Regulation under Asymmetric Information in Road Transport Networks∗ Jung S. You† and Ju Hyun Kim‡ February 16, 2015 PRELIMINARY AND INCOMPLETE DO NOT CIRCULATE

Abstract This paper is concerned with the analysis of regulations on a monopolistic public enterprise. We construct a game-theoretical contract model between the government and a public enterprise where the regulated firm holds private information about its cost-efficiency. Based on the theoretical model, we evaluate the welfare consequences of regulatory policies in road transport network. Our empirical analysis is based on the novel panel dataset comprised of traffic information, toll prices, governmental subsidies, and construction and maintenance costs of expressways in South Korea. We estimate our structural contract models including the travel demand function, cost function, and optimal subsidy function, and use the estimates to study the welfare implications of price and subsidy-based regulation policies. Keywords: Procurement, contract, regulation, transport network.

JEL Classification H20; L32; L38; L44; L51; L91; R41; R42; R48.

∗

We are grateful to Isabelle Perrigne and Quang Vuong for their comments and discussions. We thank the Korea

Institute of Public Finance and Korea Expressway Corporation for their great help in collecting data. We would like to thank seminar participants at Korea University and California State University, East Bay. † California State University, East Bay, Hayward, CA, U.S.A. Email: [email protected] ‡ University of North Carolina, Chapel Hill, NC, U.S.A. E-mail: [email protected]

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1

Introduction Korea Expressway Corporation (KEC), a public enterprise, has constructed and maintained

expressways in Korea since 1969 on behalf of the Korean government. The corporation began building expressways in the 1960s and plans to expand the expressways by 2020. The fact that a single firm serves almost the entire country justifies public intervention on pricing. The maintenance cost is fully covered by toll charges. The construction cost is partially covered by toll charges, but the rest is covered by governmental subsidy. The toll prices the public enterprise charges drivers is determined by the government, and the firm receives subsidies from the government to offset its construction costs. The purpose of the subsidy is to help KEC maintain nonnegative profit. The principles of expressway pricing is based on the Act on the Management of Public Institutions. According to the Act, the regulator aims to maximize the consumer welfare of expressway users. To determine price and subsidy amount, the government requires KEC to report the construction cost of expressways since the regulator is unlikely to be perfectly informed. His investigation to uncover the cost is expensive. The enterprise intends to maximize profit, which implies it would like to receive subsidies as much as possible, even by misreporting the construction cost. Therefore, the government designs the price and subsidy contracts not only to maximize consumer welfare, but also to incentivize the enterprise to report the cost truthfully. We construct and estimate a game-theoretical model of procurement contract between the government and the monopolistic public enterprise where the regulated enterprise holds private information of construction cost. Furthermore, we will quantify the extent to which the political and legal tensions affect the governmental subsidy to the regulated enterprise to deviate from the socially optimal level of subsidy. This deviation can occur under constraints faced by the regulator that arise from politics, legal systems, and external environments (Perrigne and Vuong, 2011).1 For example, if a party that is against the idea of regulating and subsidizing public enterprises dominates the parliament, the governmental regulator is under pressure to decrease the subsidy amount. Such a circumstance will be observable to both regulator and regulatee. In our model, the deviation of realized subsidies from optimal levels will be explained by introducing an additive random quantity to the subsidy equation. Estimating econometric models, we derive policy implications from the 1

According to the interview with the regulator and regulatee, they both think the observed subsidies are not optimal due to political circumstances.

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comparison between optimal and observed subsidies. The difference between optimal and observed subsidies indicates the intensity of political and legal tensions in the decision making of regulated transport network industries. In this paper, we consider the complexity of the contract model on a road network in our econometric analysis. Up to our understanding, empirical study on procurement contract under asymmetric information has mostly adopted a reduced form approach (Chiappori and Salani´e, 2003, Joskow and Rose, 1989). Moreover, the complexity of network is not well reflected in existent analysis. For the case of road network, construction, maintenance and subsidy are based on a road interval between two consecutive toll gates. On the other hand, traffic volume is measured by the number of cars that pass a specific pair of departure and exit toll gates. The cost and subsidy data are misaligned with traffic volume. Thus, we need to construct the demand between two consecutive toll gates to match up with cost and subsidy data. Our analysis attends to this aspect and this makes our work more unique. Our empirical analysis is based on the novel dataset comprised of traffic information, toll prices, governmental subsidies, construction and maintenance costs of major expressways over the last ten years. Using this dataset, we estimate expressway demand function, cost function, optimal subsidy function and the subsidy deviation from the optimal subsidy due to political and legal tension. The subsidy deviation is supposed to be independent of asymmetric information. To the best of our knowledge, this is the first study to do both game-theoretical and empirical analysis on a road vehicle transport network. Our game theoretical model is close to Baron and Myerson (1982), which pioneered the mechanism design approach to regulation and contract. Like our model, Baron and Myerson (1982) study regulatory price and subsidy design, which both maximize consumer welfare under nonnegative profit constraint. However, in their model, the regulated firm is a private entity who has an option to exit the market. Unlike theirs, the regulatee in our model is a public enterprise who does not have an opt-out option. As a public enterprise is more heavily regulated than a private one, the decision of subsidy level is susceptible to external environments that do not necessarily relate to the intrinsic characteristics of the firm. Brocas, Chan and Perrigne (2006) construct the game-theoretical model of regulating water utilities in California where the plants of water utilities are privately informed of their labor efficiency. In their model, the regulator’s objective is to maximize consumer surplus while guaranteeing the rate of return on 3

