Working Paper: Cost of Regulatory Error when Establishing a Price Cap

Transcription

1 Working Paper: Cost of Regulatory Error when Establishing a Price Cap January

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3 1 Cost of Regulatory Error when Establishing a Price Cap Abstract Regulators often motivate their wish to impose a price cap by care for consumers. We use a simple model to demonstrate that a price cap maximising consumer surplus can be nonoptimal from the perspective of total social welfare. Further, the regulators might obtain incorrect evaluation of price dynamics or incomplete information on producer cost which is likely to aggravate the negative impact of a mis-specified price cap on both consumer surplus and social welfare. Politicians have long discussed the idea of energy price caps. This paper investigates the impact of an incorrectly imposed price cap on social welfare. Unsurprisingly, a socially suboptimal price cap reduces welfare, the magnitude depending not only on the scale of error, but also on the correct assumptions about demand range and producer s cost. In a base model with monopoly and uncertain demand, there is only a small range of price cap values that increases social welfare. A very strict cap close to marginal cost is likely to reduce welfare. Since the true level of production cost is private information, pinning down a socially optimal price cap is not an easy task where a small mistake would generate substantial welfare losses not only to producers but also to consumers. We first present a tractable analytical model to examine the impact of price cap on producer s decision and social welfare. We then examine the reduction in social welfare 1.1 Base model Monopoly producer and uncertain demand We assume a monopoly producer with zero fixed cost and the following profit function: π(θ, Q) = Q(P c) where π profit, P monopoly price and c constant marginal cost. The inverse linear demand function is given by P(θ, Q) = θ Q, where θ is a reservation price. The monopoly s optimisation problem under certain demand is standard, with the optimal output given as Q = (θ c)/2, and the equilibrium price given as P = (θ + c)/2. Uncertainty is modelled as a price shock or a varying reservation price. 1 The θ parameter is distributed on the interval [θ L, ] according to some cumulative distribution function, cdf, F(θ), that is θ~f(θ), F: [θ L, ] [0,1]. Hence, f(θ) probability distribution function, pdf, with f(θ)dθ = 1. θ L The expected value of price shock is given as 1 Alternatively, θ can reflect variation in time. For example, electricity demand has daily, weekly and yearly fluctuations

4 E(θ) = θf(θ)dθ θ L Monopoly s output decision under uncertainty is similar to that under certainty. Since demand is not known in advance, the monopolist prefers to fix the output and absorb the price shock. The expected profit equals to and the equilibrium output becomes E(π) = π(θ, Q)f(θ)dθ = Q(E(θ) Q c) θ L A single price realisation is now given as Q = (E(θ) c)/2 P = θ Q = (2θ E(θ) + c)/2 while its expected value still resembles the price under certainty: Price cap E(P ) = (E(θ) + c)/2 A price cap p breaks the demand function in two pieces: horizontal piece and downward sloping piece (see Figure 1 below, left panel). The demand function changes to P(θ, Q) = min {p, θ Q}. The range of price shocks [θ L, ] also breaks in two intervals: [θ L, θ ] and [θ, ] where θ = p + Q. When the equilibrium price P happens to exceed the cap p, the producer has to increase its output to Q(p ) = θ p and, hence lower the price, to comply with the cap (Figure 1, right panel). Figure 1: Capped demand curve (left) and capped marginal cost (right). The expected profit breaks in two parts: θ E(π) = Q(P(θ, Q) c)f(θ)dθ + Q(p c)f(θ)dθ θ L θ p +Q θ = Q( θ Q c)f(θ)dθ + H θ Q(p c)f(θ)dθ L p +Q The producer now behaves as a monopolist in the area of downward sloping demand and as a competitive player in the area of horizontal demand. His maximisation decision as a monopolist affects the monopoly price and hence the probability whether the price cap is binding or not, i.e. the relative size of θ intervals

5 1.1.3 Optimisation problem given uniform distribution Since analytical solution to the maximisation problem with uncertainty and price cap is not possible, we investigate an illustrative example with a uniform distribution: f(θ) = 1 θ L. The expected value of price shock equals to the middle of the full interval [θ L, ]: E(θ) = + θ L 2, The expected value of shock conditional on a price cap equals to the middle of the uncapped interval [θ L, θ ]: E(θ p ) = E(θ θ < θ ) = θ + θ L 2 Monopolist s expected profit: turns to be a weighted average of two components: p +Q dθ dθ E(π) = Q( θ Q c) + Q(p c) θ L p +Q = θ θ L Q(E(θ p ) Q c) + θ Q(p c) = θ θ L π mon + θ π cap The first term θ θ L θ L is the probability of monopoly outcome and (E(θ p ) Q c) is expected profit margin. The second term θ θ L is the probability of price-capped outcome and (p c) is certainty profit margin. Smaller output means higher price and eventually higher monopoly profit but it also increases chances of being capped. As Q, π mon but θ θ L θ L. In a similar way, a lower price cap increases probability of the capped profit but decreases its amount. As p, θ θ L and π cap. The analytical solution to the maximisation problem exists, it is a positive root of a quadratic equation and so not very intuitive. However, it is a global profit maximum for Q > 0 and therefore it is unique. Q = 2 3 (p θ L) p 2 + θ L 2 8p θ L + 6p 6c( ) The sign of Q p and, hence, the impact of p on Q is ambiguous. A numerical example (see below) shows that a smaller price cap may actually reduce output Producer and consumer surplus Producer surplus is the expected monopoly profit at the equilibrium output: PS = E(π ) = θ θ L Q (E(θ p ) Q c) + θ Q (p c) Consumer surplus is a weighted average of CS in case of monopoly and CS in case of perfect competition: CS = p +Q θ L 1 2 Q (θ P(θ, Q dθ )) + p +Q 1 dθ 2 Q p - 3 -

