Transaction Costs and Institutions: Investments in Exchange

2.1 Preferences and Production Technology

The economy is populated by a countable infinity of agents, i∈I. All agents have the same utility function, uc, with a constant degree of relative risk aversion. Each is endowed with the same amount of capital, 0 < k < ∞. Agent i produces amount λiyi of the non-storable consumption good, where yi≤k is the amount of capital used in production and λi the idiosyncratic production technology. The set of possible technologies, Λ, is finite and bounded away from infinity. λi is distributed i.i.d. across agents with pλi denoting the probability of any agent drawing λi, and ∑λ∈Λp(λ)=1.

Let ω represent the state of nature, i.e., a list of λi for all i∈I. Let Ω be the set of all possible states of nature, and pω the probability of some ω occurring and ∫ω∈Ωp(ω)=1.^[5]

2.2 Diversification and Intermediation

To diversify against risk, agents can form markets in a star network around a single intermediary, as in Townsend (1978). A market is a set of agents M⊂I with cardinality #M < ℵ0; so a market is finite sized. The set of agents with whom agent i exchanges directly is denoted Nⁱ. Agent h∈M is an intermediary if Ni=h for i∈M∖h and Nh=M∖h. So defined, markets are disjoint and agents only exchange with one intermediary. Mh denotes the market intermediated by agent h∈H where H⊆I is the set of all intermediaries in the economy.

Agents can exchange some of their endowment of capital on a one-for-one basis for shares in the consumption portfolio compiled by the intermediary, less their contributions to the total costs of transactions. Each agent in a market exchanges twice with the intermediary and pays an equal share of market-wide exchange costs. So for a given exchange cost, α, the per-agent cost of exchange in a market of size #M is 2α#M−1#M.

2.3 Institutional Capital and Exchange Costs

Intermediaries form markets using institutional capital as an input to an exchange cost technology (ECT) which determines the cost of exchange in that market. There is free-entry to intermediation (i.e., the ECT is accessible to all agents). Institutional capital takes two forms: There is a market-specific institutional capital (S-capital) and a general economy-wide institutional capital (G-capital). We assume that the specific capital is excludable to a market and agent i allocates a proportion τsi of the endowment as a private investment in this local capital. The general capital is a public good, each an agent i making a public investment τgi of the endowment. S-capital thus reflects activities specific to the transaction being made – such as organizational choices, standardization of quality, private infrastructures or learning about property rights – and is excludable to the market. On the other hand, G-capital reflects the general legal enforcement of contracts, fiduciary duties, public infrastructures or competition policy and is, by contrast, a public good.

In a market intermediated by agent h, the exchange cost, αh, is determined as follows:

where F() is the ECT, which is concave, continuous and increasing in each of its arguments and satisfies some Inada-type conditions.^[6] The range of F is the unit interval, so 0≤αh≤k. Since the exchange cost is proportional to k it is akin to an “iceberg cost”. We assume that S-capital is the average contribution of agents in a market and G-capital is the average across the whole economy.^[7] Given that all agents are identical ex ante, all will make the same allocations and so we generally omit superscripts. Clearly, then, S≤τsk and G≤τgk. The ex ante allocations, τs+τgk, combined with the ex post costs, 2α#M−1#M, make up total transaction costs.

Since the intention of this paper is to explore the general equilibrium implications of agent investments in reducing exchange costs, the equation for the exchange cost, α, is left as a reduced-form expression. The expression for α imposes: i) that zero ex ante spending on institutions means no trade; ii) that greater ex ante investments reduce the costs of individual ex post trades; iii) there are decreasing returns to investing in the institutions, and iv) zero exchange costs require infinite investment. Nevertheless, there are a number of specific mechanisms that may generate transaction costs in a way captured by eq. [1].

First, we may consider the transaction cost to the be product of the “transfer of property rights” (Allen 2000, 901). That is, there are frictions associated with a transaction because of the “costs of bookkeeping, the cost of enforcement, the cost of monitoring…” (Townsend 1983, 259). That “cost” is related, however, to the institutional capital from which the agents in the exchange can draw; the form of eq. [1] imposes a natural relationship between investments in that institutional capital and the consequent cost of conducting exchange. The cost of enforcing a contract is a product of the quality of the legal system, for example. Consider, for example, the problem of trading an item of unknown quality. This mechanism can be motivated using an example in Langlois (2006). Prior to the coming of the railroad to the American Midwest in the mid-nineteenth century, the trade of grain could be conducted on a personal basis with quality levels maintained through the observed reputation of individual farmers. Once the railroads vastly increased the scale of trade, individual farmer grains became mixed and the grains were traded as commodities in a way detached from their original producer, thus breaking the prior system of quality control. In response, the Chicago Board of Trade (CBOT) was formed to standardize the nature of the grain trade. Setting up such standards was costly but there were also continuing costs of inspecting each transaction for conformance. The CBOT reduced the risks involved in making an individual exchange of grain, reducing the margin required by traders engaged in buying and selling grain.

