the various insurance markets, hence causing various degrees of com- petition with respect to various types of insurance, and to failure of insurance markets ...
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Chapter 5 Competition in Financial Markets
1. Introduction In the short run competition is specified exogenously, i.e., we have perfectly competitive markets, monopoly markets or oligopolistic markets, all of which are, specified exogenously. In the long run there are a large number of potential entrants not all of whom enter a particular industry (Hahn (1985)). Besides, what competition means in financial markets like the insurance market is not clear. Competition in an industry depends on the number of participants (buyers and sellers) in that particular industry. Since participation in an industry is costly it seems natural that agents will consider the costs and benefits of participating as either buyers or sellers in the various commodities of that industry, and not everybody can participate. Hence, there will arise various degrees of competition depending upon to what extent there is participation within the industry. In the context of the insurance industry there will arise various degrees of competition with respect to the various types of insurance. The purpose of this paper is to derive the endogenous participation rate from the consideration of the costs of such participation and expected utility of such participation with respect to any particular insurance market. Since
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93
participation indexes competition, this chapter provides a model for endogenously determined competition.
Uncertainty arises from ignorance - ‘Knightian’ uncertainty. Thus, under uncertainty agents invest in both information and insurance to maximise expected utility of state contingent consumption. The state space when no uncertainty has been resolved is the set of strictly positive integers. Investment in information reduces the dimension of uncertainty or the number of possible states of nature facing an individual and each agent plans for contingent consumption with respect to this reduced number of possible states of nature. While considering investment in information the maximum expected utility derivable from insuring against this possible set of states and the cost of learning which reduces the size of uncertainty to this set of states is taken into account. Since insurance is a contingent commodity, level of learning determines the number of state-contingent commodities or the various types of insurance agents choose to buy. In a two period model heterogeneity in initial wealth levels and complete ordering of agents in terms of intertemoral wealth levels may cause difference in the demand or supply of various agents with respect to the level of learning and hence access to the different types of insurance, which determines participation in the various insurance markets, hence causing various degrees of competition with respect to various types of insurance, and to failure of insurance markets due to deficient demand or supply.
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Financial Economics We do not model the insurance firm in this paper, our model is one
of pure exchange. While we do not provide any policy implications, we discuss some policy issues in the conclusion. Section 2 sets up the two period model, Section 3 provides the characterisation in the logarithmic utility case, Section 4 concludes. 2. The Model : There are finite number of agents N > 1 in this economy
(1)
Agents plan over two periods date 0 and 1. There is no uncertainty at date 0 (the first period), all the uncertainty is at date 1. The uncertainty is with respect to the state of the world. 2.1. Investment in information : Each agent i’s initial belief at date 0 regarding the states of the world (which are ordered) that might realise at date t=1 is given by s = 1, 2, , ∞, s = 1 is the true state at each date, but this fact is not known to any agent. Learning partitions the initial state-space into finer partitions. If the agent i invest li (in physical units of the only consumption good) in learning (which we shall use interchangeably with information processing) then his updated state partition ΩPi becomes 0
0
ΩPi = ΩPi U (ΩPi )c 0
given ΩPi = {1, 2, , si } 0
(ΩPi )c = {si + 1, si + 2, , ∞}
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95 0
such that P { Some sΩPi occurring at t = 1} =1 and the agent assigns uniform probability weight to each state within ΩPi