capital to utilities to meet revenue requirement. They assume the observed subsidies are at the optimal levels, which differs from our model. Their data consists of aggregate demand for each water plant, thus there is no complexity of water network. However, our data is more refined since it has traffic volume and price for every pair of toll gates. In addition, cost and subsidy data exist for road intervals between specific pairs of toll gates. Perrigne and Vuong (2011) provide a nonparametric identification method of a game-theoretical model of procurement contract. The information asymmetry in their model consists of both the regulatee’s labor efficiency and cost reduction effort. The complication of identification in their paper generates from the identification of the cost of public funds. Our model is simpler than theirs as we only consider unknown type of labor efficiency and do not insert the cost of public funds in our model. Both our model and theirs allow the observed subsidy to differ from the optimal subsidy. The rest of the paper is organized as follows. Section 2 describes the contract model and shows the closed form of optimal price and subsidy. Section 3 explains our dataset and Section 4 presents our estimation strategy and results. Section 5 concludes with a short summary of results and their implications.

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Model Demand function is denoted by q(p) where p is the unit price. The realized demand is affected

by random shock d in demand side, thus demand function is written as q(p, d ). Production cost C is a function of the level of inefficiency, the quantity produced and random shock c in supply side. Cost function C(θ, q, c ) is increasing in θ, which implies the less efficient firm generates greater cost. The inefficiency level θ is private information of the firm and is unknown to the regulator. However, ¯ with a density f (·) > 0 the regulator knows the type distribution F (·) of θ on the support of [θ, θ] where θ denotes the most efficient firm and θ¯ the least efficient one. The regulator determines the menu of (p(θ), T (θ)) where T is the money subsidy from the regulator to the regulatee. The objective of the regulator is to maximize consumer welfare under the constraints that the company reports its cost truthfully and makes nonnegative profit. As price and subsidy are determined by the regulator after the firm reports its inefficiency parameter θ, price and subsidy are written as p(θ) and T (θ). Since the subsidy is funded by tax money which consumers pay, net consumer

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surplus is written as

R∞

p(θ) E[q(s, d )]ds

− T (θ).

When the regulated firm chooses to report θˆ while its true type is θ, its objective is to maximize expected profit. The expected profit U from reporting θˆ with true type θ is written as ˆ θ) = p(θ)E[q(p( ˆ ˆ d )] − E[C(θ, q(p(θ), ˆ d ), c )] + T (θ). ˆ U (θ; θ),

For notational simplicity, we denote U (θ; θ) by U (θ). The regulator determines price and subsidy functions of θ which maximizes expected net consumer surplus under the constraints that they incentivize the firm to report θ truthfully and guarantee the expected profit from truthful report to be nonnegative. The problem the regulator solves is described as follows: Z

θ¯  Z ∞

max p(θ),T (θ) θ

 E[q(s, d )]ds − T (θ) f (θ)dθ

(1)

p(θ)

subject to ˆ θ) for all θˆ ∈ [θ, θ], ¯ U (θ) ≥ U (θ;

(2)

¯ U (θ) ≥ 0 for all θ ∈ [θ, θ],

(3)

ˆ θ) = p(θ)E[q(p( ˆ ˆ d )] − E[C(θ, q(p(θ), ˆ d ), c )] + T (θ). ˆ U (θ; θ),

(4)

and

where

The following theorem shows the optimal solution (p(θ), T (θ)) of the regulator’s problem. Theorem 1. The optimal solution (p∗ (θ), T ∗ (θ)) to the problem (??)-(??) satisfies the following:2 p∗ (θ) = 2

E[Cq (θ, q(p∗ (θ), d ), c ) · qp (p∗ (θ), d )] E[Cθq (θ, q(p∗ (θ), d ), c ) · qp (p∗ (θ), d )] F (θ) + , (5) E[qp (p∗ (θ), d )] E[qp (p∗ (θ), d )] f (θ)

The lower index refers to the partial derivative of the function.

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T ∗ (θ) = −p∗ (θ)E[q(p∗ (θ), d )] + E[C(θ, q(p∗ (θ), d ), c )] +

Z θ

θ

E[C θ (s, q(p∗ (s), d ), c )]ds.

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ˆ θ) and Proof. According to incentive compatibility condition (??), we have U (θ) ≥ U (θ; ˆ ≥ U (θ; θ). ˆ The first inequality is rewritten as U (θ)  ˆ θ) = U (θ; ˆ θ) ˆ − E[C(θ, q(p(θ), ˆ d ), c )] − E[C(θ, ˆ q(p(θ), ˆ d ), c )] . U (θ) ≥ U (θ;

(7)

The second inequality is rewritten as  ˆ ≥ U (θ; θ) ˆ = U (θ; θ) + E[C(θ, q(p(θ), d ), c )] − E[C(θ, ˆ q(p(θ), d ), c )] . U (θ)

(8)

