Some countries such as China and the USA have multiple, competing incubators. Many other countries seem to have monopolistic incubator systems. Why should that be so? This paper argues that countries with competing incubators may simply believe that they have low risk (market and regulatory). On the other hand, countries with a monopoly incubator do not necessarily have high risk, they may be uncertain about the prevailing risk level.

### Knight's typology of probabilities

Prominent economist Frank H. Knight, in his 1921 book *Risk, Uncertainty, and Profit*^{2} famously recognized a distinction between “three different types of probability.” He named those (1) “*a priori probability*,” (2) “statistical probability” and (3) “estimates.”

He wrote that *a priori* probability is "absolutely homogeneous classification of instances completely identical except for really indeterminate factors," meaning a theoretical probability, which is independent of any particular experiment. An example would be the probability for obtaining a Heads or a Tails at the throw of a fair coin, assumed to be 1/2. He described his second category, statistical probability, as "empirical evaluation of the frequency of association between predicates," meaning frequency distributions derived from empirical observations.

Knight described his third category, "estimates," as probability dealing with "no valid basis of any kind for classifying instances." An example he presented is "the 'probability' of error … in the judgment" when a manufacturer makes an estimate about "the advisability of making a large commitment in increasing the capacity of his works." Knight argued "it is manifestly meaningless to speak of either calculating such a probability a priori or of determining it empirically by studying a large number of instances. The essential and outstanding fact is that the 'instance' in question is so entirely unique that there are no others or not a sufficient number to make it possible to tabulate enough like it to form a basis for any inference of value about any real probability in the case we are interested in." Thus, Knight's third type of "probability" describes a situation in which an economic or social actor cannot assign either an *a priori* probability, or an empirical frequency to an "instance." In this article Knight's expression "instance" is treated synonymously with "a state of the world."

In the discussion below, "risk" means an empirical probability, i.e. the second type of Knightian probability, as in the phrase "historical market risk." "Uncertainty" refers to Knight's third type of probability. Knight's first type is referred to as a "theoretical prior." To simplify, think of Knight's first and second types as "known probabilities," and of his third type as "unknown probability." This distinction can have significant implications when formulating and testing hypotheses in economics, as well as important policy implications. This paper discusses, as a specific example, the expected relation (i) between risk and "speed of innovation," *versus* (ii) between uncertainty and "speed of innovation." It is shown that the two relationships can differ starkly in terms of optimal policy.

### Results in the literature

There is limited theoretical or empirical research on the subject of risk/uncertainty and innovation speed. Moreover, the relevant articles do not always differentiate between risk and uncertainty, or clarify what they mean by either term.

The textbook model of innovation is presented by Scherer (1967),^{3} reproduced in various industrial organization textbooks hence. That model implies higher risk speeds up innovation. In contrast, Stadler (1992)^{4} shows that in a dynamic oligopoly model, higher risk slows down innovation.

Carbonell and Rodriguez (2006)^{5} present a comprehensive review of the relevant literature, and comment on it. To our knowledge, there have not been more recent findings which substantially alter their remarks. They note:

Overall, empirical research has been inconclusive regarding the moderating role uncertainty. Findings from Kessler and Bierly (2002) … show that the [positive] link between innovation speed and success is weakened when market uncertainty is high. In contrast, the study of Chen et al. (2005) … show that a timed-based strategy is more suitable under conditions of high market uncertainty. On balance, we expect that market uncertainty weakens the positive effects of innovation speed on positional advantage, and new product performance.

### Contribution of the paper

Knight's typology of probabilities is extremely important for economic policy recommendations. Risk and uncertainty can have different, even opposing policy implications. For example, the relationship between risk and innovation can be the opposite of the relationship between uncertainty and innovation. The next subsection presents an example to show that while *risk* may speed up innovation, *uncertainty* can delay innovation. This can happen because a decision rule that the society is using can yield a different result under uncertainty, than under certainty.

#### Competition speeds up innovation

This section first discusses Scherer's 1967 model of innovation, reproduced in Viscusi, Harrington and Vernon (2005).^{6} The model is intended to present the relationship between market structure and innovation and is graphically demonstrated in Figure 1. In the figure, the horizontal axis measures time and the vertical axis measures present value. There are three straight lines labeled V1, V3 and V5. Line Vn represents the present value ("PV") at time zero of profits accruing to innovation to a firm that innovates in a market with n firms. Thus, line V1 represents the PV, at time zero, of innovation to a monopolist. Line V3 represents the PV, at t=0, of innovation to an innovator in a market with three competing firms. Line V5 represents the PV, at t=0, of innovation in a market with 5 firms. On each of these lines the PV of innovation is diminishing because of time-discounting, i.e. because profits in the distant future are discounted more heavily than profits in the near future.

