The net present value of governmental policy: A possible way to find the Stackelberg solutions

Tilburg University

The net present value of governmental policy

Gradus, R.H.J.M.

Publication date:

1987

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Gradus, R. H. J. M. (1987). The net present value of governmental policy: A possible way to find the Stackelberg

solutions. (Research Memorandum FEW). Faculteit der Economische Wetenschappen.

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THE NET PRESENT VALUE OF GOVERNMENTAL POLICY: A POSSIBLE WAY TO FIND THE STACFOrLBERG SOLUTIONS

Raymond H.J.M. Gradus

FEw 274

~~S

1

THE NET PRESENT VALUE OF GOVERNMENTAL POLICY: A POSSIBLE WAY TO FIND THE STACKELBERG SOLUTIONS

by Raymond H.J.M. GRADUS ~) Tilburg University

P.O. Box 90153 - 5000 LE Tilburg The Netherlands

August 1987

Abstract.

In a paper by Kort (1987) it is shown that in deterministic dynamic models of the firm the net present value of the last dollar investment equals zero, if the firm is in equilibrium. We show that this result not only holds in the case that the firm is a single decision maker, but also if the firm is a player in a differential game. If we have a differential game in which the government influences firms by announcing a certain tax rate and the firms have to decide about whether to invest their money or pay out dividend, then we can show that for the Nash- and

Stackelberg concept firms pay out dividend at the moment that the NPV equals zero. We can also derive a net present value for the government (the stream of (discounted) cash-flows (i.e. tax payments)) due to one extra dollar investment by the firm). It turns out that in the open-loop Stackelberg (in case of certain parameter values) _{case this NPV for the} government also equals zero at the moment the firms stop investing. This is of great importance, because in this way we can derive decision rules for governmental policy and we can calculate very easily the Stackelberg solutions for more difficult models.

1. Introduction 2. The model

2.1 The firm

2.2 The government

2.3 The total model

2.4 The solution 1 2-5 2 2-3 3

3-5

3. The net present value for the government and for the

firm

6-16

3.1 Introduction 6

3.2 The net present values at switching times 6-8

3.3 A graphical illustration _8-16

4. A method to derive the Stackelberg solutions 16-17 5- Some extensions and their Stackelberg solutions 17-19

5.1 The model with a salvage value _17-18

5-2 The model with a discount rate _1g-19

6. Conclusions ₂₀

References _20-21

Appendices _21-25

Appendix 1 The derivation of Table 5 21-22

Appendix 2 The derivation of the open-loop Stackelberg 22-24 solutions

1

1. Introduction:

For the outcome of the economic process and the way firms behave governmental policy is of great importance. A crucial question in economic theory is in this way: "how can the government by tax policy, wage

regulations or monetary measures influence the decisions of the firms in such a way that the objectives of the national economy are achieved?" In a previous paper [1] we have tried to model and describe interactions between government and firm as a differential game in which different solution concepts (Nash, Pareto and Stackelberg) are possible. Zn that modlel the government has the possibility to influence the economic process by

announcing a certain tax rate, while the firms (represented by one) decide whether to invest their money or to pay out dividend.

In this paper we give this model a better economic interpretation by introducing the term net present value, which is wellknown from business economics. Moreover, by using the net present value of marginal investment, which we can calculate for both agents, we can derive decision rules for firm behaviour and governmental policy and it turns out that these decision rules can also be applied to more complex models.

In section 2 we give a brief presentation of the basic model and its solutions for the different concepts. For a more detailed description of this model we refer to [1]. Section 3 contains the calculations of the net present value of marginal investment for the government and the firm. Our interest is especially focused on the net present value at switching times, i.e. the time-points at which an agent switches to another policy. In section 4, we give a method for finding the Stackelberg solution for the basic model based on the present values. In section 5 this method is used to derive the Stackelberg solutions for the model with some extensions such as a discount rate and a salvage value. Finally, in section 6 we make some concluding remarks.