A second perspective is one of incomplete contracts, i.e., that transaction costs are “the costs of establishing and maintaining property rights” (Allen 2000, 898). Hart and Moore (2008) introduce a model in which the broad outlines of trade may be defined ex ante. Ex post, there are costs to fill in the detail of the finer points. A contract embodies a trade-off between flexibility and rigidity in which an optimal arrangement may not be fully specified over all possible states of nature. For Hart and Moore (2008), the costs from such partial incompleteness can take the form of a deadweight loss. We may also think of this in the context of complexity (cf. Anderlini and Felli 1999). In the context of our model, goods may be produced with different technologies that require specific contracts that are complex (costly) to write. To write ex ante contracts for each possible technology would be exorbitantly expensive, but to write no contracts ex ante would mean that, ex post, there would be no trade. Ex ante investments in legal and accounting systems could mean that agents are committed to trade under the rough outlines of an agreement, leaving the costs of writing specific contracts for individual trades to be incurred after the state of the world is known.

The distinction between the specific and general determinants of the costs of exchange also emerges from this literature. Williamson (1979) refers to the “governance of contractual relations”. Anderlini and Felli (1999) consider that some of what determines the extent of complexity costs is environmental, since a contract is “‘embedded’ in a larger legal system” (Anderlini and Felli 1999, 25). A second determinant is specific to the market in which the contract is formed. The distinction between S- and G-capital can also be motivated using the example of the Chicago Board of Trade. The CBOT was established by local businesses, the standards were private to the CBOT and benefited the farmers and traders that used it. However, there were also public goods that made the CBOT effective. In 1859, for example, it obtained the authority of the State legislature to appoint grain inspectors with powers of enforcement. Without the public institutional input, the sector-specific arrangement may have been less effective.

Each type of institutional capital may impact upon the other in determination of exchange costs. In our baseline case, we assume that S and G are complementary inputs, so FSG=FGS > 0. This is the form of relationship described in connection to the CBOT: Without a public institution, the S-capital would be less effective; without S-capital, there may be little point in an economy-wide institution. In our robustness analysis below we also consider the impact of their being substitutes. There is an additional form of complementarity that we will not consider: Private investments in the specific capital of one market may not be excludable, instead benefiting other markets. For example, it may have been that the establishment of the CBOT demonstrated the feasibility of such an arrangement, serving as a blueprint for additional Boards elsewhere. A version of this model with ex ante heterogeneity may naturally incorporate such a form of investment if investments in markets composed of one type of agent affected exchange costs in other markets also composed of that agent type. In this model, all agents are ex ante identical and so we leave this consideration for future work.

3.1 Optimality

All agents share an aliquot consumption payout in each market as determined by the observed average technology for that market, λˉ(ω,h)=(#M)−1∑i∈Mhλi(ω). One more assumption is required before stating Proposition 1 which establishes that well-defined solutions to eq. [2] exist. Recall that total per capita costs are 2α#M−1#M. If for all k it is the case that 1−τs−τgk≤2α, then the optimal market size will be bounded since perfect diversification would imply non-positive consumption. Hence, let an arbitrary maximum value of k be kˉ. Let K≡k_,kˉ] where k_<kˉ, be the set of potential endowments. When k=kˉ, as shown below, investments in ECT will be at their maximum level: τˉs,τˉg.^[8] A sufficient condition for finite-sized markets to be part of the optimal plan is then,

Equation [10] ensures that there exists a Mˉ≥#M(kˉ), where Mˉ is a finite integer which may serve as an upper bound on optimal market size. Let M≡1,...,Mˉ be the set of feasible market sizes. In short, eq. [10] imposes that exchange costs do not fall “too quickly” as institutional capital rises.