0
(2) where si the index of residual uncertainty is given by :
si = L(li )
(3)
L(.) is the private learning technology available to agent i. The learning technology works as follows. It uses the consumption good as input and produces a forecast of the possible state-space at date 1. In fact it partitions the initial infinite state-space to a finite set, which contains the true state, and the residual infinite set. Each agent i knows 0
that the first set (Ωpi in (2) ) contains the true state but does not know which one it is. They therefore attach probability 1 to the first set and probability zero to the other. These sets are as given in (2). The learning technology has the following properties :
L(o) = ∞, min{x : L(x) = 1} = ∞
(3a)
L(.)I+ for li > 0
(3b)
∀li0 , li > 0 s.t. li > li0 , L(li0 ) ≥ L(li )
(3c)
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Financial Economics
and, ∀li0 > 0&L(li ) ≥ 3∃li0 : ∞ > li0 > li and, L(li0 ) = L(li ) − 1
(3d)
I+ refers to the set of positive integers. Assumption (3b) is made so as to keep the choice model in discrete state space and satisfy the convention of numbering states. Assumption (3a) implies the s=1 being the true state it is never possible to have perfect knowledge a priori with finite wealth. Assumption (3c) implies that an increase in learning reduces the size of the state space in steps and (3d) implies that the exogenous learning technology which agent i uses to learn about the possibility of various states occurring is a stepwise linear function of the Leontief type. With the investment in information li > 0 the updated 0
state space Ωpi is finite and hence, the posterior probability weights 0
πi (s) - with respect to each states within Ωpi becomes, πi (s) = P { State 0
s occurs at t = 1 for sΩPi } = 1 >0 si
(4)
Notice that from (2) the effect of investment in learning by agent i i.e. li > 0 is that the probability of occurrence of the residual set of states in zero i.e. P { s{si , si + 1, , ∞}occurs at t = 1} = 0 The optimal investment is information is defined as follows. Definition 1 : Let Vi (Li ((p(s))) be the maximum expected utility
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97
derivable from consumption to agent i, where Li ((p(s))) is a feasible choice of horizon at any price vector (p(s)). L∗i ((p(s))) is an optimal choice of information by agent i at prices (p(s)) iff,
Vi (L∗i (p(s))) ≥ Vi (Li (p(s))), ∀ such Li ((p(s))) the choice model which gives rise to this value is described later.1 2.2 Contingent Consumption : There is one perishable good in the economy which can be used for consumption or for information processing. Information processing is private. The contingent consumption of the good by agent i in state s at date t is denoted by csi (t). The price of the good at date 0 is normalised to 1 while its state contingent price in state s at t = 1 is given by p(s) ≥ 0. Each agent’s decision at date t = 0 involves allocating his resources to consumption at t = 0, to learning with respect to t = 1 and purchasing contingent consumption for t = 1 with respect to the updated 0
state space Ωpi . Agent i’s state dependent utility function with respect to state s at any date is given by the function usi (.) for s{1, 2, , si }. Each usi (.) has the following properties : 1
This notion of investment in information can be extended to investment in
health care or education where the future distribution of income is determined by the present investment in health or education.
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Financial Economics usi (csi (.)) is continuous over R+ and usi (0) ≤ 0 ∀ i, s
(5a)
δusi (.)/δ arg > 0
(5b)
δ 2 usi (.)/δ arg 2 < 0
(5c)
Agent i’s rate of time discount is given by δ where 0 < δ < 1. Since the state at date 0 when there is no uncertainty is 1 the utility function with respect to date 0 is u1i (.). Since agents assign o probabilities to each of the states {s : s > si }, the expected discounted utility with respect to t = 1 is given by si X
δπi (s)usi (csi (1))
s=1
where πi (s) is the updated probability beliefs as given by (4). The price of the good at date 0 is normalised to 1. Therefore, the cost of consumption at date 0 is given by c1i (0), where c1i (0) is i’s consumption at date 0, and the cost of learning is given by li . The price of the consumption good in state s at date 1 is given by p(s), and hence the total cost of consumption with respect to date 1 is given by si X
p(s)c1i (1)
s=1
The level of wealth at date o for agent i is given by wi0,1 .