Combining (??) and (??) gives  ˆ q(p(θ), d ), c )] − E[C(θ, q(p(θ), d ), c )] − E[C(θ, ˆ ≥ U (θ) − U (θ)  ˆ d ), c )] − E[C(θ, ˆ q(p(θ), ˆ d ), c )] ≥ − E[C(θ, q(p(θ), ˆ the condition (??) is translated into the following Dividing all by θ − θˆ and taking limit θ → θ, (local) condition: U˙ (θ) = −E[Cθ (θ, q(p(θ), d ), c )]. ¯ ≥ 0 as total cost increases in θ. Individual rationality condition (??) is translated to U (θ) We define Hamiltonian as follows:

H (p (θ) , T (θ) , π (θ) , θ) "Z # ∞ = E[q(s, d )]ds − T (θ) f (θ) + π(θ) [−E[Cθ (θ, q(p(θ), d ), c )]] p(θ)

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and Lagrangean as the following:

L(p(θ), T (θ), π(θ), µ(θ), θ)   = H (p (θ) , T (θ) , π (θ) , θ) + µ(θ) U (θ) − p(θ)E[q(θ), d ] + E[C(θ, q(p(θ), d ), c )] − T (θ)

The first order necessary conditions are: ∂L = −E [q (p (θ) , d )] f (θ)−π (θ) E [Cθq (θ, q (p (θ) , d ) , c ) · qp (p (θ) , d )]−µ (θ) E [q (p (θ) , d )] ∂p − µ (θ) p (θ) E [qp (p (θ) , d )] + µ (θ) E [Cq (θ, q (p (θ) , d ) , c ) · qp (p (θ) , d )] = 0 (9) ∂L = −f (θ) − µ(θ) = 0, ∂T π˙ = −

(10)

∂L = −µ(θ), ∂U

(11)

∂L U˙ (θ) = = −E[Cθ (θ, q(p(θ), d ), c )] ∂π

(12)

with transversality conditions: ¯ (θ) ¯ = 0, π(θ) ¯ ≥ 0, U (θ) ¯ ≥ 0. π(θ)U

(13)

Condition (??), µ(θ) = −f (θ), and condition (??) together lead to π(θ) = F (θ) at setting ¯ = F (θ) ¯ = 1, we have U (θ) ¯ = 0. Thus, (??) gives us π(θ) = 0. From condition (??) with π(θ) Z U (θ) =

θ¯

E[Cθ (s, K(s), q(p(s), d ), c )]ds. θ

Arranging (??) and (??), we find the equations.  The optimal price is found as in the equation (??). We can interpret the first term in the right hand side of equation (??) as expected marginal cost. Similarly, the expression

E[Cθq (θ,q(p∗ (θ),d ),c )·qp (p∗ (θ),d )] E[qp (p∗ (θ),d )]

in (??) is interpreted as the partial derivative of marginal cost with respect to inefficiency type θ. The residual expression

F (θ) f (θ)

can be approximated by θ − θ when the distribution is close to uniform

distribution. Thus, the second term in the right hand side of (??) is a change in marginal cost from the increased inefficiency. If the market is competitive without asymmetric information, the

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price equals to the marginal cost. In our model, the market is neither competitive nor completely informed. The inefficiency hidden to the principal pushes the market price beyond the marginal cost. Inserting the optimal subsidy given by (??) and optimal price by (??), we rewrite the expected profit in (??) from telling the true type θ as the following:

U (θ; θ) = p(θ)E[q(p(θ), d )] − E[C(θ, q(p(θ), d ), c )] + T (θ) Z θ¯ E[Cθ (s, q(p∗ (s), d ), c )]ds. = θ

The marginal cost Cθ is positive since the cost function is increasing in inefficiency level θ. The ¯ This says that the expected profit increases when θ moves away from the most inefficient level θ. firm’s expected profit is not zero though the participation constraint (??) is binding at break-even profit. The expected profit actually increases when the firm is cost-efficient. The subsidy is designed to promote the firm’s efficiency. In summary, the optimal price and subsidy in Theorem 1 work as the follows: Assuming the demand is inelastic, the firm’s sales revenue increases with greater inefficiency pushing the price up. At the same time, greater inefficiency reduces the subsidy, and thus, reduces the profit. Therefore, the price and subsidy schemes incentivize the firm to report the true inefficiency level θ to the regulator.

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Data We use price, demand, cost, subsidy data from KEC and additional data from other sources.

As Table ?? shows, the KEC constructs and manages most expressways in Korea. In Table ??, “Public” indicates the total distance of expressways that are constructed and maintained by KEC and “Private” indicates the total distance of expressways that are constructed and maintained by private companies. The expressways operated by private companies do not function as a hub of road networks. Thus, we can consider KEC as a monopolistic company in the market of expressway production. As explained in Introduction, the construction costs are covered partially by toll charges and

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Total Distance (km) 3749.35

Public 3749.35 (90.73%)

Private 347.46 (9.27%)

Table 1: Public expressway vs Private expressway the rest is covered by governmental subsidies in principles. However, as Figure ?? shows, actual subsidies and toll charges are not enough to cover total construction costs each year, and the debt incurs to the firm each year. The enterprise argues that the fact it has debts indicates the current subsidies are far from the optimal levels. Also, it says that the size of debt itself is the same as that of subsidy deviation from the optimal subsidy levels. The enterprise KEC runs other businesses like operating snack corners along the expressways etc clarify this part: the components of debts. Whether the debt generates solely from short subsidies is not clear. We will decompose the debt to subsidy deviation and some other causes.