2. The model:

2.1. The

firm-We assume that the firm behaves as if it maximizes the

shareholder's value of the firm. This value consists of the sum of the

dividend stream over the planning period. Assuming a zero discount rate yields: T max. J D(t)dt, 0 (1) in which t - time T - planning horizon D(t) - dividend

Assume that the amount oF capital goods can only be raised by investment:

K(t) - I(t), (2)

in which K(t) - capital good stock I(t) - investment

We assume that profit is a linear function of the capital good stock:

0(t) - qK(t), (3)

in which 0(t): profit (before tax payment)

q: rentability of capital good stock

Assuming that profit after taxation could be used for investment or to pay out dividend, we get the next relation:

0(t) - TX(t) - D(t) t I(t), (q)

in which TX(t) - tax payment

Furthermore, investment and dividend must be non-negative:

D(t) ~ 0 ₍₅₎

I(t) ~ o (6)

2.2. The government:

For the government too we make simplifying assumptions: all the tax payments received will immediately be spent on government consumption (,which is not productive). The government is not able to spend more than it receives (i.e. no budgetary deficit):

TX(t) - G(t), (~)

3

T

max. f U(G(t))dt,

T 0 (8)

where U(G(t)) is the utility function for the government, which is defined in terms of government consumption and ~r is the profit tax rate. In this

paper we assume:

U(G(t)) :- G(t), ₍₉₎

so the government has a linear utility function. Of course, other utility functions are possible, but we have taken the linear one for simplicity. In that case ( 8) becomes:

T

max, f G(t)dt

i 0 (10)

Furthermore we assume that the tax payments are restricted by:

~lo(t) ~ Tx(t) ~ ~2o(t). ₍₁₁₎

where T1 and t2 are constants such that: 0 ( tl~ t2~ 1

2-3. The total model:

We can easily rewrite the model as follows -government

(12)

T

max.

u f qKuldt , 0 C i1C ul( i2( 1, ₍₁₃₎

1 0 -firm T max. u₂ f qK(1-ul)(1-u2)dt , 0~ u2 ~ 1 (14) 0 --state equation K - qK(1-ul)u2, ₍₁₅₎

where u1:-tax rate- p _{and u2:-investment rate- I} 0-TX

In this way we have derived the same mathematical model as Lancaster [2) but we use it to solve a completely different economic problem.

2.4. The solutions:

feedback, closed-loop) and the solution concept (Nash, Stackelberg and Pareto). As pointed out in [1] the open-loop Nash, the feedback Nash and the feedback Stackelberg (with government as leader) are equal, so we can concentrate on the open-loop solutions. The solutions are given in Table 1-4. In the Stackelberg game is assumed that the government is the leader.

Table 1: The Nash-solution if ~2 ~ ₂

---I t E[O,t) I t E[t,T] ---ul(t) I ~1 I T2 uZ(t) ~ 1 ~ 0 K(t) q(1-il)t I K~e I Kw ---Table 2: The Stackelberg- and Nash-solution if

i2 ( 2

---I t E[O,t) I t E[t,t) I t E[t,T]

---ul(t) I T1 I t2 I T2 u2(t) ~ 1 ~ 1 ~ 0 q(1-il)t . q(i-iZ)(t-t) _.. K(t) ~ K~e ~K(t)e ~ K ---Table 3: The Stackelberg-solution if T2 ~ Z

---5

Table 4: The Pareto-solution

---~ t e[~.tN) ~ t E[tN,T] ---Ui ( t) ~ T1 ~ T g [T1 ..t2] u2(t) ~ 1 ~ 0 ~i(i-T )t NNMN K(t) ~ K~e i ~ K ---where t - T - 1 q(i-t2) ---ln(2T2) t - t 4 ) , q(1-2Z tN-T- 1 q ~ 1-2i1 t - min{T,T - )} q(22-~1 t- min{T - q , T- q(i-T )} 1 N K - K~exp{q(1-T1)t} MN ~ K - K~exp{q(1-ii)t}exp{q(1-i2)(t-t)} NiFM K - KDexp{q(1-ti)t} NMi1~F N K - K~exp{q(1-Ti)t } (16) (17) (18) (19) (20) (21) (2z)

(23)

For the Pareto solution we assume that both players are in the same bargaining position (see [5]). From the Tables 1 through 4 we conclude that the problem has a bang-bang structure (i.e. the control jumps from its

to see that the period of investment is different for the 3 solution

~ ~

concepts: t( t C t. In the Stackelberg case, as a compensation for the fact that the firm goes longer on with investment, the government will postpone the application of the high rate. So, in the Stackelberg case there is a period possible in which the firm pays out dividend and the government asks the low rate, while such a period could not exist in the Nash game.

