A Model of Access in the Absence of Markets

2.1 When Exclusion is Feasible

When exclusion is feasible, efficient utilization of the resource maximizes total willingness-to-pay by granting access to the individuals with the highest valuations. Thus, the marginal valuation z defining efficient access is given by 1–F(z)=q/m. Only agents with willingness-to-pay z and above are granted access. Total willingness-to-pay corresponding to the efficient use is

where

is the expectation of an individual’s valuation conditional on exceeding z, that is, conditional on gaining access (Che, Gale, and Kim 2013).

Note that efficient access, denoted by ae, describes who receives the resource, but does not specify how this assignment is to be achieved. The efficient outcome in eq. [1] could be attained by allowing the transferability of access rights. In a competitive market, the equilibrium price for the access right would be pe=z, and the subset of individuals with the highest willingness-to-pay would end up with the access rights, regardless of the initial allocation. Hence, allocative efficiency follows. This decentralized mechanism relies on the assignment of clearly defined user rights and does not require the manager to have information on individuals’ types.

In the following sections, we pursue the idea of access to resources when exclusion is not viable, to help understand the interaction between access, sorting and valuation.

2.2 When Exclusion is Infeasible

Let the different sets of controls that the manager could use to ration the resource (e.g. lotteries for charter schools, queuing for driver’s licenses, gear restrictions and seasonal closures in commercial fishing) represent distinct access scenarios. Let Ω be the set of agents. Then each access scenario ai induces an assignment, ai:Ω×[0,q]→[0,1], which determines the probability of access for each valuation in Ω. Under access scenario ai, an assignment consists of a family of distributions of the form

where ai(v,y) is the probability of access to the yth available unit for valuation v, and ρi(η,y) the cumulative probability of access to the yth available unit for valuations η or lower. Let’s look again at our example with only three agents with willingness-to-pay v1=$$2,v2=$$4, and v3=$$6 and two available units of the public resource. The efficient assignment –scenario e– would allocate the two units to the highest valuation individuals; thus, the first unit would go to the individual with willingness-to-pay of $6, ae(v=6,y=1)=1 and ae(v=4,1)=ae(v=2,1)=0, and the second unit to the agent with willingness-to-pay of 4, ae(v=4,2)=1 and ae(v=2,2)=ae(v=6,2)=0. However, it could also happen that under a given access scenario k all valuations have equal probability to access the resource. We would then have ak(v=2,1)=ak(v=4,1)=ak(v=6,1)=ak(v=2,2)=ak(v=4,2)=ak(v=6,2)=1/3. In this latter case, it would hold for the first unit that ρk(v=2,1)=1/3,ρk(v=4,1)=2/3,ρk(v=6,1)=1 and similarly for the second unit ρk(v=2,2)=1/3,ρk(v=4,2)=2/3,ρk(v=6,2)=1. Many cases in between are possible.

For any access scenario ai, it holds that

since by assumption capacity is scarce and therefore each available unit is consumed. From eq. [4] it follows that:

where q denotes the available quantity or the resource capacity.

We now introduce the function γ(⋅), which we need in order to explicitly identify the probability that each valuation has to access the resource. For any assignment ai we define the weighting function γ(v,y) according to γ(v,y)=ai(v,y)/f(v). Thus, γ(v,y) is nonnegative and may be smaller, equal or larger than 1. A particularly interesting case arises when assignment is independent of the unit of the resource being accessed. If ai(v,y)=f(v), then γ(v,y)=1 and we call the scenario uniform random access. In this form of assignment, each user has an equal probability of access, regardless of valuation, and the expected value of each accessed unit of the resource is simply vˉ=∫0vˆvf(v)dv. For all the other access scenarios, and for each available unit, the function γ(v,y)≠1 redistributes probability mass across valuations v, thus resulting in the modified probability distribution a(v,y)=γ(v,y)f(v). It is this modified distribution that defines the probability that each valuation has to access the resource. For example, rationing by waiting (queuing) will generally result in different probabilities of access than in a lottery. From eq. [4] it must hold that ∫0vˆγ(v,y)f(v)dv=1, that is, the mean weight must equal one for each available unit y.