Figure 1: Components of Construction Cost (billion won) Figure ?? shows the subsidy proportion of construction costs for each road interval. It shows that big construction projects receive relatively smaller subsidies and small projects are more heavily subsidized. A big construction likely occurs when it is expected to draw high demand and growth of economic activities, so that it’s justified from economic reasons. On the other hand, a small construction occurs mostly in remote places for short distance. Its construction starts from a political reason, for example, promoting balanced development regardless of economic value. The map of expressways in Korea is shown in Figure ??. There are 24 expressways run by KEC. Each expressway has multiple toll gates. We define a road interval as a shortest path or 9

Figure 2: Subsidy Proportion versus Construction Moneywise Size (billion won) route between two toll gates. Our traffic and toll prices are given for all road intervals that KEC operates. We assume that demand choices on any two different road intervals are independent of each other. As road intervals have different lengths, we will normalize price, cost and subsidy data for each interval by dividing its distance. Table ?? shows the toll prices for road intervals which are normalized in distance. Our traffic volume data counts the number of cars that drive between a specific pair of departure and exit toll gates. There can be multiple pairs of departure and exit that use the same road interval. On the other hand, road construction, maintenance and subsidies are performed on physical road intervals. There is mismatch between product of roads demanded and the product supplied. To deal with this complexity, we look at the finest road intervals, that is, pairs of two toll gates between which there is no toll gate. Then, we compute traffic volume on the finest intervals. We will explain the idea in details with the example of Figure ??. Let the interval [0,3m] be a highway which has tollgates at 0, m, 2m, 3m as Figure ?? describes. For simplicity, we assume that the distances between any two adjacent tollgates are the same and that the highway allows cars to drive only in the direction from 0 to 3m. Letter A (B, C) indicates the traffic volume which departures from 0 (m, 2m) and exits the highway at m (2m, 3m), respectively. Letter D (E) indicates the traffic volume which departures from 0 (m) and exits the highway at 2m (3m), respectively. Letter F indicates the traffic volume from 0 to 3m. Letters

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A, B, C, D, E and F represent traffic of all the possible routes that cars can make on the highway [0,3m]. If a driver’s decision on departure and exit gates is explained by the exogenous variables X for all possible routes, we can write the traffic as follows: A = βX + A , B = βX + B ,· · · , F = βX + F where E[Xi ] = 0 for i = A, · · · , F . Here, route specific features can be explained by dummy variables in X. Then, demand q1 on road interval [0,m] is the total number of cars who drive through from 0 to m, which is q1 = A + D + F . Likewise, we write q2 = B + D + E + F and q3 = C + E + F . Then, we can rewrite the demands as q1 = 3βX + 1 , q2 = 4βX + 2 and q3 = 3βX +3 . We can estimate demand equation q = βX + where E[X] = 0 by using observation q1 q2 3, 4,

and

q3 3.

According to this idea, we creates demand data qi of each finest interval i. To do

so, we count the number of possible pairs of departure and exit tollgates whose path includes the interval i. Then, we estimate demand function q = βX +  with constructed observation data qi ’s. For this procedure, we assume drivers use the shortest path to reach their destination.

Figure 3: Expressways in South Korea

Figure 4: Demands on a road interval

Note that price and quantity are jointly determined in demand and supply model. To overcome this possible endogeneity problem in demand estimation, we rely on instrumental variable methods. We consider the regional population, regional wealth, local express train stations, expressway identities, observation years as exogenous variables. In detail, we use the population of departure

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Year 2003 2004 2005 2006 2007 2008 2009 2010 Total

Mean 41.6 45.94 45.9 48.78 48.46 48.64 49 48.54 47.56

Standard Deviation 13.27 11.65 11.34 12.63 12.4 14.00 13.66 13.70 13.19

Maximum 733.333 600 428.571 476.19 476.19 1285.71 1285.71 1285.71 1285.71

Minimum 19.82 21.94 21.94 23.16 23.16 23.16 23.16 23.16 19.82

Observation Number 45641 51133 53649 57219 68708 73505 93734 99952 543542

Table 2: Price Summary Statistics Year 2003 2004 2005 2006 2007 2008 2009 2010 Total

Mean 336,851.1 337,995.6 339,245.6 340,705.4 342,462.8 344,135.9 345,490.1 351,839.8 342,340.8

Standard Deviation 1,049,577 1,049,369 1,048,352 1,049,236 1,049,802 1,050,228 1,050,542 1,062,132 1,051,155

Maximum 10,174,086 10,173,162 10,167,344 10,181,166 10,192,710 10,200,827 10,208,302 10,312,545 10,201,268