3. The net present value for the Qovernment and for the firm:

3.1. Introduction:

What is meant by the present value of an investment proposal ? Every book on finance theory gives a definition. So we can find in [3, p. 26]: "an investment proposal's net present value is derived by discounting the net cash receipts at a rate which reflects the value of the alternative use of the funds, summing them over the life of a proposal minus the initial investment outlay". In our framework we can define the marginal net present value for the firm as follows: the stream of dividend ( the cash receipts for the shareholders) over the remaining part of the planning period due to one dollar extra i nvestment minus the initial investment outlay, which equals one. In the basic model we have assumed that the discount rate is zero, so we can take the (undiscounted) values of dividend. In section 5.2 we shall relax this assumption. In the same way we can define the marginal net present value for the governmment: the stream of tax earnings ( the cash receipts for the government) over the remaining part of the planning period due to one dollar extra investment by the firm minus one. In subsection 3.2 we will derive these net present values at the switching points form

investment to dividend and from low to high tax rate. In subsection 3.3 we shall derive this net present value for some examples at every time-point

for different tax regimes.

3.2. The present values at switchin;q

Table 5:the NPV at switching times ~ investment switch --- --- ---firm - government --- ----I-Nash i2~ Z 1 z2C Z Stack _{tl) 2} 0 0 0 1 1 T1~ 2' T2~ 2 0 tax switch --- ---firm government --- ---0 1 222 1-221 T -Z (1-T2)-1_{2 1} Pareto ~ -t ~ T I-T I T ---(for a derivation _{see appendix 1)}

The firm switches from investment to dividend at the moment that its net present value equals zero. Only in the Pareto case the firm still continues investing, because we have a cooperative solution and the sum of both present values must be equal to one. This is optimal for the firm because in the period before the net present value equals zero, it is greater than zero, which implies that marginal earnings are greater than marginal costs. After the switching time the net present value is less than zero, which means that the firm makes a loss if it goes on with investing.

We will now discuss the situation T2~ 2. In the Nash case the net present value for the government is still greater than zero, at the moment that the firm stops investment. The government wants more investment, but it cannot force the firm to do so. In the Stackelberg case the government can force the firm to postpone the switch from investment to dividend by announcing a longer period of low tax rate at the beginning of the planning period. Therefore, playing Stackelberg is better for the leader than

playing Nash. For the follower too, Stackelberg is better than Nash. The government fixes its switching point from low to high rate, so that at the moment the firm switches from investment to dividend, the net present value of the government equals zero. Now, the government also has reached an optimal situation. However, in the case that il)Z the low rate during the

0

whole planning period is not enough to have a net present value for the government of zero at the switching time. So, the optimal policy is to maintain the low rate during the whole period in order to tempt the firm to invest during a long period.

When T2(Z, Nash is equal to Stackelberg. Now, the government switches earlier than the firm due to the fact that its net present value sooner equals zero. This implies that the net present value for the government will not have a value of zero at the switching time from investment to dividend, even not if the government postpones the application of high rate, so there is no incentive to do so. At the

switching time of the firm the net present value of the government is less than zero, so the government cannot reach an optimal situation with respect to investment.

3.3. A Qraphical illustration:

It is also possible to calculate this net present value for marginal investment at every time-point and not only at the switching times. In Figures 1-6 we present a graphical illustration of the net present value, _{in which we used the following parametervalues:}

T1-4, 22-~ (Figures 1-2), t1-~, t2-~ (Figures 3-4), t1-b, tZ-b (Figures 5-6),q-K~-1 and T-5. For each set of parametervalues we have compared the net present values for the following four possibilities:

i) the government and the firm play Nash

i) the government (the leader) and the firm play Stackelberg

9

Table 6 :the switching times

I T1-4 ~2-~ I T1-~ T2-~ I~1-b 22-È

---I---I-

---I-

---~investment~ tax ~investment~ tax ~investment~ tax ~

---~---~---~---~---~---~---~ Nash Stackelberg only T1 only i2 I 1 I 1 I 1 I 3 I ~ I ~

₃

-

~

I

I;

I

I

I

I ;

I

I s

I

I

--... .. ... I- --~- --...- -~-....- --...

comment _{IStackelberg-only -[ll Nash-Stackelberg}

I ---With our three sets of parameter values we have three illustrative exampies of the possible solutions (see Tables 1,2 and 3).