*Figure 1-Given a market structure, optimal innovation is determined at the point where the PV line for that market structure is parallel to the cost function. Source: Viscusi, Harrington and Vernon (2005). *

At a given t, the PV decreases as the number of firms increases. For example, in Figure 1, it is seen that V1 > V3 > V5 for t > 0. This is due to two effects. First, except in a monopoly market, an innovation can be copied by the competitors, which lowers the value of innovation to the innovator. Second, the probability that the innovation will be copied increases with time. Before the innovation is copied, the innovator would continue to earn monopoly profits. Once the innovation is copied, each of the firms in the market would earn at most one-n'th of monopoly profits (in case of perfect collusion), or, worse, the equilibrium level of profits in an oligopoly market with n identical firms. As a result, n < m implies Vn(t) > Vm(t) for t > 0.

In Figure 1, the curved line represents the cost of innovation at time t, measured at t=0. This line represents the idea that R&D efficiency falls as more resources are used in a given time frame; hence R&D cost increases as a firm attempts to innovate sooner. The negative slope of this line reflects both time-discounting as well as increasing R&D efficiency (and hence lower cost) as time proceeds.

Scherer argues that given a market structure, a firm would innovate when the slope of the PV line equals the slope of the cost curve, as long as the PV of profits from innovation exceeds the cost of innovation. That is, a firm would innovate at time t* such that Vn'(t*) = C'(t*), provided Vn(t*) > C(t*). Thus, in Figure 1, the optimal innovation time for a monopolist is t1* because the slope of V1 is equal to the slope of the cost curve at t1*, i.e. V1'(t1*) = C'(t1*). Similarly in a market with three firms the optimal innovation time is t3* because the slope of V3 is equal to the slope of the cost curve at t3*, i.e. V1'(t3*) = C'(t3*). On the other hand, in this example innovation does not ever occur in a market with 5 firms because V5 lies below the cost curve at all times, i.e. V5(t) < C(t) for all t. The implication is that as long as the net PV of innovation (defined as PV minus cost) is non-negative, the optimal time to innovate is inversely related to the number of firms in the market. That is, "competition speeds up innovation."

The rest of this subsection establishes that higher risk speeds up innovation, based on Scherer's 1967 model presented above. The gist of the analysis can be summarized as follows. A higher the risk implies a higher discount rate. With a higher discount rate, PV of innovation decreases faster. As a result, the PV line for each market structure becomes steeper. Steepening of the line Vn (for a market with n firms) implies that the Vn line becomes parallel to the cost function at an earlier time, measured on the horizontal axis. Stated formally, if t* is the optimal time to innovate given discount rate d*, and t** is the optimal time to innovate given discount rate d**, then d** > d* implies t** < t*. Higher risk reduces innovation time, i.e. "risk increases innovation speed."

To formally analyze the effect of increased risk on innovation in Scherer's model, mathematical relations are fitted to three of the four lines in Figure 1 above.^{7} Figure 2 displays the fitted relations as heavy dashed lines superimposed on those in Figure 1.

*Figure 2-Analytical relations visually fitted to Figure 1.*

#### Risk speeds up innovation

Each of the fitted relations in Figure 2 assume a specific value for the discount rate, δ*. To analyze the effect of increased risk on the optimal innovation time, the discount rate was doubled to 2δ* and the fitted relations were re-computed using this higher discount rate. The resulting changes in V1, V3 and the cost function are displayed in Figure 3, as a shift from the heavy dashed line to the corresponding heavy smooth line immediately below the dashed line.

*Figure 3-Effect of an increase in the discount rate, reflecting higher risk.*

In Figure 3, the three thick arrows indicate the movement of V1, V3 and the cost function. With a higher discount rate each of the three lines pivot downward. The two thin arrows indicate how the optimal innovation time changes for each market structure. Based on a visual inspection of Figure 3 it can be conjectured that *ceteris paribus*, an increase in the discount rate would tend to lower the optimal innovation time, given the number of firms in the market. To verify this conjecture, a mathematical relationship was derived between the optimal innovation time tn* for the monopoly market and the three-firm market,^{8} and the discount rate δ. The resulting relations are displayed in Figure 4.

*Figure 4-How optimal innovation time tn* relates to the discount rateδ, in the monopoly market (n=1) and the three-firm market (n=3).*

In Figure 4, the higher of the two lines, labeled t1*(δ), corresponds to the optimal innovation time for a monopoly market as the discount rate δ varies from approximately 0.03 to 0.1. The lower line in Figure 4, labeled t3*(δ), corresponds to the optimal innovation time for a market with three firms as the discount rate δ varies from approximately 0.01 to 0.1. Both lines can be seen to be negatively-sloped, implying that as the discount rate increases, reflecting higher risk, firms find it rational to innovate sooner. Moreover, since both lines are downward-sloped, it can be stated that this effect is present regardless of the number of firms in the market – although the degree to which higher risk leads lowers optimal innovation time can depend on the market structure.

#### True uncertainty can slow down innovation

Having established that higher risk reduces optimal innovation time, turn to the next question, how might "true uncertainty" in the sense of Knightian type-3 probability affect optimal innovation time? To this end consider the following two states of the world:

Low risk (low discount rate)

High risk (high discount rate)

Assume that there is "true uncertainty" as to which state will prevail in the near future. That is, assume that the society is *uncertain* about which *risk* state will prevail in the near future. The society doesn't really have meaningful priors, so it cannot average out across the two risk states. Assume that actions available to the society are:

Set up a single incubator (monopoly),

Set up multiple competing incubators (competition).