Minimum 26,716 26,183 24,755 24,209 25,781 23,864 23,478 23,386 24,796.5

Observation Number 112 112 112 112 112 112 112 112 896

Table 3: Population Summary Statistics cities and the population of arrival cities we acquire from Korean Statistical Information Service (KOSIS).3 Table ?? shows the summary statistics of the population data. Other instrument variables are the regional wealth (reliance) of departure and arrival cities which are provided by Local Finance Open System.4 Table ?? shows the summary statistics of the regional wealth data. The dummy variables are the existence of the Korea Train Express (KTX) stations in departure and arrival cities, the major expressway each interval belongs to, and observation years. Table ?? shows yearly construction cost incurred to the company and subsidy paid by the government to the company. The subsidy varies in percentage each year, and it does not follow an optimal principle. Table ?? shows the summary statistics of fixed and variable cost data for 24 expressways. 3

http://kosis.kr/abroad/abroad 01List.jsp?parentId=A Population counts only citizens. http://lofin.mopas.go.kr/lofin finan/InfoDetail.jsp?idx=58&finan idx=3&type=title&word=&pg=1 Reliance as a measure of regional wealth is defined as the percentage ratio of a regional government’s revenue that it earns from a regional taxation and businesses to its budget which includes subsidy from the federal government. 4

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Year 2003 2004 2005 2006 2007 2008 2009 2010 Total

Mean 27.61545 27.60357 28.39107 27.65 27.29196 27.69732 28.03125 27.77589 27.75707

Standard Deviation 18.38327 18.2033 19.37647 18.63983 17.74783 17.6946 17.74479 17.20213 18.12403

Maximum 95.9 95.5 96.1 94.3 90.5 88.3 92 85.8 92.3

Minimum 8.6 8.4 6.9 8.3 7.7 8.7 9 8.7 8.2875

Observation Number 112 112 112 112 112 112 112 112 896

Table 4: Reliance Summary Statistics

Year 2004 2005 2006 2007 2008 2009 2010

Total (= Construction Cost + Subsidy) 25,925 31,868 29,783 25,900 23,437 29,961 30,004

Subsidy 13,995 13,076 10,844 9,320 8,806 13,117 10,717

Construction Cost 11,930 18,792 18,939 16,580 14,631 16,844 19,287

Table 5: Construction Cost and Subsidy (billion won)

Mean Maximum Minimum Standard Deviation Observation Number

Construction Cost 29.56 1765.12 0.00 140.90 243

Maintenance Cost 3.17 7.35 0.17 1.42 243

Table 6: Cost Data Summary Statistics (billion won/km)

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Subsidy as % 54% 41% 36% 36% 38% 44% 36%

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Estimation

4.1

Estimation Strategy

This section presents estimation methods for a demand function, a price equation and a cost function based on the models that we derive in Section ??. Our strategy to estimate these three functions is similar to Brocas et al. (2006), but also different from their approach in the sense that we take advantage of the panel data structure. We overcome the endogeneity problem in estimating a demand function using a fixed effect regression, whereas Brocas et al. (2006) use instrumental variable methods using time series data. We first estimate the demand function, then estimate the price equation based on the estimates of the demand function. Using the estimated price equation, we recover unobserved types and estimate cost function using recovered types. Then, we will compute optimal subsidies and estimate the random quantity deviation from the optimal subsidy level. By doing this, we will examine how much the actual subsidies deviate from the optimal levels of subsidies. For the functional form specification of the demand function, we consider constant price elasticity as in Brocas et al. (2006). We estimate cost function using the functional form obtained from the firm’s cost minimization under Cobb-Douglas production technology.5 We log-transform our demand equation (??) and use the fixed effects estimation method for the linearized panel data model. We consider individual fixed effects to avoid endogeneity problems between the price and time invariant unobservable, which is a function of the individual time invariant type. d1 d2 qi = exp(d0 )Zdi pi exp(di ),

(14)

The unknown type of cost function influences the level of fixed cost. This makes sense especially because the regulator subsidizes construction cost for road intervals other than maintenance cost. We rewrite this cost function per kilometer for interval i as: β

βc Ci = exp(β0 ) exp(β1 θi )qi q Zci exp(ci )

5

(15)

Since our data do not include factor prices and the level of capital stock, our cost function is set differently from their cost function.

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where θ ∼ U nif (0, 1) and c is a mean zero stochastic shock. Note that we assume that the distribution of unobserved heterogeneity θ representing labor inefficiency follows standard uniform distribution. Then θ can be interpreted as a rank. That is, whatever distribution the type θ follows with its cdf F , F (θ) ∼ U nif (0, 1) , and thus our distributional assumption θ ∼ U nif (0, 1) essentially implies the quantile or the rank of the individual labor inefficiency θ in the entire distribution affect the cost. We check robustness of this distributional assumption in Appendix. It is natural to assume β1 > 0 as we use Cobb-Douglas production function.6 The fact that our cost function includes quantity q instead of inputs implies that it already comprehends the firm’s cost minimizing behavior. Baron and Myerson (1982) set a cost function which is a linear in quantity produced. Note that our cost function ?? cannot be directly estimated because θi is unobservable to econometricians for each i. To get around this challenge, now we turn to the optimal price equation

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Here we will explain how we get cost function (??). The form is derived from cost minimization as follows: C = pL L + pK K where pL is price of labor, pK is price of capital, L is labor input and K is capital input. L is labor input, also observable to the regulator. L∗ , L∗ = L/ exp(θ), is actual labor input unobservable to the regulator as θ is labor efficiency which captures the firm’s management and production skills and is the private information of the firm. Our production function has Cobb-Douglas technology: q = P (L, K) = m(L/ exp(θ))a K b