Decisions on τs,τg and #M are taken after k is known but before agents’ productivities are revealed. Using eqs [12] and [13], define the feasible policy choices as follows: Γ:K→T×M. That is, T×M is the product space,

For each ω∈Ω, Γ is clearly a non-empty, compact-valued, continuous correspondence. A typical element of that mapping is denoted z∈Γ(k,ω). The optimization problem, therefore, involves a strictly concave criterion function and a non-empty, compact constraint set, so that the maximum is attained. Since the maximum is attained, the value function, V(k), is well-defined. It follows from the Theorem of the Maximum that V(k) is continuous. As the feasible policy set is not strictly convex, the optimal policy

need not be unique. Finally, the equivalence between the cooperative case and core allocations is established in Proposition 5.■

Some properties of efficient equilibria are immediate. Since the ECT is freely accessible, rents from intermediation must be zero so eqs [5] and [6] hold with equality. The optimal allocations to the ECT is the pair τs∗,τg∗ which satisfies,

where θ ≡ 2#M−1#M, and FX(X ≡ S,G) denotes a partial derivative of FSh,G. So, for any market size, the optimal institutional investments equate the marginal cost with the marginal gain from reducing the exchange cost. When there are no transactions, agents would not undertake such investment and the optimal choice of market size satisfies

Clearly, the optimal market size is unity, θ=0. The choices τs=0,τg=0 and #M=1 also define the unique reservation utility for any agent to be a member of any #M>1 coalition, V‾=∑λ∈Λpλu λk, ∀k. That furthermore ensure that 0≤τs+τg < 1. A direct implication of the reservation value of utility is that any equilibrium of the model in which τs > 0 and τg > 0 is one in which #M > 1, necessarily.

Remark 1 follows from analysis of eqs [14] and [15]:

The choice of any market size greater than one is determined by the ECT and agents’ attitude to risk. Equations [14] and [15] determine efficient investment in the ECT and hence total resources diverted from goods production. Agents’ risk aversion provides an upper bound on how much they are willing to pay for consumption insurance, given an alternative not to diversify; that bound is independent of the ECT. Agents will optimally form markets with #M > 1 when efficient investment in the ECT delivers transaction costs lower than that bound.^[9]

The rest of the analysis of equilibrium is contained in the following four Propositions. One can show that in general there is a critical level of k below which transaction costs dominate and agents do not diversify:

The key to understanding that Proposition is to recall that S=τsk. So, a given τ has a larger proportionate impact on α as k rises, even though α itself is proportional to k. Hence, higher levels of k permit lower ex post transaction costs and help sustain larger market sizes and the benefits from consumption risk sharing.

Proposition 2 also indicates the possibility of multiple optimal plans. Proposition 3 now shows that no more than two such alternative plans can exist.

As a corollary, it follows that for any #M > 1 which is optimal, subsequent market size increases are single steps for (sufficiently large) increases in k. The final Proposition and following remarks establish that multiple equilibria can arise although they are, in a sense, special cases.

where τ1 ≠ τ2,α1 ≠ α2, M1h ≠ M2h and where maximized utility is identical under both programs.

The set of values of k which result in multiple equilibria is of measure zero. That is, such values of k∗ correspond to pairs of θ′ s; these θ′s are a proper subset of the rationals and hence themselves drawn from a set of measure zero.

To summarize: Low endowment economies may resort to autarky (Proposition 2). However, for economies with larger endowments (k > k_) it is optimal to invest in the ECT and to form markets (Proposition 1 and Remark 1). Equilibrium plans need not be unique (Proposition 4) but those equilibria are, in a sense, of limited interest (Proposition 3 and the brief discussion following Proposition 4). Finally:

6.1 Equilibrium with Exogenous τg

The distortive behaviour centres on the resources turned over to G-capital via the tax τg. Such distortions might reflect deeper political tensions, perhaps between the electorate and a “political class”. In an environment where the definition and measurement of the costs of exchange are not well understood, it may also be simply that the objective of tax policy is not straightforward to design. Alternatively, deviations from optimality may reflect “irrational” voters, as in Caplan (2008). First, we look at the consequence of varying τg over some interval. In SubSection 6.2, we look at particular policy objectives that may drive specific deviations.

Figure 2 reports the consequences of varying τg over the interval [0,0.4]. Responses in τs and #M are unconstrained but can be, of course, affected by deviations in τg from its efficient level. This can result in significant compensating changes in agent behaviour, as Figure 2 shows. When τg deviates from τg∗ agents can respond by changing τs and/or market size. If market size does not change, then τs varies positively with τg. This follows from eq. [14] and the assumption that S and G are complementary inputs; a higher G increases the marginal return to S-capital investment. For a sufficiently large increase in τg, the constrained optimal choice of market size may also change by increasing or decreasing. Increasing τg holding #M fixed clearly reduces net endowment since the optimal τs also increases. One response is to recover some lost productive capacity by reducing the number of exchanges (lower #M). Another response is to take advantage of the lower exchange cost created by the higher τg and diversify further (higher #M). For this parameterization, market size is hump-shaped in the level of τg. At low levels of τg, there remains a substantial residual endowment and so the gain from additional risk-sharing dominates: A sufficiently large increase in τg lowers the exchange cost such that increasing market size (further risk-sharing) is attractive enough to incur the cost of a higher κ. However, for τg≫τg∗ the residual endowment is severely diminished. Diverting additional resources away from production is more costly than the gain from further diversification; as such, market size and τs both fall because that is the only way to reduce κ.

