ
τ υξ µ
θ
−
− −
=+ + − +
−
 t
KY
C
er K w u L C F u L H     (13) 
where υ and µ are co-state variables associated with the state variables K and τ respectively. 
The control variables are c and  Y u . A flowchart of the economy is shown in Fig.1. 
K rK
wuL τ ττ π +
τ P ττ
V rK Y uL YY wuL
K 
uL τ
V 
Firms 
Households 
Financial 
Intermediaries 
Financial 
Innovators 
 
Fig 1. Flowchart of the model 
 
 
4. Equilibrium, Solutions and Results 
 
We define the balanced growth path or steady state of the model as one in which all variables 
grow at constant rates. As there is no technological progress, aggregate output, aggregate 
capital and aggregate consumption grow at the same rate as the labor force in the steady state. 
Defining  / ≡ kK L ,  / ≡ cC L  and  / ≡ yY L , we can show that the steady state solutions to the 
model are, in sequential fashion:   6
     
** 1 τ
Γ
= =−
Γ+Φ
Y uu ,       ( 1 4 )  
     
1
* 1
*
λ φ
τ ξ
κ
− ⎛⎞
=⎜⎟
⎝⎠
Fu
n
,       ( 1 5 )  
      
1
* 1
**
φ ξα
ρδ
− ⎛⎞
= ⎜⎟ + ⎝⎠
Y
A
ku       ( 1 6 )  
      
*
*
*
() ρδα δ
α ξ
+− +
=
nk
c ,      ( 1 7 )  
where  () αλκ δ Γ≡ + n  and  () ( )[ ] 1( 1 ) αρδρ κ Φ≡ − + − − n . These solutions imply that: 
1.  An increase in the financial sector spillover effect, φ , increases the steady-state proportion 
of labor employed in the financial sector, 
*
τ u . That is, 
* /0 τ φ ∂ ∂> u . 
2.  An increase in the rate of time preference, ρ, and the degree of risk aversion, θ, decreases 
*
τ u . That is, 
* /0 τ ρ ∂∂ < u  and 
* /0 τ θ ∂∂ < u . 
3.  The steady state growth rate of the economy is independent of the characteristics of the 
financial sector. In the steady state,  /// = ==   YY KK CC n  and  / / / 0 ===   yy kk cc . 
  To discuss the properties of the model away from the steady state, we need to reduce 
the dimensionality of the model by assuming that the share of labor in the financial innovations 
sector,  u , and the physical investment rate, s , are constant and exogenous. The resulting 
0 =  k  and  0 ξ =   schedules are given by 
      ()
1
* 1
* 1
α ξ
δ
− ⎛⎞
= − ⎜⎟ + ⎝⎠
sA
ku
n
,      ( 1 8 )  
      ()
1
1
* 1
λ α φ
ξ
λ
− ⎛⎞ −
= ⎜⎟
⎝⎠
Fu
n
.     ( 1 9 )  
Figure 2 illustrates the dynamics of the model in  ξ − k  space and the impact of a rise in the 
productivity of financial innovators due, for example, to financial liberalization. Steady state 
intermediation efficiency, ξ
*, and capital per worker, k
*, are both permanently higher after the 
shock. 
 
   
Fig.2 Dynamics of the model and the impact of an increase in F.   7
5. Extensions 
 
5.1 Financial Innovations and Technological Innovations 
 
We can incorporate endogenous technological progress into the model by adding an R&D 
equation, such as that in Jones (1995) but including a spillover effect for τ:  
()
η β ψ τ = 
A A BuL A,       ( 1 9 )  
where   A denotes the quantity of financial innovations per unit time, uA is the fraction of the 
labor force engaged in R&D, B is a productivity parameter,  (0,1) ψ ∈  measures the extent of 
spillovers from previous R&D activities, while  (0,1) β ∈  measures the impact of financial 
development on R&D productivity. The idea is that some financial innovations, such as 
convertible loan notes and redeemable convertible preference stock, are used by venture 
capitalists to fund risky R&D projects with potentially high payoffs and may therefore raise the 
rate of technological innovation. We can show that the growth rate of the economy is now 
given by:  
()
* /1
1
βλ
γ ηψ
φ
⎛⎞
= +− ⎜⎟ − ⎝⎠
A n . 
This growth rate depends in part on the parameters characterizing the financial sector, and 
neatly demonstrates the inter-dependence between financial and technological innovations in 
generating long-run growth. 
 
6. Conclusion 
 
In this paper, we demonstrated a simple and succinct way of incorporating the financial sector 
into an economic growth model. This is particularly useful as a pedagogical tool for explaining 
the real macroeconomic impact of the financial sector in undergraduate economics courses.
2 In 
our model, the financial sector produces financial innovations, which raises the efficiency of 
financial intermediation, thereby facilitating capital accumulation. We derived the steady state 
solutions of the model, explained its implications, and studied its transitional dynamics. 
Finally, we also showed how the model may be extended to include endogenous technological 
progress. 
 
References 
 
Ramsey, F. (1928): “A Mathematical Theory of Saving,” Economic Journal, 38, 543-559. 
Cass, D. (1965): “Optimum Growth in an Aggregative Model of Capital Accumulation,” 
Review of Economic Studies, 32, 233-240. 
Herrera, H., and E. Schroth (2001): “Implications of Non-Patentable Financial Innovations,” 
New York University Stern School of Business Working Paper BCES-01-03. 
Jones, C.I. (1995): “Research and Development Based Models of Economic Growth,” Journal 
of Political Economy, 103, 759-784. 
Koopmans, T.C. (1965): “On the Concept of Optimal Economic Growth,” in The Economic 
Approach to Development Planning. North-Holland, Amsterdam. 
                                                 
2 For teaching purposes, it is easier to assume a fixed saving rate (s) as in the Solow (1956) model. Then the key 
equations are: 
1 ()
α α − = Y YA K u L ,  () ξ δ =−  Ks YK , and  ()
λ φ
τ τ τ =  Fu L  where  /
κ ξτ ≡ L . For 
introductory courses, a description of the model and a graphical illustration like fig.1 would suffice.   8
Romer, P. M. (1990): “Endogenous Technological Change,” Journal of Political Economy, 98, 
S71-S102. 
Solow, R. M. (1956): “A Contribution to the Theory of Economic Growth,” Quarterly Journal 
of Economics, 70, 1, 65-94. 
Tufano (1989): “Financial Innovation and First-Mover Advantages,” Journal of Financial 
Economics, 25, 213-240. 

2 For teaching purposes, it is easier to assume a fixed saving rate (s) as in the Solow

A Contribution to the Theory of

A Mathematical Theory of Saving,”

Financial Innovation and First-Mover Advantages,”

On the Concept of Optimal Economic Growth,” in The Economic Approach to Development Planning.

Research and Development Based Models of Economic Growth,”

A PEDAGOGICAL TOOL FOR ILLUSTRATING THE REAL IMPACT OF THE FINANCIAL SECTOR

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A PEDAGOGICAL TOOL FOR ILLUSTRATING THE REAL IMPACT OF THE FINANCIAL SECTOR

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