Curvas de rechazo en función de los flujos transmembrana para el cuarto

2. The aggregate number of ships and cargo quantity demanded are exogenously deter- mined.

3. Oil traders live until they make a shipment and then “die,” replaced by a new trader drawn from a truncated normal distribution of trade demand. They are impatient; there are opportunity costs for waiting to ship oil because each week they buy oil and have to pay a storage cost for the days until the ship arrives at port.

4. Ships can only match at discharge and waiting areas, and once they are matched, they cannot match to another trader until they discharge the cargo at the discharge area. 5. There are no search costs for matching.

6. The market is competitive. Ships consider market conditions, but do not focus on analyz- ing how rival ships will respond if they take particular decisions.

7. Where there are multiple routes, ships travel on the route most travelled for the area-area pair based on a distribution.

8. Agents are rational and maximize profits.

4.3

The theoretical static matching model

I first develop the theoretical static matching model for the tanker market. The static model is defined as one period model and is the same as solving for the terminal period T in a finite horizon dynamic model.

4.3.1 Supply side of the market

The supply side of the market is defined as follows:

• A ship j has characteristics defined by its type xjtdescribed in section 4.1.

• The vector n(xjt, t) = [n(x1t, t), ..., n(xJ t, t)] holds the quantities of each ship type xjt

at time t.

• If a ship j matches with a trader i, it receives a payoff Wx_(x

jt, t) equal to:

max

yi

[P (xjt, yi, t) − C(xjt, yi, t) + βxWx(xj,t+1, t + 1)] (4.2)

1. P (xjt, yi, t) is the freight rate in lumpsum units ($).

2. C(xjt, yi, t) is the shipment cost which includes the fuel and opportunity costs as-

sociated with traveling from the ship’s current location to the discharge location. 3. Wx(xj,t+1, t + 1) is the expected future payoff from the discharge location, known

as the option value for ship i of type xj,t+1 defined by the function g(xjt, yi, t)

which determines the ship’s discharge location.

The option value depends on the policy employed. Policy 1 approximates the option value using a “quasi-myopic” approximation. In shipping, it is almost always the case that ships must ballast empty to another load area after they have dropped off the cargo at the discharge location and discounting the future completely would ignore this cost. The option value in Policy 1 is therefore the discounted cost of returning to the original load area. I include this policy in the model for two reasons; the first is simply for comparison purposes, and the second is because we do not know whether shipowners are forward- looking. In contrast, Policy 2 includes the value of employment in future periods, but because the freight rate is volatile, there is uncertainty about future payoffs. Therefore the shipowner only looks ahead one period.

The forward-looking policy differs from the quasi-myopic policy in two ways. First, the repositioning cost from the current period’s match in Policy 1 is the repositioning cost from the discharge location back to the same load area where the ship matched, whereas in Policy 2 the ship considers other load area locations such that it is an expected repositioning cost. The second difference is the option value in Policy 2 also includes the expected future payoffs if it matches to a trader or if it doesn’t match to a trader and has to relocate to a waiting area. It is important to include these future payoffs when considering different matching options because there could be instances in which the repositioning costs are similar between two matches, but the future matching payoffs are different because they are specific to the load and/or waiting area. The option value is discounted by βxwhich equals 1/(1 + rx)d(xjt,yi)_{, where r}x_{is the discount rate and the}

discounting is over the duration d(xjt, yi) of the current voyage.

• If a ship j doesn’t match with a trader i, it has to relocate to a waiting area w ∈ W. In the model, this is equivalent to matching to a dummy trader ∅yand the payoff (the ship’s

surplus) of this match is equal to:

4.3. The theoretical static matching model 63 where g(xjt, ∅y) denotes the function determining the ship’s type which includes its new

location when it remains unmatched. In the model, there are two waiting areas: one in Fujairah in the Arabian Gulf and the other in West Africa and the surplus from these matches are: s(xjt, ∅y1, t) = ˜s(xjt, ∅y1, t) + β xjt,d(xjt,∅_y1)_Wx_(g(x jt, ∅y1), t + 1) s(xjt, ∅y2, t) = ˜s(xjt, ∅y2, t) + β xjt,d(xjt,∅y2)_Wx_{(g(xjt, ∅y} 2), t + 1)

where ∅y1 represents the dummy trader in Fujairah and ∅y2 is the dummy trader in West

Africa. The maximum value of these options will be the preferred unmatched option:

s(xjt, ∅y, t) = max [s(xjt, ∅y1, t), s(xjt, ∅y2, t)] (4.4)

4.3.2 Demand side of the market

The demand side of the market is defined as follows:

1. A trader i has characteristics defined by its type yidescribed in section 4.1.

2. The vector n(yi, t) = [n(y1, t), ..., n(yI, t)] is a vector holding the quantities of each

trader type.

3. If a trader i matches with a ship j, it receives a payoff Wy(yi, t) equal to:

max

xjt

[π(xjt, yi, t) − P (xjt, yi, t)] (4.5)

where π(xjt, yi, t) is the profits from the sale of the oil, and P (xjt, yi, t) is the freight

rate in lumpsum units ($) if a trader of type yimatches to a ship of type xjt.

4. If a trader i doesn’t match with a ship j then it has to pay a storage cost to store the oil in the load area until it can match with a ship. In the model, traders which are unmatched match to a dummy ship ∅xwith a payoff (surplus) equal to:

s(∅x, yi, t) = ˜s(∅x, yi, t) + βy,d(∅x,yi)Wy(yi, t + 1) (4.6)

where ˜s(∅x, yi, t) is the storage cost and Wy(yi, t + 1) is the expected future payoff in

4.3.3 Pairwise surplus function

The combined payoff of a match between a ship and trader is known as the pairwise surplus function. It provides the total valuation of a ship of type xjt matched to a trader of type yi and

equals the sum of the payoffs Wx(xjt, t) and Wy(yi, t) from equations 4.2 and 4.5 respectively:

s(xjt, yi, t) = max zt [ π(xjt, yi, t) − P (xjt, yi, t)
+ + P (xjt, yi, t) − C(xjt, yi, t) + βxWx(xj,t+1, t + 1)
] = π(xjt, yi, t) − C(xjt, yi, t) + βxWx(xj,t+1, t + 1) = s(x˜ jt, yi, t) + βxWx(xj,t+1, t + 1) = Wx(xjt, t) + Wy(yi, t) (4.7)

Because the pairwise surplus function includes a transfer of P (xjt, yi, t) from the trader to

the ship, the freight rate cancels out. The surplus function can be split into the current period’s surplus ˜s(xjt, yi, t) and the surplus in the future period βxWx(xj,t+1, t + 1).