In the following we analyze how the production side of the economy determines the return to different types of innovation—the demand for innovation. The next section then discusses the other side of this equation, the cost of different innovations or the supply side of innovations.
Average profits from capital goods selling for old innovators in sectormread
…m;tD.1/1C1p
11
m;t MtQm;t (63)
and in sectorz
…m;tD.1/1C1p
11
z;t ZtQz;t: (64)
The net present discounted value of expected profits for young innovators in sector j, denoted byVj;t,j2 fm;zg, are then given by
Vj;tD EtŒ…j;tC1 rt
: (65)
one can show that theV0s are equal to Vm;tD0p
11
m;tC1MtC1Qm;tC1Rm;1tC1 Vz;tD0p
11
z;tC1ZtC1Qz;tC1Rz;tC1 1: (66) where0 WD .1/1C1. So the larger Vz;t in relation to Vm;t, the greater is the reward to develop Z-augmenting capital goods(see Acemoglu 2002, p.789).
The two profit equations given in (66) suggest that capital goods producers’
profits increase with greater output prices, with greater factor use and with greater productivity in the intermediate sectors and decrease with larger interest rates on business credit. The latter effect simply follows from the relationship between credit interest rates and success-probabilities given in (40).
To show and discuss these determinants in more detail, take the ratio ofVz;tand Vm;t. This gives
Vz;t
Vm;t
D pz;tC1
pm;tC1
11
„ ƒ‚ … price effect
ZtC1
MtC1
„ƒ‚…
market size effect
Rz;tC1
Rm;tC1
1
„ ƒ‚ … risk effect
Qz;tC1
Qm;tC1
„ ƒ‚ … productivity
effect
: (67)
The conclusions drawn from Eq. (67) represent the central results of this paper. The next two theorems comprehend these main findings.
Theorem 7 The direction of technical change—whether technical change will favour relatively scarce or abundant factors—is determined by four different market forces: theprice effect, themarket size effect, theproductivity effectand therisk effect.
• The price effect: Since 20; 1Œ, the relative profitability of inventing new Z-augmenting capital goods is increasing in the relative price pz;tC1=pm;tC1. Therefore, the higher this relative price is, the greater is the return on developing newZ-complementary technologies (Acemoglu2002, p.789).
Naturally, relatively scarce factors are relatively more expensive. Thus, the price effect directs technical changes to technologies or sectors that complement scarce input factors and thus command higher commodity prices.
• Themarket size effect: The relative profitability of inventingZ-complementary technologies increases in the relative supply of the factorsZtC1=MtC1. Therefore, the larger this relative factor supply, the greater is the return on the development of newZ-complementary technologies (Acemoglu2002, p.789).
In this study, the market for a technology is determined by the factors that use this technology. An increase in the supply of a factor leads to a larger market for capital goods that complement this factor. The market size effect encourages innovations in sectors that use the more abundant factor. Hence, this effect works in the opposite direction compared to the price effect.
• The risk effect: The risk effect results from the fact that Vz;t=Vm;t decreases inRz;tC1=Rm;tC1. The lower the relative probability of successful innovation in sectorz,z=m, the higher is the corresponding relative loan rateRz=Rmand the risk of default on external credit reflected in the interest rate on loans. Thus, the risk effect directs innovations to sectors with a higher probability of research success. This illustrates the significance of the assumption, how the currentj’s relate to the number of previous innovations in any industry#jand sectorm,z with respect to the direction of technical change.
• Theproductivity effect: The ratioVz;t=Vm;tincreases inQz;tC1=Qm;tC1and thus the productivity effect encourages innovations in sectors with a higher productivity.
The result established in theorem 7 resembles the effects stated amongst others in Acemoglu (2002) plus the risk effect. Altogether the net effect of these partly counteracting forces determines the direction of technical change. The following theorem establishes the second fundamental result:
Theorem 8 If innovators are capital constrained and R&D is financed by loans, then credit interest rates influence the direction of technical change.
First, credit interest rates contain a risk premium given by the inverse of the probability of successful innovation. The risk premia and the loan rates relate inversely to another: the lower the probability of successful innovation in any one sector, the higher is the risk of default on credit obligations by innovators who try to invent new capital goods in that sector. Consequently, banks compensate for the higher risk and demand increasing risk premia and thus charge innovators higher interest rates on credit. A relatively higher interest rate—in, say, sectorz—ceteris paribus implies relatively increasing credit costs for innovators whose plan is to invent new Z-complementary capital goods. To keep profits constant, innovators would have to cut R&D investment. This would result in a lower probability of success in sectorz. The expected profitability of developing newZ-complementary capital goods would decline and innovators would direct R&D effort to the relatively less expensive sector: The presence of banks adds an additional component to the determinants of directed technical change through the relative interest rate charged on loansRz;t=Rm;t, which is equivalent to the inverse of the probabilities of successful innovation.z;t=m;t/1.