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99
The wealth process at t = 1 is {wi1,s } where s is a state and hence the value of total wealth within the decision horizon at t = 1 is given by si X
p(s)wi1,s
s=1
All wealth levels for any agent are bounded (5d) Since, as has already been described, each agent i treats all states s > si as having probability 0, hence he does not take these wealth realisations into consideration in his optimising problem. Therefore, the budget constraint of agent i is given by, c1i (0)
+ li +
si X
p(s)(csi (1) − wi1,s ) − wi0,1 = o
s=1
Agent i’s learning-investment decision at date 0, therefore, is :
max
{(ci ≥0),li ≥0}
u1i (c1i (0)
s.t.c1i (0) + li +
+
si X
δπi (s)usi (csi (1))
s=1 si X
p(s)(csi (1) − wi1,s ) − wi0,1 = 0
s=1
si = L(li ) πi (s) =
1 si
(6)
where, (ci ≥ 0) represents the set {c1i (0) ≥ 0, {csi (1) ≥ 0}}, c1i (0), csi (1) represents his consumption at date 0 and date 1 state s respectively,
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wi1,s is his state contingent endowment is state s, si is given by (3), and πi (s) is given by (4). 2.3 Endogenous competition and market failure in the insurance market : Insurance is modeled as Arrow contingent claims The contingent consumption by agent i of the only good at date 1 state s is denoted by csi (1, (p(s)), where (p(s)) is the price vector. The imperfectly competitive insurance market is characterised as follows : Insurance a is a contract which pays off 1 unit of the consumption good is state a and 0 units in all other states. Thus, if the agent wishes to transfer q units of consumption from state i to state j then his supply of insurance i is q and demand for insurance j is q. The supply and demand at t = 0 for insurance paying off in any state η at t = 1 at any set of prices (p(s)) arises in the following way: if, cηi (1; (p(s)) − wi1,η < 0 for some i there is positive supply of insurance η by agent i If, cηi (1; (P (s)) − wi1,η > 0 for some i there is positive demand for insurance η by agent i I shall say that at t = 0 supply or demand for insurance η by any
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agent i 6= j does not exist (to be contrasted with supply or demand for insurance η is zero) iff cηi (1; (p(s)) is undefined for all agents i 6= j
(7b)
I shall say that cηi (1; p(s)) is undefined for agent i iff η > s∗i (p(s))
(7c)
where s∗i (p(s)) is the optimal choice of learning by agent i at prices p(s)) (see Definition (1). I shall say that agent j faces an imperfect insurance market at t = 0 at prices (p(s)) if for some s there is positive supply or demand of insurance s by agent j but supply or demand of insurance s by any agent i 6= j, does not exist.2
(7d)
I shall say that the insurance market at t = 0 is imperfect at prices (p(s)) if there exists at least one agent j such that j faces an imperfect insurance market at t = 0 at prices (p(s))
(8)
3. The Logarithmic utility case : 2
The nature of the demand curve for insurance might appear to have a kink as in
Oligopoly theory. However, this resemblance is only apparent as the discontinuity in demand for State contingent consumption is in the dimension of the non-autarkic demand vector unlike in kinked demand analysis.
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Agents are heterogeneous only with respect to wealth levels. State 1 is realised at date 0 before agents make decisions. The utility profiles are given by,
usi (.) = ln(.),
s≥1
The individual optimising problem takes the form :
max
{li ≥0,(ci ≥0)}
ln(c1i (0)
s.t.c1i (0) + li +
+
si X
δπi (s) ln(csi (1))
s=1 si X
p(s)(csi (1) − wi1,s ) − wi0,1 = 0
s=1
si = L(li ) πi (s) =
1 si
(60 )
Where, c1i (0) is consumption at date 0 when s = 1 is realised and the wealth that is realised is therefore w10,1 and csi (1) is consumption at date 1 in state s. Definition 2 The function f (x, α) : xX ∈ Rn and α is a parameter vector is said to be subdifferentiable in α, if it is convex in α, and if for each α1 ∃ a vector, say fα (x, α1 ), s.t. the inequality f (x, α0 ) − f (x, α1 ) ≥ fα (x, α1 )(α0 − α1 )∀x ∈ X holds ∀α0 . If f is concave, the inequality is reversed. (Anderson & Takayama (1979)).