In the first situation, where T1-~ and z2-4, it is easy to see that the government will be better off, when Stackelberg is played, because the NPV at the initial time is greater. The Stackelberg's NPV is also greater than the NPV's of 'only i2' and 'only T1'. In fact the government tries to choose a tax switch such that its net present value at the initial time is as big as possible. The NPV of Nash is the greatest possible

outcome under the condition that the situation with low tax and no investment cannot appear.

In the second situation, where t1-~ and T2-~, it ís optimal for the government to stick to its low tax level. Each situation with only a little interval t2 will give a lower NPV at the initial time.

~,1 - 1 r4 Z2 a 3I4 N P V J 1 i

1

i

11

FIGURE 2: THE NET PRESENT VALUES FOR THE FIRM

73

FIGURE 4: THE NET PRESENT VALUES FOR THE FIR~d

FIGURE 5: THE NET PRESENT VALUES FOR THE GOVERNMENT

Z1

-

~ r6

z2

-

2r6

TIME

4 3

75

FIGURE 6: THE NET PRESENT VALUES FOR THE FIRM

rate. So there is no reason for a time-interval with low tax rate and no investment and there is no difference between Stackelberg and Nash. 4. A method to derive the Stackelberg solutions:

In this section we give a solution method for finding the open-loop Stackelberg solutions for the model specified in (13)-(15). In the previous section we have argued that in the case ti(2 and 22)2 the net present value at the investment switch for both particípants is zero. IF the investment switch take places at t(u2(t)-1,te[O,t);u2(t)-0,te[t,T]) and the tax switch take place at t(ul(t)-Tl,te[O,t);ul(t)-TZ,tE[t,T]), we can derive, taking into account that t~ t:

NPVf(t)-0 --~ NPVf(t) - J ~(t)(1-T(t))dt - 1 t t T - J q(1-T1)dt . f q(1-TZ)dt - 1- 0 t -t -~ (t-t)q(1-T1)t(T-t)q(1-T2)-1 ₍₂₅₎ NPVg(t)-0 -~ (t-t)qtl.(T-t)qt2-1 ₍₂₆₎

From (25) and (26) can be calculated t and t, because there are 2

17

Summarizing we can give the following method for finding the open-loop Stackelberg solution for the basic model:

1) Find the Nash solution and calculate the parameter values, when NPVg(t)(0. For these values Nash-Stackelberg.

(comment: also the Nash solution can easily be found with this NPV-method: NPVf(t)-0 -~ t, if NPVg(t) ~ 0 then t - tax switch - t

if NPVg(t) C 0 then NPVg(t)-0 ~ t) 2) Find the solution of the two equations NPVf(t)-0

NPVg(t)-0 ~ t and t.

3) In the situation that t)T there will be no tax switch and the government asks the low rate during the whole period.

From an economic point of view we can argue that this method can also be used for the extensions, we take into consideration.

In appendix 3 we have given a scheme of the solution procedure. 5. Some extensions and their Stackelberg solutions:

Of course the model we have presented in section 2 is simple and has some unrealistic features. In a previous paper [1, section 5] we have

generalized this model in several ways by incorporating a discount rate, investment grants, depreciation etc.. In that paper we give the Nash solution for these extendend models. In this paper we give the open-laop Stackelberg solutions for two important extensions. To find the solutions for more extensions ( and combinations) is a topic of future research. 5.1. the model with a salvage value:

If we incorporate a salvage value for the gove-rnment and the firm the model becomes, ( OCa,bCl):

T

G: m~t) J qK(t)ul(t)dt ~ aK(T) , T1 C ul( i2. ₍₂₇₎

1 0

T

K - 9K(t)(1-ul(t))u2(t) ₍₂₉₎ and the solution has the following form:

Table ~: The Stackelberg solution if T~ 1-a₂ 2-a-b ---I t E Co.tl) I t E Ctl.tl) ~ _{t e Ct1,T]} ---ul(t) ~ T1 ~ T2 ~ i2 u2(t) ~ 1 ~ 1 ~ 0 ---Table 8: The Stackelberg solution if i~ 1-a_{2 - 2-a-b}