6 = θ θ L 1 2 (Q ) 2 + θ 1 2 Q p Finally, social welfare is the sum of producer and consumer surpluses: where α is the relative weight of consumer surplus. W = α CS + (1 α) PS As is the case with output, the impact of price cap on surplus and social welfare is not defined. The numerical example shows that the price cap may increase or decrease welfare. 1.2 Base model numerical example The parameters have the following values: = 2, θ L = 1, c = 1. The price cap p ranges from 1 to 2, with a step of 0.1. The numerical example shows that output can look like a bell-shaped function of price cap (Figure 2). A very high price cap is no binding so the producer behaves as if it was standard monopoly with uncertain demand. Once the cap is lowered and becomes binding, the producer increases its output. The output reaches its maximum at p = 0.5 but the increase is not very substantial. Then, as the cap is tightened, the producer reduces output. The intuition is that a smaller cap increases the probability of capped profit but reduces the capped profit margin (p c). The latter effect dominates, and the producer prefers to reduce output. An extremely low price cap drives output, in fact, down to zero. In other words, setting a cap close to marginal cost reduces incentives for a monopolist to produce anything. Figure 2: Equilibrium output as a function of price cap - 4 -

7 Figure 3 exhibits producer profit which falls naturally as the cap becomes tighter. Figure 3: Producer profit By contrast consumer surplus first increases and only then start to decrease as the cap is tightened (Figure 4). Consumer surplus is the highest at a moderate price cap p = 1.3. Total social welfare behaves in a similar way (Figure 5) although its increase is not so pronounced since the increase in consumer surplus is dampened by decrease in producer profit. In addition, the welfare maximum is achieved at a different price cap p = 1.4. The total welfare function remains unaffected largely unaffected by the choice of α, i.e. the relative weight of consumer vs. producer surplus. Consumer surplus is above its value in the standard monopoly case for a relatively wide range of price cap, p [1.1, 1.6]. However, the total welfare lies above its benchmark monopoly level over a different and slightlysmaller range of the price cap p [1.3, 1.7]. In short, the regulator faces a trade-off between consumer and total welfare. Maximising one of the values leads to the suboptimal level of another value. Figure 4: Consumer surplus - 5 -

8 Figure 5: Total welfare (alpha = 0.5) 1.3 Impact of regulatory error Scenarios Any set of parameter values would generate its own optimal price cap so that making an error in parameters would lead to a suboptimal price cap and lower welfare. This section investigates the impact of such error on welfare change, under various scenarios. We assume that the regulator has: 1. Correct assumption about the mean shock value but wrong estimate of the shock range. 2. Correct assumption about the mean monopoly price but wrong estimate of the shock range. 3. Wrong assumption about the shock range. 4. Wrong assumption about the producer s cost. In other words, in the first and second scenarios the regulator obtains correct estimates of the mean values but wrong estimate of shock variance. In the third scenarios, the regulator under-estimate the mean value. Finally, the forth scenario models cost information asymmetry. In the first three cases we would assume c = θ L so that under the lowest price shock the equilibrium price equal cost and the monopolist prefers to produce nothing: P(θ L, Q = 0) = c. In the fourth case, we would allow c > θ L to reflect the fact that the regulator holds a correct view on demand distribution but has wrong perception of producer s cost. Constant shock mean wrong shock range (variance) This scenario assumes that the unrestricted mean value of price shock is estimated correctly: E(θ) = const while the variance is not. Under uniform distribution, E(θ) = +θ L and V(θ) = ( θ L ) 2. Together, it implies that the shock range 2 12 [θ L, ] becomes longer or shorter by adding or subtracting the same amount on either end of the interval.: + θ L = 0, where θ i = θ actual i θ estim i, i {L, H}