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103
In deriving our results we shall make use of this proposition which is adapted from Anderson and Takayama (1979), (Proposition 1 case (v)) Proposition Consider the problem (M )Maxx f (x, α) s.t.gj (x, β, γ) ≡ βj − hj (x, γ) ≥ 0, j = 1, 2, , m and xX ⊂ Rn , where f and gj s are real value functions and where α, β and γ signify vectors of shift parameters. Let, A be the set of (α, β, γ) in which the solution of (M) exists. We assume A(&X) is non-empty. Given (α, β, γ) let x(α, β, γ) denote the solution of (M). Corresponding to a shift of parameters we define the function xj by
xj = x(αj , β j , γ j ) Suppose, f is convex and subdifferentiable in α with fα denoting the subgradient vector with respect to α. Then, if
g(x1 , β, γ 0 ) ≥ (γ 1 − r0 ).x1
(i)
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Financial Economics g(x0 , β, γ 1 ) ≥ (γ 1 − γ 0 ).x0
(ii)
fα (x0 , α0 )4α1 + λ11 4γ 1 x0 ≤ F (α1 , β 1 , γ 1 ) − F (α0 , β 0 , γ 0 ) ≤ fα (x1 , α1 )4α1 − λ01 4α1 − λ01 4γ 1 x1 0 ≥ 4α1 {−fα (x1 , α1 ) + fα (x0 , α0 )} + 4γ1 (λ01 x1 + λ11 x0 )
(I) (II)
if g(x2 , β, γ 0 ) ≥ (γ 0 − γ 2 )x2
(iii)
g(x0 , β, γ 2 ) ≥ (γ 0 − γ 2 )x0
(iv)
fα (x0 , α0 )4α2 + λ02 4γ2 x2 ≤ F (α2 , β 2 , γ 2 ) − F (α0 , β 0 , γ 0 ) ≤ fα (x2 , α2 )4α2 − λ12 4γ2 x0 0 ≥ {−fα (x2 , α2 ) + fα (x0 , α0 )}4α2 + 4γ2 (λ02 x2 + λ12 x0 )
(III) (IV )
if, g(x1 , β 0 , γ 0 ) ≥ (β 0 − β 1 ) + (γ 1 − γ 0 )x1
(v)
g(x0 , β 1 , γ 1 ) ≥ (β 1 − β 0 ) + (γ 1 − γ 0 )x0
(vi)
fα (x0 , α0 )4α1 + λ13 4β1 + λ13 4γ1 x0 ≤ F (α1 , β 1 , γ 1 ) − F (α0 , β 0 , γ 0 ) ≤ fα (x1 , α1 )4α1 + λ03 4β1 − λ03 4γ1 x1
(V )
0
0 ≥ 4α1 {−fα (x1 , α1 ) + fα (x0 , α0 )} + 4β1 (λ13 − λ03 ) +4γ1 (λ03 x1 + λ13 x0 )
(V I)
where, F (αi , β i , γ i ) = maxxi f (xi ) s.t. the budget constraint where m αi , β i , γ i are the parameters, ∀i, j, λij R+ and fα is given in definition
(2). (where whether xj represents the function x(αj , β j , γ j ) or the variable xj is clear from the context). Proof : See appendix
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105
The following Lemmas are necessary in order to apply the above Proposition in the proof of the theorem. Lemma 1 : Suppose conditions 3(a) - 3(d) hold. Then, L−1 (Li ) = min{xR+ : L(x) = Li } is bounded, monotonically decreasing and continuous for Li {I+ /1}U {∞} Proof : See Appendix. 3.1 Value of information : Investment in information improves the accuracy of the forecast by reducing the dimension of uncertainty i.e. the number of states to which the agent attaches positive probability. This reduces the dimension of the second period wealth vector considered in the budget constraint. At the same time it increases the probability weight on the utility derived from each state contingent consumption. Thus, the reduction in the dimensionality of the demand vector for state contingent consumption has to be compared with the shift in the demand levels with respect to each of the state contingent commodities within the possible set of states for valuing various levels of investment in information. Lemma 2 : The value of information V (L−1 (Li ); {p(s)}, wi0,1 ) at date 0 to agent i is given by,
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Financial Economics
V (L−1 (Li ); {p(s)}, wi0,1 ) = ln(χ(L−1 (li ))) +
Li X
δ(Li )−1 ln((p(s)Li )−1 χ(L−1 (Li )))
s=1
where, {wi0,1 − L−1 (Li ) + χ(L (Li )) = 2 −1
PLi
p(s)w1,s }