Second, R&D expenditures are proportional to (expected) profits. Credit costs determine equilibrium profits and thus the amount of resources devoted to R&D in equilibrium. This in turn influences indirectly the probability of successful innovation and thus therateof technical change in both sectors.12
12As a side note, the literature on induced innovation (e.g. Hicks1932; Habakkuk1962; Kennedy 1964; Dranakis and Phelps1965; Samuelson1965) states that relative factor prices influence the type of technological progress. In particular, the literature argues that innovations are directed at
“more expensive” factors. In the discussion here, I confine myself to the role of (output)-prices, market size effects et cetera. However, we can show the similarity between the present approach and a formulation that considers factor costs as determinants to develop new technologies rather than output prices: Combine the capital goods demand stated in Eqs. (21a) and (21b) with the first order conditions of the intermediate firms with respect to the factorsMandZgiven in (20a). Using these expressions we can rewrite (63), (64) as
…mDwMM and …zDwZZ: (68)
Then one can express the relative profitability of developingZ-complementary capital goods in terms of factor costswM;wZand market sizesMandZand the relative profitability of developing newZ-complementary capital goods then reads
…z
…m
D wZ
wM
Z
M: (69)
For purposes of a compact notation, I employ the following notation to all relative equilibrium variables from now on: Letam andazdenote arbitrary sector-specific variables for sectormand sectorZrespectively. Then we define therelative variable with respect to sector ZasaQ WD aaz
m. Then relative expected profits of innovation in sectorz, given in Eq. (67), equal
VQtD Qp
11
tC1ZQtC1QQtC1RQtC11: (70)
To get final expressions of relative profits and relative productivity, we have to eliminate endogenous pricespm;t,pz;tand also loan ratesRm;tandRz;t.
Using (34), insertNjWD.1/1C1p
11
j Jforj2 fm;zg, use (56) and take the ratio gives
QtD
Nz;tC1
Nm;tC1
1˛˛ D
pQ
11
tC1ZQtC1
1˛˛
: (71)
Note that the factorsMtandZtare constant in supply, so as long as the relative price pQtis constant,Qtis also constant. Since relative credit interest rates relate inversely to the ratio of success-probabilities, relative credit interest rates read
RQtD
pQ
11
tC1ZQtC1
1˛˛
: (72)
To highlight the role of the risk effect in determining the direction of technical change, we eliminate the relative pricepQt, as given in (53), from (72) in a first step.
The risk effect then reads RQtD
Q"ZQtC11QQtC11 .1˛/˛
; (73)
where WD 1C.1/."1/is the elasticity of substitution between the two factorsMandZ.13 Inspection of (73) reveals that the response of the risk effect to
Equation (69) indicates a higher incentive to innovate for factors that are more expensive. This result shows the equivalence of the approach presented here (and for instance by Acemoglu2002) that considers output prices, and the approach in the induced innovation literature cited above, which concentrates on factor input prices.
13For the purpose of this study it is important to distinguish the cases where"71and thus71. For the relevant parameter values of"and, the exponent is always positive, i.e.1=.1˛/ > 0, since˛20; 1Œand > 0by assumption.
an increase in the relative supply of factorsZQ WD Z=M depends on the size of (since1˛ > 0). In fact (73) implies the following proposition:
Proposition 1 If factors M and Z are gross complements," < 1, < 1, the risk effect increases with an increase in the relative supply of factorZ.Q
If factors M and Z are gross substitutes," > 1, > 1, the risk effect decreases with an increase in the relative supply of factorZ.Q
After elimination ofpQt, as given in (53), relative expected profits from innovation in sectorz, given in Eq. (67), read
VQtD
Q"ZQ1QQ1RQ 1
: (74)
Ignore the risk effect in Eq. (74) for the moment (and suppose thatRQ D 1). The parameter has a crucial impact on the direction of technical changes. If > 1,M andZare gross substitutes and anincreasein the relative factor supplyZQ WD Z=M (either because the supply ofZincreases or equivalently, the supply ofMdecreases) willincreasethe relative profitability of inventingZ-complementary capital goods VQt. On the one hand, ifZQ increases, factorZ becomes relatively more abundant.
This translates into a larger market for the capital goods that complementZ, which increasesVQ (this is referred to as the “market-size effect”). On the other hand, ifZQ increases, factorM becomes relatively scarce and thus relatively more expensive.
This translates into higher prices of goods that useM-complementary capital goods in the production process and thereby increase the profit from inventing thoseM- complementary capital goods. ThusV decreasesQ (this represents the “price-effect”).
Consequently, the price effect and the market-size effect work in opposite directions.
If > 1,V, as given in (74),Q increases with an increaseinZQ and we can conclude that the market size effect dominates the price effect. If < 1,V decreases withQ an increaseinZQ and the price effect dominates the market-size effect. Together this implies that the parameterregulates whether the price effect dominates the market size effect.
Next consider the response of relative expected profits to a change in the relative factor supplyZ, given that the risk effect is an additional determinant in the directionQ of technical changes (and compare this case to the hypothetical situation above, where we treated the risk effectRQ as equal to 1). For that, insertRQ from Eq. (73) into (74):
VQtD
"ZQtC11QQtC'1 .1˛/1
; 'WD1.1˛/: (75)
Inspection of (75) shows first that relative expected profits basically respond to an increase in the relative factor supplyZQ in the manner described above: If > 1, VQt, as given in (75), increases with an increase inZQ and the market size effect dominates the price effect. If < 1,VQtdecreases with an increaseinZQand the price
effect dominates the market-size effect. So the result derived by Acemoglu (2002) that the elasticity of substitution plays a crucial role in determining the direction of technical change does not change if we consider capital constrained firms and financial intermediation.
Additionally, comparing (74) (and still assumeRQ D 1there) with (75) reveals that the response of relative profits to an increase inZQ changes, if we additionally account for the risk effect in the determinants of technical change:
Proposition 2 Relative expected profitsVQtincluding the risk effect respond stronger to an increase in relative factor supplyZ. This result holds independent of the sizeQ of the elasticity of substitution.
Taken together, Propositions1 and 2 imply that the elasticity of substitution between factors regulates the way in which innovators’ incentive to invent Z- complementary capital goods responds if the relative factor supply changes in the first place and additionally, this response changes if the risk effect of private sector lending enters the determinants of directed technical change.14