(16)

where a > 0, b > 0. The firm tries to minimize total cost under constraint q = P (L, K). Lagrangian function is G = pL L + pK K + µ(q − m(L/ exp(θ))a K b ). Its first order conditions are ∂G 1 ∂G ∂G = pL − µam(L/ exp (θ))a−1 K b =0 = pK − µbm(L/ exp (θ))a K b−1 = 0 = q − m(L/ exp (θ))a K b = 0. ∂L exp (θ) ∂K ∂µ L . This equation is rearranged as K = From the first order conditions, we have ppK = aK bL
bpL b a+b (??), we have q = m apK exp (−aθ)L . Then, we write this as

1 − a+b

L= m



bpL apK

−

b a+b

 exp

b pL a pK

L. Plugging this to

 1 a θ q a+b . a+b

Total cost is written as    − b   a+b 1 a b 1 b a + b − a+b b a C= pL L 1 + = m pL a+b pK a+b exp θ q a+b . a a a a+b Setting exp (β0 ) =

1 b a+b m− a+b ( ab )− a+b , a

β1 =

a , a+b

βq =

1 , a+b

a

(??) is written as

b

a+b C= exp (β0 )pLa+b pK exp (β1 θ)q βq .

As we do not have observations of pL and pK , our cost function is rewritten as C= exp (β0 ) exp (β1 θ)q βq Zcβc exp (c ) a

b

a+b where E[pLa+b pK Zc−βc |Z] = 1.

15

(17)

that can be derived from our cost function specification. Note that given (??) and (??), we have qp (p, d )= d2 exp (d0 )Zdd1 pd2 −1 exp (d ), Cq (θ, q, c )= βq exp (β0 ) exp (β1 θ)q βq −1 Zcβc exp (c ), Cθ (θ, q, c )= β1 exp (β0 ) exp (β1 θ)q βq Zcβc exp (c ), Cθq (θ, q, c )= β1 βq exp (β0 ) exp (β1 θ)q βq −1 Zcβc exp (c ),

where subscripts refer to partial derivatives. From these equations, we can write the optimal price equation (??) as:   1 A + d1 (βq − 1) ln Zd + βc ln Zc + ξ(θ) ln p = 1 + d2 (1 − βq ) (θ)
where A = ln βq + β0 + (βq − 1)d0 + ln E[exp (c + βq d )] and ξ(θ) = β1 θ + ln 1 + β1 Ff (θ) .

We now estimate the price equation.   1 A + d1 (βq − 1) ln Zdi + βc ln Zci + ξ(θi ; γ1 , γ2 ) ln pi = 1 + d2 (1 − βq )

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(θi )
where A = ln βq + β0 + (βq − 1)d0 + ln E[exp (c + βq d )] and ξ(θi ; γ1 , γ2 ) = β1 θi + ln 1 + β1 Ff (θ . i)

Remember that we have estimates for d0 , d1 , d2 in our estimation of the demand equation ??.

(θi ) Second, we estimate (??) whose error term is β1 θi + ln 1 + β1 Ff (θ i)




and its moments are

unknown. Using the assumption θ ⊥ ηd , we have moment conditions as:  E

   1 A + ξ(θi ; γ1 , γ2 ) η di = 0. 1 + d2 (1 − βq )

Note that we do not know the joint distribution of ηd and θ. Thus, we cannot directly compute the expectation of the above questions. Instead, using (??), we rewrite them as:  E

  1 1 ln pi − d1 (βq − 1) ln Zdi − βc ln Zci η di = 0 1 + d2 (1 − βq ) 1 + d2 (1 − βq )

where d1 , d2 and ηdi can be replaced by their estimates dˆ1 , dˆ2 and ηbdi in the step of estimating (??). 16

Then, the second step provides estimates for βq and βc . It remains to estimate β1 , β0 + ln E[exp (c + βq d )] and (γ1 , γ2 ). If we write (??) as ln pi = ψ(θi ; β1 , β0 + ln E[exp (c + βq d )], γ1 , γ2 ), we have moment conditions as: E[(ln pi )j ] = E[ψ j (θ; β1 , β0 + ln E[exp (c + βq d )])]

where j = 1, 2, 3, 4. The left hand side of the above question is the sample moment of ln pi . The right hand side is the moment of ψ(θ). However, E[ψ j ] for j = 1, 2, 3, 4 does not result in a computationally tractable equation. Computing numerical integral of the ψ j (θ), we run Simulated Method ˆ (c + βq d )]. Plugging these estimates to Moment (SMM). It provides estimates βˆ1 , β0 + ln E[exp (??), we can recover every θi .

Finally, we estimate (??). Using E[ci ] = 0, nonlinear GMM estimator is based on

E[ci ] = E[log Ci −β0 − β1 θi − βq log qi − βc log Zci ] = 0. This step provides estimate βˆ0 .

