8. THE PRODUCER'S OPTIMUM UNDER INCREASING RETURNS TO SCALE

In the current economy we are observing an unprecedented phenomenon. For some technologies (e.g. those facilitating the provision and intermediation of information) a firm will pay high fixed costs (and potentially overcome other barriers to entry to the market). Subsequently however, practically any increase in the volume of services it provides (due, for example, to a growing customer base) will increase its revenue, while its costs increase only slightly, if at all. In such case, its marginal costs are zero and its returns to scale as output increases are increasing over the entire domain of the production function (contrary to the assumption of decreasing returns to scale made in standard microeconomics).

Standard neoclassical microeconomics does not concern itself too much with this situation. For a viable technology with increasing returns to scale the optimal volume of production (in the sense of maximum profitability) tends to infinity in conditions of increasing returns to scale with given prices. In a monopoly situation, the optimal production volume is q*, at which marginal revenue is equal to (in this case zero) marginal costs MC = 0. The optimal price p* and the optimal production volume q* are both given by the demand function D(p) and marginal revenue MR(q), which can be derived from the demand function. Profit is maximized at the output volume at which marginal revenue equals marginal costs. Figure 40 illustrates this for a linear demand function.

Figure 40: The monopolist's optimum E given zero marginal costs in standard microeconomics

A different situation arises if the monopolist is threatened by the entry of a competitor hitherto deterred by the high fixed costs. This is not directly a standard game situation — the "opponent" is merely a suspected possibility. The threshold price (profitability) that attracts the competitor is unknown to the decision-taker. It involves a sort of choice under uncertainty.^{[1]}

Standard economics does not take into account the risks associated with the potential entry of an unknown competitor into a monopoly industry and the related aversion to situations with a high degree of such risk in a producer's decisions. Yet risk aversion and aversion to the unknown are among the most important characteristics of people's choices in the theory of human motivation (self-actualization).^{[2]}

In standard economic models, risk aversion is modelled on the basis of a strictly concave function of the expected utility of income^{[3]} or by considering the size of the perceived loss in relation to the risk by applying a criterion in the for m of a weighted average of the expected value and the variance, referred to as mean-variance utility (ibid.). Another possibility is the stochastic approach to understanding risk.^{[4]}

We assume that firms have the option of entering the industry. However, this entry is not cost-free but is conditional on payment of a high entrance fee (taking the form of fixed costs). The producer in our model operates in conditions of increasing returns to scale and zero marginal costs. It does not try to maximize its instantaneous profit, as an extremely high profit might attract another agent able and willing to pay the entrance fee.

If a high price or high profit does attract a competitor, the firm's revenue will fall dramatically. Let us assume that a second agent will enter when the price exceeds ph, namely the threshold price at which it pays the second agent to pay the high entry costs. We will also assume that even after the second agent enters the market, the incumbent monopolist will continue (at least temporarily) to have a price-setting advantage over the newly arrived producer. This means that the new competitor will set the same price as the incumbent monopolist.^{[5]} Let us assume that the total quantity demanded D(p) will be split equally between the two oligopolists, so the demand of the decision-taker d(p) is halved if a competitor enters the market (i.e. d(p) = D(p)/2 for p > ph) and is equal to total demand in the opposite case (i.e. d(p) = D(p) for p < ph). The slope of the marginal revenue curve will be one-quarter of the slope of the total demand curve D(p) for p> ph and Just one-half for p ≤ ph.

Figure 41: Threat of entry of a second agent: individual demand d{p) and marginal revenue MR(q)

Point E, which is the optimum in the standard approach (Figure 41), is not a feasible solution here. Point E1 at price ph and point E2 at price pz > ph come into consideration for the choice. However, our firm (i.e. the firm whose choice we are modelling) does not know the threshold price ph and is threatened with a sudden drop in output combined with a fatal decrease in profitability.