Li = L(li ) L−1 (Li ) = min{xR+ : L(x) = Li } Proof : See Appendix. 3.2 Solution to the continuous extension of the problem : The function u({csi }, L−1 (θi )), θi I+ is discontinuous with respect to li whose range is R+ . Hence, we have to transform the utility function into a continuous extension with respect to li . Consider the following extension of the function u({csi }, L−1 (θi )) denoted by u˜({csi }, li ) as follows : u˜({csi }, li ) = u({csi }, li )if, li = L−1 (θi )for some θi I+ = τ u({csi }, li−1 ) + (1 − τ )u({csi }, li+1 ), if li 6= L−1 (θi )for any θi I+ . where, li = τ li−1 + (1 − τ )li+1 for some τ : 0 < τ < 1 and, li−1 = L−1 (θi − 1)for some θi I+
Chapter 5
107 and li+1 = L−1 (θi )
The maximisation problem (60 ) can be restated using u˜() as the utility function as follows :
max
{(ci ≥0),li ≥0}
s.t. wi0,1 +
Li X
u˜({csi }, li , πi )
p(s)(wi1,s − csi (1)) − c1i (0) − li ≥ 0 (600 )
πi = 1/L(li ) where, u˜({csi }, li , πi ) = u({csi }, li , πi )if u˜({csi }, li ) = u({csi }, li ) = τ u({csi }, li−1 , πi ) + (1 − τ )u({csi }, li+1 , πi ) if u˜({csi }, li ) = τ u({csi }, li−1 ) + (1 − τ )u({csi }, li+1 ) and, u({csi }, L−1 (θi ), πi ) = ln c1i (0) +
θi 1 X ln csi (1) L(li ) s=1
we shall establish the equivalence between (60 ) and (600 ) in the following lemma. Lemma 3 : ({ci ∗}, li∗ ) is a local optimum of (600 ) ⇔ ({c∗i }, li∗ ) is a local optimum of (60 ). Note (2) : In proving this lemma we have used concavity of u({csi }, li ) with respect to {csi } only. Note (3) : If u({csi }, L−1 (θi )) is concave (convex) with respect to a certain range of θi I+ , u({csi }, li ) is concave (convex) with respect to to li lying within the range of L−1 (θi ) for such θi . Hence we shall
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only concern ourselves with problem (60 ) where, the choice set of li is restricted to L−1 (θi ), θi I+ . 3.3. Existence of solution to individual’s optimising problem As discussed in sec (3.2) a change in the investment in information by any agent alters both the present value of wealth as also the probability weights on the set of contingent consumption, the dual impact of which given a particular learning technology and the probability updating rule (4) depends on the shape of the state contingent utility functions, the distribution of state contingent wealth over the State-space as also the particular price vector under consideration. A sufficient set of conditions which ensures as interior solution to the above investment problem is derived in the next two lemmas. Lemma 4 : If conditions 3(a) - 3(d) hold with respect to L(.) and if (i)wi1,s = w1,s > 0 f ors ≤ si < ∞ = 0 for s > si ,
∀i
where, {w1,s }∞ s=1 is a sequence of strictly positive real numbers and (iii) ∞ > p(s) > 0∀s Then, the subgradient of u({csi }, li ) denoted by ui ({csi }, li ) and given by
ul ({csi }, li )
L(li )−1 i) X X δ δ l(l 4L 1 ln ci (s) − ln c1i (s)] ] = [ L(li ) − 1 L(li ) 4li
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exists. Where, li = L−1 (θi ) for some θi I+ and 4L = θi − (θi − 1), 4li = L−1 (θi ) − L−1 (θi − 1) Proof : see Appendix. Note (4) : The subgradient ul ({csi }, li ) defined in Lemma 4 is the maximum of all the subgradients of u({csi },
) at li , since u({csi },
)
is convex in li (hence concave in Li ) by the conditions of the Lemma (see Rockfeller (1970) pg. 229) Lemma 5: If ∃ an integer L0i ≥ 2 s.t. L0i −1
Y
and
0
(p(s))δ > p(li0 )δ(Li −1)
0
Li Y
0
(p(s))δ ≤ p(L0i + 1)δLi
then, L0i is an interior optimum for agent i if i. conditions 3(a) - 3(b) hold w.r.t. (L.) ii. wi1,s = w1,s
for s ≤ si < ∞ =0
for s > si , ∀i
where {w1,s }∞ s=1 is a sequence of finite strictly positive real numbers.