Several countries including China and the USA have established multiple incubators that compete with each other in a broad sense.^{9} Many other countries have single-incubator systems. It is the hypothesis of this paper that each of these choices of market structure may in fact be optimal given the interplay between risk and uncertainty specific to a country. To analyze the effect of uncertainty on innovation time, a "game against Nature" is presented below. In this game, Nature decides the risk level that a country could face as either Low or High.

Suppose that given low risk and alternatively high risk, the optimum time to innovate t* under each market structure (monopoly vs. competition) is:

*Table 1-Demonstrative optimal innovation times for monopoly and competition under low risk vs. high risk*

Further, suppose the socially optimal time to innovate is s* = 7 and is not influenced by the risk level, perhaps because the society spreads risk over many projects, market and nonmarket.^{10} Assume that when faced with uncertainty the society applies the minimax principle to minimize its maximum regret. To analyze this, the regret levels are calculated as |t* – s*|:

*Table 2-Regret values computed as the absolute differences between optimal innovation times in Table 1 and the socially optimum time of 7 years *

The first step of the minimax principle is to find the maximum regret for each choice of market structure. For monopoly, the maximum regret is 3, under low risk. For competition the maximum regret is 4, under high risk.

*Table 3-Application of the minimax method to the regret values in Table 2. *

The second step of the minimax principle is to find the minimum of the two maximum regret levels calculated in the first step. The minimum of 3 and 4 is 3, meaning that the minimax method indicates that under uncertainty, the society should set up a monopoly incubator.

In the context of the above example, a country which knows it has low risk would choose "competition" from the outset and innovate in "5 years" (t*=5). A country which knows it has high risk would choose "monopoly" and innovate in "6 years" (t*=6). Assume that when faced with uncertainty the society uses the minimax principle, and minimizes its maximum regret. Given the regret levels in this example, the society chooses monopoly over competition. Innovation takes place in "10 years" if risk turns out to be low, and in "6 years" if risk turns out to be high. The result of this paper is that even though *risk* speeds up innovation given the market structure, *uncertainty* (weakly) delays innovation. This happens because in this example, when the society faces "true uncertainty," it chooses monopoly over competition.^{11}

This result illustrates that the distinction between risk and uncertainty is not merely an intellectual curiosity and can have important policy implications.

### Conclusion

This paper endeavors to highlight the important conceptual point that risk and uncertainty may have different policy implications. This point is illustrated by way of an example about the speed of innovation. It is demonstrated that while risk may increase the speed of innovation, uncertainty may delay innovation. Thus it is important to discriminate between "risk" and "uncertainty" when formulating hypotheses in economics.

It is argued that countries such as USA and China which have competing incubators may simply believe that they have low risk while countries with a monopoly incubator may be uncertain about the prevailing risk level. To exhibit the potentially different effects of risk versus uncertainty on the speed of innovation, this paper uses a highly stylized "textbook" example. That model is not espoused as the definitive instrument to analyze innovation. However, as a demonstrative tool, it helps to make the point that risk and uncertainty can have different policy implications.

- CRETC Competition & Regulation Economics Testimony and Consulting, LLC. The author thanks Dr. James Langenfeld for suggesting the opportunity to contribute to Euromoney, and acknowledges research input of Nikhil Manohar and comments from Dr. Paul Godek. Any remaining errors are the author's.
- Frank H. Knight.
*Risk, uncertainty and profit*. Courier Corporation, 2012 - Frederic M. Scherer. "Research and development resource allocation under rivalry."
*The Quarterly Journal of Economics*81, no. 3 (1967): 359-394. - Manfred Stadler. "Determinants of innovative activity in oligopolistic markets."
*Journal of Economics*56, no. 2 (1992): 137-156. - Pilar Carbonell and Ana Isabel Rodriguez. "The impact of market characteristics and innovation speed on perceptions of positional advantage and new product performance."
*International journal of research in marketing*23, no. 1 (2006): 1-12. - W. Kip Viscusi, Joseph E. Harrington, and John M. Vernon.
*Economics of regulation and antitrust*. MIT press, 2005. - Those relations are

V_{1}(t)=a⁄ (1+d)^{t-1}

V_{3}(t)=[a(1-t/10)+(a/3)(t/10)]⁄(1+d)^{t-1}

C(t)=((b t^{-k})⁄(1+d)^{t-1}

where*a*= 15.5, b = 12, d = 0.0425 and k = 0.425. - The fitted relations stated in footnote 7 above were used.
- Throughout this paper the term "incubator" is used inclusively, meaning the totality of endeavors that help to initiate and/or to develop innovation, including, for example, "accelerator" and "exit" functions.
- If this is too strong an assumption, the example can be modified to obtain the same qualitative result, provided that risk affects the social discount rate and the private discount rate differently.
- This result does not qualitatively change when the optimal innovation times are altered to reflect those shown on Figure 3. With a judicious choice of the socially optimum time to innovate, this qualitative result continues to hold.