9 Let us denote the change at the top of the interval by δ = so that θ L = δ. When the interval is estimated to be shorter, the variance is also underestimated. The actual variance is higher than the assumed one: V(θ) = δ( estim θ L estim +δ) 3 > 0. Constant price mean wrong shock range (variance) This scenario assumes that the unrestricted mean price is estimated correctly: E(P ) = E(θ)+c = const. The 2 shock range [θ L, ] expands or contracts while preserving the ratio: + 3 θ L = 0 (note that we assumed c = θ L above), that is = δ and θ L = δ/3. The difference between the actual and estimated mean shock and variance equal to E(θ) = δ/3 and V(θ) = 2δ(3 estim 3θ estim L +2δ), respectively. 27 Shifting the shock interval In this case the whole interval would shift to the left or to right by some amount: δ = = θ L. The shock mean would change by the same amount, E(θ) = δ, while the mean price would change by the half of the shift value: E(P ) = δ/2. Note that the shock variance V(θ) would remain the same. Higher producer cost The final case assumed the difference between actual and estimated producer s cost: δ = c = c actual c estim. This is the only scenario where we allow > θ L. The mean value and variance of the shock would remain the same, but the mean price and all other parameters would clearly be affected Comparison The following table summarises the four cases. A shift parameter δ is used across the scenarios to compare the change of initial parameters, mean price and shock values and shock variance. Table 1: Parameter change under different scenarios Scenario θ L c E(θ) V(θ) E(P ) Constant shock mean δ δ 0 0 δ(θ estim H θ estim L + δ) 0 3 Constant price mean δ δ/3 0 δ/3 2δ(3θ estim H 3θ estim L + 2δ) 0 27 Shifting the shock interval δ δ δ/2 Higher producer cost 0 0 δ 0 0 δ/2 We are interested in how the parameter δ > 0 affects the optimal price cap and welfare. The regulator would use the old set of parameters (section 1.2) and the associated welfare function to generate an optimal price cap: p reg = arg max W(p θ i estimate, δ = 0) However, the true state of the economy is characterised by a new set of parameters and the actual optimal cap: p actaul = arg max W(p θ i estimate, δ > 0) = arg max W(p θ i actual ) - 7 -

10 The difference between the two caps is simply: p = p actual p reg However, a welfare loss due to the regulatory error equals to the difference between the welfare functions with the actual versus the regulator s optimal price cap, given the actual parameters: W = W(p actual θ i actual ) W(p reg θ i actual ) 1.4 Impact of regulatory error numerical result We use numerical simulation to quantify the difference in price caps and the welfare loss. The existing set of parameters is given by = 2, θ L = 1, c = 1, and the regulator s optimal price cap p reg equals to We set δ ranging from 0 to 0.4 with a step of In the third scenario increasing δ beyond 0.4 would still produce p actual but with θ L actual > p reg (1.4>1.39) it would be impossible to estimate W(p reg θ i actual ). In the fourth scenario δ does not exceed 0.3, for a similar reason: The actual cost of 1.35 is too close to the regulator s price cap of 1.39 and the monopoly would simply produce nothing. Figure 6 exhibits the excess of the actual price cap relative to the regulator s benchmark, in % terms. The actual optimal cap is always higher, or less strict, than the regulator s value. The finding suggests that a mistake in parameter estimates leads to a more stringent price cap than actually needed. The excess is the highest, or the regulator s cap is the most stringent, when the regulator misestimate the shock/demand range. Alternatively, an initially optimal regulator s cap might become suboptimal if the overall demand (shock interval) shifts upward due to economic development or other factors. For example, a severe winter and a sufficiently low price cap would generate high demand for energy which, in the extreme case, might have to be rationed. Figure 6: Excess of actual optimal price cap over the regulator s cap A mistake in parameters has a different impact on welfare losses in the scenarios 1-2 and 3-4 (Figure 7, left and right panes respectively). The welfare loss is relatively small in the first two cases, no more than 6%, although in absolute terms, the loss might be substantial. The welfare suffers much more from the regulatory - 8 -

11 error in the third and fourth scenarios. In the 4 th case with δ = 0.3 the loss reaches extreme 75% or 3/4 of welfare with an actual optimal cap. A milder error (δ = 0.2) would create losses of c %. Figure 7: Welfare loss, due to regulatory error. In the first two scenarios, the excess of price cap and welfare loss combined together, suggest that the regulatory cap can be set relatively high, or not so strict. It would not affect the welfare much but it would reduce variation in demand uncertainty, and hence prices (cf. Table 1, column V(θ)). In the second two scenarios, alleviating the price cap, that is moving it towards the actual optimal value, would dramatically improve welfare. 1.5 Conclusion A simple theoretical model, together with a numerical example, shows that a price cap has an ambiguous impact on output, consumer surplus and total welfare. First, a relatively high cap is never binding at all and as such does not affect producer s output decision. Second, a moderate price cap increases output, consumer surplus and total welfare. However, this positive range is not very large for the social welfare. Finally, a tight cap severely distorts producer s monopoly incentives and so drives output almost to zero, and also decreases producer and consumer surpluses and social welfare. Implementing a price cap as a practical political decision might be a difficult task and is likely to be welfarereducing. A regulator might obtain incorrect estimates of demand distribution, either variation or demand range, or producer s cost. The resulting regulator s price cap would be suboptimal and could be increased, to reflect uncertainty without affecting welfare, or to improve welfare in general. Alternatively, the parameters might change over time, and the regulator s cap being initially optimal might reduce welfare later on, especially if the overall economic situation or specific demand changes in a substantial way. A price cap revision would typically lag behind