Los Impuestos sucesorios y las razones de Justicia

Let ( * +) denotes the present value of the UFB contract for Chorus at the node

( ) when the firm is in state* +, where and equal the numbers of construction periods that have been completed in HL, HH, LH and LL cities respectively. Since all the cities in our basic model take one period to complete, the values of and will be either 0 or 1, where 1 means that the city has been finished and vice versa for 0. For example,

( * +) represents the present value of the UFB contract for Chorus at the node

( ) when the firm has already finished both the HL and HH cities.

If no city has been completed at the expiry date N then the UFB contract has no value, so that

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at all nodes ( ).

Now consider the more general case where either Chorus has built up a fibre network in some cities or it has not started yet. From date N onwards the development right is worthless since the right to develop has expired. However, the firm still receives revenue from those completed cities in each post N period. Therefore

( * +) ( * +)

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( * +) , ( ) ( )- ( ) ( )

for all . At each earlier date, Chorus must choose between constructing and waiting. If it decides to wait, then there is cash flow from cities already completed at date , and at date the present value of the fibre contract will be ( * +) if an up move occurs and ( * +) if a down move occurs. The payoff from waiting is therefore

( * +) ( ) ( * +) ( ) ( * +) ( )

In contrast, if Chorus chooses to construct, it is in one of four categories below at date , depending on the values of and .

State category D (when the state indicator * + equals * +, * +, * + or

* +):

For example, if Chorus is in the state * +, there is only one remaining city for it to construct at date (which is city HL), hence it pays ( ) and immediately receives ( * +), where

( )

( * +) ( ) ( )

At date , the UFB contract will be worth ( * +) if an up move occurs and

( * +) if a down move occurs. The payoff to Chorus from constructing in state * + is therefore

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43 ( * +) ( )

( ) ( * +) ( ) ( * +)

From equation 5, we learn that the payoff to Chorus from waiting in state * + should be

( * +) ( ) ( * +) ( ) ( * +)

Since Chorus must choose between constructing city HL and waiting when it is in this state, it should adopt whichever action has the higher payoff, so that the present value of the UFB contract for Chorus at node ( ) equals

( * +) { ( * +) ( ) ( * +) ( ) ( * +) ( * +) ( ) ( ) ( * +) ( ) ( * +) } ( )

The evaluations for all other states in category D follow the same process as state * +.

State category C (when the state indicator * + equals * +, * +, * +,

* +, * + or * +):

Similarly, we use one of the specified states to demonstrate the evaluation process in this category. For example, if Chorus is in the state * +, there are two remaining cities for

the firm to construct at date , hence it can choose from the following three actions—pay

( ) to build up city HL, pay ( ) to build up city HH or pay

( ) to build up two city together. No matter which decision Chorus makes it immediately receives ( * +), where

( ) ( ) ( ) ( ) ( * +) ( )

If Chorus chooses to construct city HL only, at date , the UFB contract will be worth

( * +) if an up move occurs and ( * +) if a down move occurs. The payoff to Chorus from constructing city HL is therefore

( * +) ( )

( ) ( * +) ( ) ( * +)

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However, if Chorus chooses to construct city HH only, at date , the UFB contract will be worth ( * +) if an up move occurs and ( * +) if a down move occurs. The payoff to Chorus from constructing city HH is therefore

( * +) ( )

( ) ( * +) ( ) ( * +)

Besides, if Chorus chooses to construct both cities together, at date , the UFB contract will be worth ( * +) if an up move occurs and ( * +) if a down move occurs. The payoff to Chorus from constructing city HL is therefore

( * +) ( )

( ) ( * +) ( ) ( * +)

From equation 5, we learn that the payoff to Chorus from waiting in state * + should be

( * +) ( ) ( * +) ( ) ( * +)

Since the firm must choose from constructing city HL, constructing city HH, constructing both cities and waiting, it should adopt whichever action has the higher payoff, so that the present value of the UFB contract for Chorus at node ( ) equals

( * +) { ( * +) ( ) ( * +) ( ) ( * +) ( * +) ( ) ( ) ( * +) ( ) ( * +) ( * +) ( ) ( ) ( * +) ( ) ( * +) ( * +) ( ) ( ) ( * +) ( ) ( * +) } ( )

The evaluations for all other states in category C follow the same process as state * +.

State category B (when the state indicator * + equals * +, * +, * +

or * +):

The payoffs of each construction action could be created in a same way as above; therefore we do not list them all here. Since the firm must choose from seven constructing actions and

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one waiting action, it should adopt whichever action has the higher payoff, so that the present value of the UFB contract for Chorus at node ( ) equals

( * +)

{ * + _{* +} } ( )

The evaluations for all other states in category B follow the same process as state * +.

State category A (when the state indicator * +equals * +):

If the firm is in this category, it will face all sixteen choices. Rather than list them all here, we just note that they are similar to those in equation 7, where

( * +)

{ * + * + } ( )

Actually, the general evaluation formula for all categories will be

( * +)

{ * + _{* +} } ( )

This equation, together with the terminal condition (equation 4), completely determines the present value of the UFB contract for Chorus at date 0.