Given access scenario ai and the induced assignment ai defined by expression [3], the expected value of the yth unit drawn is

Thus, for a given distribution of willingness-to-pay F(v), the expected marginal value functionEV(y) in eq. [6] completely characterizes the value of the accessed resource units. Embedded in EV(y) is the sorting of agents with different valuations across resource units, that is, how likely is each valuation v to access the first unit of the resource, how likely to access the second unit, and so on. In turn, EV(y) and its curvature properties are fully determined by assignment ai

Figure 1:

The expected marginal value function corresponding to assignment ai_.

Next, note that in Figure 1 the efficient allocation achieves the maximum value associated with the consumption of any number of units. Indeed, ∫0qEV(y)dy<∫0qV(y)dy for all q<m. Only in the absence of scarcity, when rationing is unnecessary and the entire demand m can be satisfied, is the welfare associated with both scenarios equal. Graphically, this implies that the area under the curve V(y) is always larger than that under EV(y), unless q=m in which case both become equal regardless of the sorting on valuations.

Next we introduce the function Γ, which will allow us to rank different access scenarios and, in turn, rank the rationing policies that induce them.

Figure 2:

Graphical representation of Γ(⋅)_.

We say that access scenario aivalue-dominatesaj if ∫0vˆΓi(v)dv<∫0vˆΓj(v)dv. In words, ai grants access to greater quantities of the resource to high-valuation individuals than aj does. For example, envision an scenario in which in-state anglers represent the lowest quartile of users’ valuations, but access half of the total allowable catch due to permit surcharges and additional restrictions on out-of-state fishermen. Then, this status quo is value dominated by an alternative scenario in which all anglers are charged the same license fee and are subject to identical catch restrictions, regardless of place of residence. Note that while value dominance implies higher expected value, it is a weaker condition than second-order stochastic dominance (SSD), which requires ∫0ηΓi(v)dv≤∫0ηΓj(v)dv for all η, and unlike SSD it provides a complete ordering of access scenarios. SSD is a sufficient condition for value dominance, but it is not necessary for ranking different scenarios. ai and aj are value-equivalent if ∫0vˆΓi(v)dv=∫0vˆΓj(v)dv. Finally, the uniform random assignment, characterized by γi(η,y)=1 for all η,y, awards equal access probability to all users, regardless of their valuations (i.e. zero-price lottery). Writing the welfare function associated with capacity q and access scenario ai as

we have the following result

Figure 3 illustrates how a given number of available units (or capacity) q generates different levels of welfare depending on the sorting induced by the access scenario. Under the efficient allocation ae the total expected value associated with q is given by the area(0qevˆ), while under access scenario i total value is given by the area(0qda), and under uniform random access by the area(0qcb), with (0qevˆ)>(0qda)>(0qcb). Naturally the efficient allocation, given by the area under V(y), yields the highest value, because the individuals with the highest marginal values access first. In terms of our earlier definition, by construction Γe(η) value dominates any other access scenarios, and according to the result above any access scenario ai that is value-dominated by ae will result in lower welfare than We in eq. [1]. Notice that this is the case for any degree of rationing q<m. Take for example q′. We know that when entire demand is satisfied, welfare associated with use of the resource equals mvˉ regardless of the sorting of valuations, that is, regardless of the order in which the available units are accessed. Graphically, this means that in Figure 3 the areas under V(y) and EVi(y) and up to m are identical. For this to hold, however, it must be the case that area(vˆah)>area(hfg), which in turns implies that area(0q′fvˆ)>area(0q′ga).

Figure 3:

Welfare associated with quota q for different access scenarios.