Figure 2:

Equilibrium with distortive institutions.

At very low levels of τg, exchange costs are so high that no diversification is worthwhile (agents set τs=0 and #M=1). When τg is very high, the loss to productive capital is such that no diversification with τs=0 and #M=1 is better than some diversification with the small residual endowment, even though the endowment are still being taxed. In short, market size is hump-shaped in τg and, for some low and high values of τg, agents optimally choose not to diversify. The general pattern of robustness to parameters discussed in the previous section holds with regard to the effect of distortive institutions; in particular, the non-monotonic responses of τs and #M to changes in τg holds across a range of endowment levels and risk aversion parameters. The non-monotonicity of #M is also robust to making S and G capital input-substitutes in the exchange cost technology, although the relationship between τs and τg is, of course, negative when we assume they are substitutes (aside from increases in τs that result from increasing #M).

Given that Section 11 has established the efficient outcome, we can calculate the consumption equivalent loss for each level of τg relative to that efficient frontier. In particular, if uτg′ is the utility obtained under policy τg′, then the percentage loss in consumption that is equivalent to moving from τg∗ to τgd can be calculated as [1−u−1 (uτgd)/u−1(uτg*)]×100. As Figure 3 shows, deviation in τg from its optimal level can have significant welfare costs. A τg that is 0.05 higher than optimal is equivalent to a 2.7% drop in consumption; a τg that is 0.15 higher than optimal is equivalent to a 15.1% drop in consumption.

Figure 3:

Welfare costs of distortionary institutions.

6.2 Specific Transaction Cost Policies

The discussion of the empirical evidence above points to the complexity of classifying and measuring transaction costs in reality. In the light of that, it is reasonable to consider an environment where the optimal tax policy is unclear. The literature on transaction cost economics suggests that optimality is where “transactions are aligned with governance structures so as to effect a (mainly) transaction cost economizing outcome” (Williamson 2010, 681). The interpretation of that optimality condition in terms of policy may be complicated since there are multiple observable and potentially non-observable components: The number of exchanges (market size), the investments in reducing the cost of exchange, and the cost of exchange itself. As such, a policy maker may have as its objective some minimum or maximum of each of these and think it a reasonable interpretation of what is optimal: First, institutions could deliver zero exchange costs; second, institutions should maximise risk-sharing; third, the cost of individual exchanges should be minimized^[18]; and, fourth, institutions should minimize the size of the transaction sector as a whole.

In our model, zero exchange costs are not feasible; institutions which deliver an infinite number of trades at zero cost are themselves infinitely costly. However, the analysis above shows that we can consider the second, third and fourth type of policy prescription. The second is a rule that maximizes the (constrained optimal) choice of market size; the third minimizes the cost of individual exchanges; the fourth minimizes size of the transaction sector. The market size policy is given by,

That is, τgM is the lowest tax required to induce the maximum market size, given that agents optimally choose market size and τs in response to τg. The tax that minimizes the exchange cost is given by,

Finally, the tax policy which minimizes the transaction sector is given by,

For each policy the percentage change in certainty equivalent consumption is calculated using the efficient frontier established in Section 3. Table 1 describes various features of equilibrium under the different rules.

None of the rules are optimal in a framework with endogenous transaction costs; the τgM rule delivers too little directly productive capital and τgκ too much consumption variability. The τgα rule delivers a worse outcome in both regards. Under the τgα rule, the policy to minimize individual exchange costs involves setting the tax to the highest level without shutting down all exchange, regardless of the effect on total transaction costs or on the amount of consumption variability. The economy is pushed toward low-market size (high consumption variability) and low productive capital. This results in a consumption equivalent loss far in excess of the other rules. Nonetheless, it is useful to analyse which of τgM and τgκ is the more costly, and why. Consider first the τgM rule. Larger markets means, in short, higher taxation to lower the costs of bilateral exchange. The transaction sector is larger as a whole and its composition has shifted toward ex ante investments. τg cannot increase by too much, however, since agents always have the option of shifting resources to goods production by reducing market size and τs. For the τgκ rule, minimizing transaction costs requires exchange costs to be too high. A low τg means that agents optimally form smaller markets and make smaller investment in S-capital, both actions reducing κ.