The profit function (the relation between profit П and output price p) has the shape depicted in Figure 42. It is discontinuous and has two local maxima — at price ph and at price p^ These local maxima correspond to points E1 and E2 in Figure 41. Because we assume a threat due to the entry of a competitor, it is sensible to count on the local maximum at price pz being less profitable (if at all). This is because it represents a substantially lower number of customers, since the considerable decrease in the volume of output is associated with a significant increase in average costs.

Figure 42: The profit function П(p), discontinuous at point ph

If, therefore, the firm knows the marginal price for the entry of the second agent ph (a competitor will enter the market when this price is exceeded), the optimal (profit-maximizing) strategy is to choose this marginal price ph with output volume qh (i.e. point E1 in Figure 41). The optimal pricing strategy in this case balances on the very margin of survival. If, though, a firm "with a self-preservation instinct” does not have such information, it will not try to get even close to price p = ph, because it would face a high risk of extinction. However, it will not choose the local profit maximum at price pz (point E2 in Figure 41) either, because that is a loss situation.

So, what pricing strategy does the firm prefer in this situation? It is reasonable to expect the decision-taker to choose the strategy it regards as optimal from the point of view of the firm's probability of survival.^{[6]} In this case, the firm tries to avoid risky situations and has a tendency to pull back from its extinction zone. Figure 43 illustrates the extinction zone Ψ. The area below the horizontal axis represents extinction due to low profitability while the area to the right of the p = ph line represents the entry of competitors attracted by the high price.

Figure 43: Extinction zone Ψ º {(p, π); (П < 0) V (p > ph)}

The optimal price will lie somewhere within the open interval (pa, ph). Let us assume that the two threats (entry of competitor and low profit) are independent. We will denote the probability of extinction at output price p by η1(p), η2(p), where η1(p) relates to the threat of ruinous entry of competition and η2(p) is the probability of extinction due to profit falling below zero.

[1] The issue of choice when the price is unknown is studied mainly in consumer theory (see, for example, the review article McMillan, J., Rothschild, M.: Searching for the Lowest Price when the Distribution of Prices is Unknown. Journal of Political Economy 82, 4(1974), 689–711, but also in general microeco-nomics (see, for example, Newbery, D. M. G., Stiglitz, J. E.: The Theory of Commodity Price Stabilization: A Study in the Economics of Risk. Oxford: Oxford University Press, 1981). For general information on this area of microeconomics see also Gravelle, H., Rees, R.: Microeconomics. London: Longman, 1992.

[2] See, for example, Maslow, A. H.: Motivation and Personality. New York: Harper and Row, 1970, or Hlaváček, J. et al.: Mikroekonomie sounáležitosti se společenstvím. Praha: Karolinum, 1999, section 2.1.

[3] The Arrow-Pratt measure of local risk aversion is defined as r(d) = –u´´(d)/u´(d), where d is income and u is expected utility. The Arrow-Pratt measure of relative risk aversion is defined as ρ(d) = –d • u´´(d)/u´(d). See Varian, H. R.: Microeconomic Analysis. New York: W. W. Norton, 1992.

[4] See, for example, Stiglitz, J.: Incentives, Risk and Information: Notes Toward a Theory of Hierarchy. Bell Journal of Economics 6, 2(1975): 552–79, or Diamond, P., Stiglitz, J. E.: Increases in Risk and Risk Aver-sion. Journal of Economic Theory 8, 3(1974): 337–60. For an empirical analysis of risk aversion, see Applebaum, E., Katz, E.: Measures of Risk Aversion and Comparative Statics of Industry Equilibrium. American Economic Review 76, 3(1986): 524–29.

[5] This is the “leader–follower” relationship from the Stackelberg model applied to price-setting. See Gravelle, H., Rees, R.: Microeconomics. London: Longman, 1992.

[6] The general formulation of the producer's decision-making model in conditions of multiple threats is given in Hlaváček, J. et al.: Mikroekonomie sounáležitosti se společenstvím. Praha: Karolinum, 1999, pp. 100–11.

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