---~ t E C~.tl) ~ t E Ctl.tl) ~ _{t e Ct1.T~} ---ul(t) ~ il I T1 I TZ u2(t) ~ 1 ~ 0 ~ 0 ---where t1- T - 1-b_q(1-Tz) 1 1-a}(t2~(1-T2)) t1- tl-q(1-T2)ln{((2-b)(T2~(1-z2)))} ' 1-a-(2-a-b)il t1- min{T,T - } q(T2-~1) t- min{T - 2-a-b T- 1-b } 1 q q(i-~1)

(for a derivation see appendix 2)

(30)

(31)

(32)

(33)

The switches will be at a later time-point than in the basic model.

5.2. The model with a discount

rate-An interesting case is when we incorporate a discount rate. Let us assume that the government and the firm have the same discount rate. In that case the model becomes:

T

max.

G: ul(t) f~ qg(t)ul(t)e-itdt _{~ il c ul( T2,}

19

T

F: u~t) f qK(t)(1-ul(t))(1-u2(t))e-ltdt _{, 0~ u2 ~ 1 (35)}

2

0

-K - q-K(t)(1-ul(t))u2(t) ₍₃₆₎

We assume that i(q(1-T2), otherwise the marginal return on dividend is greater than the marginal return on investment, so the firm will never invest. Now we have two possibilities again:

Table 9: The Stackelberg solution if i2~ 2

---~ t E CO.t2) ~ _{t e[t2.tZ) ~ t e Ct2,T]} --- -c2 ---ul(t) I T1 I TZ I u2(t) ~ 1 ~ 1 ~ 0 ---Table 10: The Stackelberg solution if

t2~ 2

---~ t E[O.t2) I t E[t2,tZ) ~ _{t e[t2.T]}

---Z- ---ul(t) ~ T1 ~ tl ~ T u2(t) ~ 1 ~ 0 ~ 0 ---where t2- T~ iln{1-_q(1-T2)1 }

₍₃₇₎

(39)

t2- min{T 4 iln(1-2q), T t iln(1- 1 )} (1}0) q(1-zl)

(for a derivation see appendix 2)

6. Conclusions:

In a paper by Kort (1987) it is shown that in dynamic

deterministic optimal control models of the firm the following decision rule holds: the firm invests at its maximum if the net present value of marginal investment is greater than zero and if it is equal to zero the firm stops investment. Now, marginal earnings are equal to marginal costs which implies that the optimal situation is reached. The firm tries to reach the point, where the net present value equals zero, as soon as possible. But in our framework where the firm is faced with corporate tax

this time-point depends on the tax policy of the government.

If we have a differential game with a tax maximizing government as one player and the firm as the other player, it turns out that in the Nash-and Stackelberg case the firm stops investment at the moment that the net present value equals zero. But the government, which gets its money only by corporate tax, also wants investment until the moment that its net present value equals zero. However, in most cases the government is not able to announce a tax policy such that the switch will take place at the moment

that its net present value equals zero. Only in the open-loop Stackelberg solution (Tl~2,i2)2), where the government is the leader and in a stonger position, will this happen.

This is of great interest, because in this way we have given a better economic interpretation of the results and we have derived decision rules for governmental policy and firm behaviour. These decision rules can be used to derive the open-loop Stackelberg solutions for more complex models.

References:

[1] R.H.J.M. GRADUS (1987), "The reaction of the firm on govermental policy: a game-theoretical approach" in G. Feichtinger (ed.), Optimal Control and Economic Analysis 3,(North-Holland, Amsterdam), forthcoming [2] K. LANCASTER (1973)~ "The dynamic inefficiency of capitalism", Journal of political economv (81), 1092-1109

21

[4] P.M. KORT (1987)."The net present value in dynamic models of the firm",

paper to be presented at the SOR conference 198~ in Passau

[5] M. POHJOLA (1984). "Threats and bargaining in capitalism: a

differential game view", Journal of Economic Dvnamics and Control (8), 291-302

Appendix 1. the derivation of Table 5: 1. Nash:

i)If T2) 2 then t- investment-~tax switch - T- 1 q(1-T2) and NPVf(t) - J ~(t)(1-T(t))dt - 1 t (A1.1) T - J q(1-T2)dt - 1- q(1-z2)(T-t) - 1- 0 (A1.2) t 2T -1 NPVg(t) - qt2(T-t) - 1- 1-~ ~ 0 2

ii)If Y2C Z then t - investment switch - T- 1

q(1-~2) ln(2t2)

and t- tax switch - t}

q(1-T2)