We now look at a concrete example to illustrate how proposition 1 can be readily used to establish the conditions under which one mechanism is superior or equivalent to another. We focus on the case of hybrid assignment rules previously studied in the literature. As noted in Evans, Vossler, and Flores (2009), several U.S. states have recently used hybrid mechanisms to allocate big game permits, a setting in which there is often a small number of permits available relative to total demand. In one such hybrid mechanism, a few permits are auctioned off with the remaining permits allocated via lottery. An alternative hybrid mechanism rations permits using a lottery with one-part or two-part entry tariffs (Scrogin 2005). A question of interest for policy makers is the relationship between the fraction σ of available permits to auctioned off in the first mechanism and the entry tariff τ in the second mechanism that makes both hybrid mechanisms equivalent from a welfare standpoint. To explore this matter and according to the proposition, we require the mechanisms to be value-equivalent, ∫0vˆΓi(v)dv=∫0vˆΓj(v)dv.

By the efficiency of the discriminatory and uniform auction formats (for single-unit demands and independent private values), we know that the σq permits that are auctioned off in the first mechanism end up in the hands of the highest valuation bidders. The lowest valuation v∗(σ) to acquire a permit through the auction is defined by [1−F(v∗)]=σqm. Thus, for these permits Γi(η)=0 for η<v∗ and Γi(η)=σ[F(η)−F(v∗)]1−F(v∗) for v∗<η≤vˆ. The remaining (1−σ)q units are subsequently allocated in a lottery in which only valuations v∈[0,v∗) participate. Therefore Γi(η)=(1−σ)F(η)F(v∗) for η<v∗ and Γi(η)=1 otherwise. Turning now to the second hybrid mechanism, note that the tariff τ discourages any valuation v<τ from entering the lottery. Since those that enter face the same probability of obtaining a permit, it is immediate that Γj(η)=F(η)−F(τ)1−F(τ) for η≥τ, and Γj(η)=0 otherwise. Using the definition of value equivalence, it is straightforward to show that the two hybrid mechanisms result in identical welfare if E[v|v≥τ]=(1−σ)E[v|v<v∗]+σE[v|v≥v∗], condition that implicitly defines τ=τ(σ). In words, the tariff should exclude enough low willingness-to-pay individuals for the average valuation in the lottery to equal the mean valuation in the first mechanism, defined as a weighted average of the mean willingness-to-pay in the auction and the ensuing lottery. Furthermore, from the implicit function theorem, it follows that the tariff is increasing in the fraction of available permits that is auctioned off under the first mechanism, τ′(σ)>0. In particular, as σ→1, τ becomes the market clearing price.

We finish this section by characterizing a class of well-behaved assignments, monotone assignments. This definition will later allow us to derive optimality conditions for allocation of access under these assignments, assignments that Mumy and Hanke (1975) depicted only in graphical form.

1. monotone increasing if access to they′thunit is second-order stochastically dominated by access to they′′thunit, ai(⋅,y′′)≥SSDai(⋅,y′), for ally′<y′′, or

In words, monotone decreasing assignments grant higher willingness-to-pay units a larger probability of being satisfied before lower willingness-to-pay units. Monotone increasing assignments, on the other hand, grant lower willingness-to-pay units a larger probability of being satisfied first than higher willingness-to-pay units (see appendix for proof). The following lemma ranks monotone increasing and decreasing assignments

Figure 4 depicts a marginal value function and the associated expected marginal value functions for access scenarios that induce different monotone assignments: EV1, the expected value function corresponding to access scenario 1, is downward-sloping as this scenario induces a monotone decreasing assignment and ai(⋅,y′′)≥SSDai(⋅,y′) for all y′<y′′ implies ∂EV1(y)/∂y≤0; similarly EV2, the expected value function corresponding to access scenario 2, is upward-sloping as this access scenario induces a monotone increasing assignment, a2. Monotone decreasing assignments are easy to envision. With a bit of price discrimination we can induce an assignment similar to EV1 in Figure 4. While the intuitive nature of EV2 is obvious, it’s less clear how this would arise. One possible case occurs when the arrival rates of individuals with different valuations changes over time, but favors low valuations early on. Lemma 1 allows us to rank different monotone assignments without the need to compute the expected welfare associated with each of them individually. In Figure 4, for example, we know immediately that access scenario a1 value dominates a2 –i.e. generates higher welfare– for any level of rationing.

Figure 4:

Expected marginal value function for different monotone assignments.