The last step computes optimal T ∗ (θi )’s according to the following: d1 d2 +1 ¯ βq Z βc Ti∗ = − exp (d0 )Zdi pi + exp (β0 + β1 θ)q ci i

and compares them with observed Ti ’s. Following Perrigne and Vuong (2011), we can write observed transfer T as T = T ∗ + T˜, that is, it may deviate from the optimal subsidy T ∗ by a random quantity T˜, which is uncorrelated with the firm’s type θ given exogenous variables Z. This is expressed as E[T˜|θ, Z] = E[T˜|Z]. We let T˜ = m(Z) + t where E[t |Z] = 0. The function m(·) represents the deterministic component that is common knowledge. The component t is a random term unknown to both the regulator and the regulatee. The function m(·) captures possible systemic departures from the game theoretical model in Section 2 due to legal, institutional and political constraints (Joskow and Schmalenesee, 1986, Joskow 2005, Perrigne and Vuong, 2011). For example, if a parliament is dominated by a party which is against the idea of regulating and subsidizing public 17

enterprises, the governmental regulator is under pressure to decrease the subsidy amount. Such a circumstance will be observable to both regulator and regulatee. Thus, we model subsidy deviation T˜ to be mainly explained by common knowledge. Note that even if the observed subsidy differs systematically from the optimal subsidy, the firm doesn’t have any incentive to change its truth-telling strategy. Given that the observed price equals to the optimal price p∗ , the expected profit is written as: ∗ ˆ ˆ θ) = p∗ (θ)E[q(p ˆ ˆ d ), c )] + T (θ) ˆ U (θ; (θ), d )] − E[C(θ, q(p∗ (θ), Z θ¯ = E[Cθ (s, q(p∗ (s), d ), c )]ds + m(Z) + t . θˆ

The deterministic deviation term m(Z) is correlated neither with the true type θ nor with reported ˆ In addition, the firm does not have an opt-out option of stopping production. Both incentive type θ. compatibility constraint (??) and individual rationality constraint (??) are intact.

4.2

Estimation Results

Following the estimation strategy discussed, we first estimate a traffic demand equation (??). We include population and distance as a set of control variables Zd . Note that p is potentially endogenous since θ possibly affects d and p. Since θ is an interval specific and time invariant type, we can consistently estimate the parameters in the demand equation by using a fixed effect regression. Table ?? reports the estimates of the demand equation when both individual fixed effects and time fixed effcts are included. According to our estimates, 1% increase in price is associated with 0.86% decrease in the traffice demand. In most cases, Drivers have options to choose various free local roads and this seems to result in the significantly positive price elasticity of demand. Also, the traffic volume is positively associated with the average population in the departure and exit locations of the paths that include the corresponding interval as a part, whereas the distance between departure and exit toll gates has a negative impact on the traffic demand. Table ?? documents regression estimates for a variety of other specifications. Next, from the sequence of estimation procedures discussed in Subsection 4.1, we obtain the estimates for the cost function. Estimation results for the cost function are displayed in Table ??. As expected, the traffic demand has a significantly positive impact on the cost; as a traffic 18

d11 d12 d2 d0

Variable Population Distance Price Constant

Coefficient 0.1487∗∗ −0.3439∗∗ −0.8560∗∗ 13.7517∗∗

s.e. .0484 .1015 .3736 1.6184

Table 7: Estimation Results for Demand Function

β1 βc βq β0

Variable Theta (type) Regional Income Traffic Constant

Coefficient 0.8471∗∗ 2.1247 × 10−11 1.9876∗∗ −0.5420

s.e. 0.1998 0.3475 × 10−10 0.7897 8.4006

Table 8: Estimation Results for Cost Function volume increases by 1 percent, the maintenance cost increases by 2 % approximately. If we assume the distribution of the type as a standardized uniform distribution, the type parameter θ can be interpreted as rank of any distribution. Since θ represent rank in labor inefficiency, the coefficient on theta is interpreted that as the efficiency of the firm increases by one percent point in its distribution, the cost on average decreases by 0.85 %.

5

Counterfactual Analysis

5.1

Scenario 1. Price cap regulation

Theorem 2. In the price cap regulation without subsidies, there exists θ˜ such that for any ˜ the monopolist charges pM (θ) where θ < θ,

pM (θ)E[qp (pM (θ, d ))] + E[q(pM (θ), d )] = E[Cq (θ, q(pM (θ), d ), c )qp (pM (θ), d )]

(19)

˜ it charges p¯ that satisfies and for any θ > θ, ¯ q(¯ p¯E[q(¯ p, d )] = E[C(θ, p, d ), c )]. ˜ = p¯. The threshold θ˜ is such that pM (θ)

19

(20)

Proof. A price cap p¯ leads to monopoly pricing for θ less than some θ˜ and to the price p¯ above ˜ The firm chooses its price pM (θ) to maximize its profit. The price cap p¯ is binding for and leaves θ. ¯ = 0.  no rent to the most inefficient type. Such a price cap satisfy U (θ) The equation (??) implies that if the monopolist is efficient, it sets the price pM (θ) where marginal revenue equals the marginal cost. Equation (??) implies that if the monopolist is inefficient, its price will be the price cap level that is equal to the average cost. For the demand and cost functions we use in this paper, equation (??) is rewritten as: 

M

p (θ) =

1  1−d2 (βq −1) d2 βq βc d1 (βq −1) E[exp (c + d βq )] exp [β0 + β1 θ + d0 (βq − 1)]Zc Zd d2 + 1

and equation (??) is rewritten as:  p¯ =

5.2

1  1−d2 (βq −1) d1 (βq −1) βc ¯ exp [β0 + β1 θ + d0 (βq − 1)]Zd Zc E[exp (d βq + c )] .