(A1.3) (A1.4)

(A1.5)

t q(1-T2)(t-t) T q(1-t2)(t-t)

1

NPVf(t) - Jq(1-t2)e dt4Jq(1-T2)e _{dt - 1- 2~ (A1.6)}

t t 2

t q(1-t )(t-t) T q(1-T )(t-t)

NPVg(t) - JqT2e 2 dt t Jqi2e 2 dt - 1- 0 (A1.7)

t t

We have this exponent because of reinvestment 2. Stackelberg:

ii)If 22~

2, 21~ 2 then t- investment switch - T- q (A1.8)

1-221 t tax switch T -q(22-21) NPVf(t) - q(1-21)(t-t) ~ 9(1-t2)(T-t) - 1- 0 NPVg(t) - q21(t-t) 4 q22(T-t) - 1- o 1-221 NPVf(t) - 9(1-22)(T-t) - 1 - _22-21(1-2 ) - 1₂ ' ' 1-22 NPVg(t) - q22(T-t) - 1- 2T122 - 1 2 1

iii)If 21~ 2 then t- investment switch - T- 1

q(1-21) t - tax switch - T NPVf(t) - q(1-21)(T-t) - 1 - 0 22 -1 NPVg(t) - q21(T-t) - 1' 1-2 ~ 0 1 3. Pareto.

In the Pareto-case the tax and investment switch will take place at

t~-T- 1 q w ~ NPVf(t ) - q(1-2)(T-t ) - 1 - -T w ~ NPVg(t )- qt(T-t )- 1- 2- 1

Appendix 2. the derivation of the open-loop Stackelberg solutions 1. The model with salvage value.

-Step 1:

23

For these parameter values Nash is equal to Stackelberg -Step 2:

(A2.4)

As argued in section 3.3 for t ~ 1-a t

-2- -2-s-b (- investment switch) ~ t (-tax switch), so that

NPVf(tl) - 0~ _{(tl-tl)q(1-il) t(T-tl)qÍl-T2) t b- 1 (A2.5)}

1 - - -

-NPVg(tl) - 0--~ _{(tl-tl)4i1 t(T-tl)qT2 t a- 1 (A2.6)}

Zt is easy to derive from (A2.5) and (A2.6):

` 1-a-(2-a-b)T1 tl- T - q(T2-T1) t1- T - 2-a-b q -Step 3: (A2.7) (A2.8)

If (1-a) -( 2-a-b)T1 C 0 or T1 C 21áab then tl(calculated by A2.7) ) T

So t1- T and t1- T- 1-b q(1-il)

2 2T2-1

NPVg(t2) C 0~ _1-2 - 1 C 0-~ ~c2~ 2 2

For these parameter values Nash is equal to Stackelberg -Step 2:

(A2.14)

As argued in section 3,3 for T~ 1-_{2- 2 t(-investment switch) ) t(-tax} -switch), so that 2 - t2 -i(t-t ) T -i(t-t ) NPVf(t2)-0 ~ _{f q(1-T1)e 2 dt ~ f q(1-i2)e 2 dt - 1- 0} 2 - t2 -i(t-t ) T -i(t-t2) NPVg(t2)-0 ~ J qTle _{2 dt t f qi2e dt} -t2 -t2 or 1 - 0

q(1-T1) _{-i(t2-t2) q(1-t2) -i(t2-t2) -i(T-t2)}

i {1 - e } t i _{{e - e } - 1 (A2.15)}

qTl -i(t2-t2) _qT2 -i(t2-t2) _-i(T-t2)

i {1 - e _{} t i {e - e } - 1}

(A2.16) From (A2.15) and (A2.16) we can derive:

t2- T t iln(1 - 21)

q

- 1-2i1 iq

t2- T - i{ln(1tq(T -T )Xq - 2i)}_{2 1}

-Step 3:

If 1- 2t1( 0 or T1) 2 then t2(calculated by A2.17) ~ T.