3.1 The Efficient Allocation

Let vje>0 denote the critical valuation in sector j such that (i) 1−F(vje)=qj/mj for a given quota qj to sector j. Denote z the value that simultaneously satisfies condition (i) for all sectors. Eliminating q1,…,qn it follows that z solves ∑jmj[1−Fj(z)]=X. Valuation z is the marginal valuation in each sector that defines the efficient allocation (i.e. as prescribed by the equi-marginal principle). In this efficient allocation only individuals with valuations z or higher gain access. To see that this is necessarily the case, note that if this condition did not hold, that is, if vie≠vke, then it would be possible to reallocate quota between these two sectors and increase total welfare, making the initial allocation a suboptimal solution. In the efficient allocation, sector j receives qje=mj[1−F(z)] in access rights. Thus, total welfare corresponding to the efficient allocation is

where ∑jqje=X and, as before,

is the expectation of an individual’s valuation in sectors j conditional on receiving quota.

3.2 Allocation when Exclusion is Infeasible

We now derive the optimality condition for the allocation of capacity among multiple sectors for any access scenarios, when exclusion of low willingness-to-pay agents is infeasible. The optimal allocation maximizes total welfare, which is given by

where q1+…+qn=X.

According to this proposition, total capacity (or quota) should be allocated until the expected value of the last unit is equal across sectors (condition SC-1), with the expected value of that last unit in each sector given by the corresponding probability distribution of access to that unit across valuations. ^[6] Condition SC-2 is necessary to guarantee a local maximum. This proposition characterizes optimality over a wide range of access settings as it does not impose any restrictions on the shape of the expected value functions. Uniqueness is not guaranteed unless monotonicity assumptions are made on the functions EVi(y) (see lemma below); consequently, the search for the optimal allocation may entail comparing total welfare for the various critical points. Finally, note that conditions SC-1 and SC-2 characterize an interior solution; to rule out potential corner solutions in which some sectors receive zero quota, it is sufficient for the first unit of quota allocated to any of the sectors to be more valuable than the same unit optimally allocated instead across the remaining sectors: limε→0EVi(ε)>EVk(qk∗∗), where qk∗∗ is defined by the conditions EVk(qk∗∗)=EVz(qz∗∗)∀k,z≠i and ∑j≠iEVj(qj∗∗)=X−ε.

The following lemma characterizes the optimal allocation for the special case of monotone assignments. For both decreasing and increasing assignments uniqueness is guaranteed as monotonicity implies, respectively, global concavity and convexity.

(ii) LetΛ1denote the set ofwi, wherewi=∫0miEVi(y)dyfori=1,…,n, and defineΛk+1=Λk∖{max{Λk}}andΩk=Λk∖Λk+1. Let the collectiona1,…,anof access scenarios induce monotone increasing assignments: vectorq∗maximizes total welfare if and only if

According to the lemma, the optimal allocation under monotone decreasing assignments is defined by the vector of quotas that equates marginal expected benefits across all sectors (condition (i)). ^[7]Figure 5 illustrates the case for the allocation between two sectors. The optimum is found at the point of intersection of the expected value functions. On the other hand, under monotone increasing assignments, that is, when high valuations are expected to access last, condition (ii) states that the optimal allocation is a corner solution that satisfies the entire demand of the sectors where access generates the largest welfare and excludes all other sectors. Figure 6 depicts this case for the allocation of access between two sectors. Total welfare in sector 1, when it receives the entire quota X, exceeds that in sector 2 when its entire demand is satisfied. Accordingly, the entire demand of sector 1 is satisfied, and sector 2 receives no quota. If, for example, Figure 6 depicted access to campgrounds along the Inca trail and sector 2 represented independent backpackers, then optimality would call for commercial, professionally guided trips only. The rationale for the result is straightforward. Backpackers would otherwise displace some high-value guided tourism by getting to campsites first. ^[8]

Figure 5:

Optimal allocation for two sectors under monotone decreasing assignments.

Figure 6:

Optimal allocation for two sectors under monotone increasing assignments.