Scenario 2. Symmetric Information

Suppose that the regulator observes the monopolist’s efficiency type θ and sets the price p(θ) and subsidy T (θ) to maximize the expected consumer surplus subject to the constraint that the firm earns at least break-even profit. The problem the regulator solves is described as follows: for ¯ any θ ∈ [θ, θ], Z

∞

max p(θ),T (θ) p(θ)

E[q(s, d )]ds − T (θ)

(21)

subject to U (θ) = p(θ)E[q(p(θ), d )] − E[C(θ, q(p(θ), d ), c )] + T (θ) ≥ 0. Theorem 3. In the case of symmetric information where the efficiency type θ is known to the regulator, the optimal price and subsidy are given as follows:

p~ (θ) =

E[Cq (θ, q(p~ (θ), d ), c )qp (p~ (θ), d )] E[qp (p~ (θ), d )]

T ~ (θ) = −p~ (θ)E[q(p~ (θ), d )] + E[C(θ, q(p~ (θ), d ), c )].

Proof. To maximize consumer surplus in (??), the constraint should bind, that is, U (θ) = 0.

20

Then, the problem (??) changes to the problem of finding the optimal price. Differentiating the consumer surplus gives the solution p(θ).  When the efficiency type is known to the regulator, the optimal price is a competitive price that is equal to marginal cost and optimal subsidy makes the monopolist’s profit break-even. For the demand and cost functions we use in this paper, the optimal price and subsidy under symmetric information are written as follows:

~



p (θ) =

βq exp [β0 + β1 θ + d0 (βq −

d (β −1) 1)]Zd 1 q Zcβc E[exp (d βq

1  1−d2 (βq −1) + c )]

and d β

T ~ (θ) = − exp (d0 )Zdd1 p~ (θ)d2 +1 + exp (β0 + β1 θ + d0 βq )Zd 1 q p~ (θ)d2 βq Zcβc E[exp (d βq + c )].

6

Conclusion

7

Appendix 1 2 3 4 ∗∗ ∗∗ ∗∗ Population 0.1487 .1315 .1707 .1777∗∗ (.0484) (.0659562) .0532 .0559 ∗∗ ∗∗ ∗∗ Distance −0.3439 −.2890 −.4601 −.4414∗∗ (.1015) (.1259) (.0953) (.0964) Price −0.8560∗∗ −.2616 .0236 .1158 (.3736) (.4171) (.2140) (.2030) Constant 13.7517∗∗ 11.1775∗∗ 10.2475∗∗ 9.7958∗∗ (1.6184) (2.1086) (1.1776) (1.1827) Individual effects? Yes Yes No No Time effects? Yes No Yes No F-Stats and p-Values Testing Exclusion of Groups of Variables Individual effects=0 16544.80 6.55 (0.00) (0.00) Time effects=0 8.64 7.88 (0.00) (0.00) 2 R 0.73 0.6516 0.3081 0.2690 N 358 358 358 358 Table 9: Estimation of Demand Function for Various Specification

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References David P. Baron and Roger B. Myerson, 1982 “Regulating a Monopolist with Unknown Costs.” Econometrica, Vol. 50, No. 4, pp. 911-939 Isabelle Brocas, Kitty Chan and Isabelle Perrigne, 2006 “Regulation under Asymmetric Information in Water Utilities.” The American Economic Review, Vol. 96, No. 2, pp. 62-66 P. A. Chiappori and B. Salanie, 2003 “Testing Contract Theory: A Survey of Some Recent Work.” Advances in Economics and Econometrics: Eighth World Congress, Vol. I, ed. by M. Dewatripont, L. P. Hansen, and S. Turnovky. Cambridge, U.K.: Cambridge University Press, pp. 115-149 Daniel Garrett, 2013 “Robustness of simple menus of contracts in cost-based procurement.” Games and Econmoic Behavior, in press: http://dx.doi.org/10.1016/j.geb.2013.06.004 N. Joskow and N. Rose, 1989 “The Effects of Economic Regulation.” Handbook of Industrial Organization, Vol. 2, ed. by R. Schlamensee and R. Willig. Amsterdam, Netherlands: North Holland, pp. 1449-1506 P. Joskow, 2005 “Regulation and Deregulation After 25 Years: Lessons Learned for Research in Industrial Organization.” Review of Industrial Organization, Vol. 26, pp. 169-193 Joskow, P., and R. Schmalensee, 1986 “Incentive Regulation for Electric Utilities.” Yale Journal on Regulation, Vol. 4, pp. 1-49 Tracy Lewis and David Sappington, 1988 “Regulating a Monopolist with Unkown Demand.” The American Economic Review, Vol. 78, No. 5, pp. 986-998 Isabelle Perrigne and Quang Vuong, 2011 “Nonparametric Identification of a Contract Model with Adverse Selection and Moral Hazard.” Econometrica, Vol. 79, No. 5, pp. 1499-1539 William Rogerson, 2003 “Simple Menus of Contracts in Cost-Based Procurement and Regulation.” American Economic Review, Vol. 93, No. 3, pp. 919-926 John Rust, 1987 “Optimal Replacement of GMC Bus Engines: An Empirical Model of Harold Zurcher.” Econometrica, Vol. 55, No. 5, pp. 999-1033 22

Frank A. Wolak, 1994 “An Econometric Analysis of the Asymmetric Information, Regulator-Utility Interaction.” Annales Deconomie et de Statistique, No. 34, pp. 13-69

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