(A2.17) (A2.18)

So t2-T and t2- T t iln(1- 1 ) _(A2.19)

25

Appendix 3. A solution scheme:

Let t- investment switch in case of Nash t- tax switch in case of Nash

t- investment switch in case of Stackelberg t- tax switch in case of Stackelberg

IN 1986 REEDS vERSCHENIIV 202 J.H.F. Schilderinck

Interregional Strvcture of the European Community. Part III 203 _{Antoon van den Elzen and Dolf Talman}

A new strategy-adjustment process for computing a Nash equilibrium in a noncooperative more-person game

204 Jan Vingerhoets

Fabrication of copper and _{copper semis in developing countries. A} review of evidence and opportunities

205 _{R. Heuts, J. van Lieshout, K. Baken}

An inventory model: what is the influence of the shape of the lead_{time demand distribution?} 206 _{A. van Soest, P. Kooreman}

A Microeconometric Analysis of Vacation Behavior 207 _{F. Boekema, A. Nagelkerke}

Labour _{Relations, Networks, Job-creation and Regional Development. A} view to the consequences of technological change

208 R. Alessie, A. Kapteyn

Habit Formation and Interdependent Preferences in the Almost Ideal Demand System

209 T. Wansbeek, A. Kapteyn

Estimation of the error components model with incomplete panels 210 A.L. Hempenius

The relation between dividends and profits 211 J. Kriens, J.Th. van Lieshout

A generalisation and some properties of Markowitz' portfolio selecti-on method

212 _{Jack P.C. Kleijnen and Charles R. Standridge}

Experimental design and regression analysis in simulation: an FMS case study

213 _{T.M. Doup, A.H. van den Elzen and A.J.J. Talman}

Simplicial algorithms for solving the non-linear complementarity

problem on the simplotope

214 A.J.W. van de Gevel

The theory of wage differentials: a correction 215 _{J.P.C. Kleijnen, W. van Groenendaal}

Regression analysis of factorial designs with sequential replication 216 T.E. Nijman and F.C. Palm

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217 P.M. Kort

The firm's investment policy under a concave adjustment cost function 218 J.P.C. Kleijnen

Decision Support Systems ( DSS), en de kleren van de keizer ..

219 T.M. Doup and A.J.J. Talman

A continuous deformation algorithm on the product space of unit simplices

220 T.M. Doup and A.J.J. Talman

The 2-ray algorithm for solving equilibrium problems on the unit simplex

221 _{Th. van de Klundert, P. Peters}

Price Inertia in a Macroeconomic Model of Monopolistic Competition 222 Christian Mulder

Testing Korteweg's rational expectations model for a small open economy

223 _{A.C. Meijdam, J.E.J. Plasmans}

Maximum Likelihood Estimation of Econometric Models with Rational Expectations of Current Endogenous Variables

224 _{Arie Kapteyn, Peter Kooreman, Arthur van Soest}

Non-convex budget sets, institutional constraints and imposition of concavity in a flexible household labor supply model

225 R.J. de Groof

Internationale coi5rdinatie van economische politiek in een twee-regio-twee-sectoren model

226 Arthur van Soest, Peter Kooreman

Comment on 'Microeconometric Demand Systems with Binding Non-Ne-gativity Constraints: The Dual Approach'

227 _{A.J.J. Talman and Y. Yamamoto}

A globally convergent simplicial algorithm for stationary point problems on polytopes

228 Jack P.C. Kleijnen, Peter _{C.A. Karremans, Wim K. Oortwijn, Willem} J.H. van Groenendaal

Jackknifing estimated weighted least squares 229 A.H. van den Elzen and G. van der Laan

A price adjustment for an economy with a block-diagonal pattern 230 M.H.C. Paardekooper

Jacobi-type algorithms for eigenvalues on vector- and parallel compu-ter

231 J.P.C. Kleijnen

232 A.B.T.M. van Schaik, R.J. Mulder On Superimposed Recurrent Cycles 233 M.H.C. Paardekooper

Sameh's parallel eigenvalue algorithm revisited 234 Pieter H.M. Ruys and Ton J.A. Storcken