The loss from τgM is greater than that from τgκ, but both are eclipsed by the τgα rule.^[19] Expected utility is a product of the portion of the endowment that is productive (1−κ) and the amount of consumption variability (#M). The extreme costliness of the τgα rule results from neglecting both of these. In this sense, our distinction between the costs of exchange and the total costs of the transaction sector is particularly important: An apparently sensible policy to minimize the costs of exchange, while neglecting the resources required to obtain that minimization, is highly distortive. The comparison between τgM and τgκ is also informative: Institutions fostering smaller exchange costs and bigger markets appear to be the more damaging in welfare terms. By targeting market size, the government “distorts” choices of both #Mandτg, leaving agents, in effect, with only one instrument to respond, τs. The alternative τgκ, which minimizes the sum of all resources allocated to transactions, leaves private agents with the choice of market size and τs. The option for agents simply to consume their own endowment constrains policy such that the outcome is not too far from that which maximizes expected utility. That additional flexibility appears to reflect some of the empirical findings of Acemoglu and Johnson (2005). That paper finds that bad property rights institutions are more damaging that contracting institutions since individuals can respond to contractual distortions by using a variety of formal and informal mechanisms. In the case of this model, agents can respond via choice of τs and market size in the face of the τgκ rule, but have only a limited range of response in the case of the τgM rule.

6.3 Transaction Taxes

Policies are sometimes designed to address the negative externalities from trade (such as trades of a pollutive item), or to raise revenue from a sector characterized by high-frequency trading (as in a Tobin-type tax). In such policies, individual exchanges are taxed. The European Commission analysis of its proposed Financial Transaction Tax (FTT), for example, finds “in a nutshell… very positive impacts on the functioning of the single market for financial instruments” (European Commission 2013, 16). The consequences of such a tax are not necessarily obvious; however, since agents may respond by increasing investment in private exchange cost reduction or may dramatically change the number of trades. Moreover, the robustness of a market to the imposition of a tax, i.e., its revenue-generating ability, may not be fully understood since those markets may shut down or relocate to avoid the tax.

In order to understand the impact of a transaction tax, we need a framework in which the nature of trade (i.e., the number of exchanges and investments in the cost of exchange) is endogenous. As such, our model can form a basis for a preliminary analysis of transaction taxes. We can consider a tax, t > 0, that simply makes individual exchanges more costly,

Figure 4 demonstrates the effect of a transaction tax where transaction costs are endogenous (solid line). Agents respond first by increasing investments in institutions in order to dampen the effect of the tax on the cost of exchange. Relative to an environment where transaction costs are exogenous (dashed line),^[20] market size is more robust to the introduction of a transaction tax. In each environment, agents can avoid the tax by simply resorting to autarky, recovering the investment in exchange and avoiding all tax, whenever the participation constraint is no longer satisfied. The important difference when transaction costs are endogenous is that agents have individually allocated a portion of their endowment to support institutions; in the exogenous case this investment has not occurred and so cannot be retrieved by agents. That means that although agents initially appear less affected by the tax, they will opt for autarky at a level of the tax far lower than that suggested when transaction costs are considered exogenous.^[21] The implication for transaction tax policy is that, while markets respond as would be expected to the imposition of a tax by reducing market size, agents will resort to autarky earlier than may expected if the individual agents’ own investments in that exchange are not taken into account. In case of the FTT, for example, this may mean that a transaction tax leads to the shifting of financial activities to geographies outwith the FTT’s jurisdiction at much lower levels of the tax than may be anticipated. Revenues from such a tax may fall short of projections.

Figure 4:

The effects of a transaction tax.

We can again calculate the losses from imposing a tax on transactions. Figure 5 depicts losses associated with Figure 4. A 5% transaction tax leads to 3.7% consumption equivalent loss in both the exogenous and endogenous cases, which is somewhat higher than an equal deviation in the τg. An increase in τg above optimum is at least reducing exchange costs; imposing a tax on exchange means that, even if agents can respond with higher τs and lower market size, the cost of exchange is increasing. In the case where agents can respond by removing their own investments in reducing exchange costs, the consumption equivalent loss is capped at 4.9% once the transaction tax induces no exchange.

Figure 5:

Welfare costs of transaction taxes.