Preferences revealed by the choice of friends 235 C.J.J. Huys en E.N. Kertzman

Effectieve belastingtarieven en kapitaalkosten

236 A.M.H. Gerards

An extension of KSnig's theorem to graphs with no odd-K4 237 _{A.M.H. Gerards and A. Schrijver}

Signed Graphs - Regular Matroids - Grafts

238 Rob J.M. Alessie and Arie Kapteyn Consumptíon, Savings and Demography

239 A.J. van Reeken

Begrippen rondom "kwaliteit"

240 Th.E. Nijman and F.C. Palmer

Efficiency gains due to using missing data. Procedures in regression models

241 S.C.W. Eijffinger

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IN 198~ REEDS VERSCHENEN 242 Gerard van den Berg

Nonstationarity in job search theory 243 _{Annie Cuyt, Brigitte Verdonk}

Block-tridiagonal linear systems and branched continued fractions 244 J.C. de Vos, W. Vervaat

Local Times of Bernoulli Walk

245 _{Arie Kapteyn, Peter Kooreman, Rob Willemse} Some methodological issues in the implementation of subjective poverty definitions

246 _{J.P.C. Kleijnen, J. Kriens, M.C.H.M. Lafleur, J.H.F. Pardoel}

Sampling for Quality Inspection and _{Correction: AOQL Performance} Criteria

247 D.B.J. Schouten

Algemene theorie van de internationale conjuncturele en strukturele afhankelijkheden

248 _{F.C. Bussemaker, W.H. Haemers, J.J. Seidel, E. Spence}

On (v,k,a) graphs and designs with trivial sutomorphism group 249 Peter M. Kort

The Influence of a Stochastic Environment on the Firm's Optímal Dyna-mic Investment Policy

250 R.H.J.M. Gradus Preliminary version

The reaction _{of the firm on governmental policy: a game-theoretical} approach

251 _{J.G. de Gooijer, R.M.J. Heuts}

Higher order moments of bilinear time series processes with

symmetri-cally distributed errors

252 P-H. Stevers, P.A.M. Versteijne Evaluatie van marketing-activiteiten

253 _{H.P.A. Mulders, A.J. van Reeken}

DATAAL - een hulpmiddel voor onderhoud van gegevensverzamelingen 254 P. Kooreman, A. Kapteyn

On the _{identifiability of household production functions with joint} products: A comment

255 B. van Riel

Was er een profit-squeeze in de Nederlandse industrie?

256 R.P. Gilles

Economies with coalitional structures and core-like equilibrium

257 P.H.M. Ruys, G. van der Laan

Computation of an industrial equilibrium 258 W.H. Haemers, A.E. Brouwer

Association schemes 259 G.J.M. van den 8oom

Some modífications _{and applications of Rubinstein's perfect} equili-brium model of bargaining

260 A.W.A. Boot, A.V. Thakor, G.F. Udell

Competition, Risk Neutrality and Loan Commitments 261 _{A.W.A. Boot, A.V. Thakor, G.F. Udell}

Collateral and Borrower Risk 262 A. Kapteyn, I. Woittiez

Preference Interdependence and Habit Formation in Family Labor Supply 263 B. Bettonvil

A formal description of discrete event dynamic systems including perturbation analysis

264 Dr. Sylvester C.W. Eijffinger

A monthly model for the monetary policy in the Netherlands 265 _{F. van der Ploeg, A.J. de Zeeuw}

Conflict over arms accumulation in market and command economíes 266 F. van der Ploeg, A.J. de Zeeuw

Perfect equilibrium in a model of competitive arms accumulation 267 Aart de Zeeuw

Inflation and reputation: comment 268 A.J. de Zeeuw, F. van der Ploeg

Difference games and policy evaluation: a conceptual framework 269 Frederick van der Ploeg

Rationing in open economy and dynamic macroeconomics: a survey 270 G. van der Laan and A.J.J. Talman

Computing economic equilibria by variable dimension algorithms: state of the art

271 _{C.A.J.M. Dirven and A.J.J. Talman}

A simplicial algorithm for finding equilibria in economies with linear production technologies

272 _{Th.E. Nijman and F.C. Palm}

Consistent estimation of regression models with incompletely observed exogenous variables

273 Th.E. Nijman and F.C. Palm

Bibliotheek K. U. Brabant

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