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Financial Economics iii. ∞ > p(s) > 0∀s iv. wi0,1 + v.
PL0 −1 i
PL0
p(s)wi1,s − L−1 (L0i − 1) > 0 and
i p(s)w 1,s wi0,1 + i 2(L0i +1)
≥ L−1 (L0i ) −
L0i L−1 (L0i 2(L0i +1)
+ 1)
Proof see Appendix Note (5) : The inequality (C*) is the proof is redundant for the purpose of this Lemma as we need to compare the value w.r.t. L0i with L0i − 1 and L0i + 1 for local optimality. Note (6) : Conditions (i), (iv) and (v) are restrictions on the shape of the learning technology. Conditions (ii) requires that given any common probability distribution over the states the expected wealth level of any agent at t=1 is bounded. Under the conditions which guarantee the existence of an interior optimal choice of investment is information for all agents we discuss market failure. 3.4 Endogenous Competition and Market failure : In order to keep the ordering of agents in terms of wealth levels complete we impose the following condition on the two period wealth levels : if, wi0,1 > wj0,1 for any i 6= j then si > sj , ∀j ≤ N Theorem :
(11)
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111
Even if the conditions of Lemma 5 are satisfied for all agents i ≤ N, the insurance market is imperfect at t = 0, if | wj0,1 − wi0,1 |> 1
∀j 6= i,
min si ≤ N i
and (11) holds. Proof: See Appendix. Note (7) : the particular notion of market clearing we have in mind when we defined imperfect insurance markets is one with restricted participation. i.e. ∗
(wi0 − c1i (0, p(s))) − li∗
X iN1 (0,1)
=
X
0,1 ∗ (c1∗ i (0, (p(s))) − wi ) + li
iN2 (0,1)
and for any s ≤ maxi L0i (p(s))) for date 1 :
X
∗ (wi1,s − cs∗ i (1, p(s))) − li =
iN1 (1,s)
X
1,s (cs∗ i (1, (p(s))) − wi )
iN2 (1,s)
where, N1 (t, s) and N2 (t, s) are the set of agents participating with respect to market s at date t, with positive supply of insurance s and positive demand for insurance s, respectively, at prices {p(s)} > 0 (see 7(a)). Here, we have ignored {(p(s)} in the argument for notational simplification.
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Note (8) : It is not necessary for our results to hold for there to be only 1 agent with wealth level of class i for each i. Our results hold with more than 1 agent in each class as long as there are more than 1 class, but the number of classes is finite.
4. Conclusion :
We have provided a model whereby the degree of competition in insurance markets, is endogenously determined. Competition ensures full employment of all resources and therefore efficiency. This endogenous derivation of the degree of competition is necessary for at least two reasons. Firstly, while in existing oligopoly theory (Varian (1992) Chap. 16) competition is treated as exogenous, either of the perfectly competitive variety or of the monopolistic variety, a variable which would depend both on the cost of entry and the benefits of entry which each agent expects to face in entering such markets and hence the consequential number of agents who decide to participate in the various markets. Since the decision to participate is made by rational agents from the expected costs and benefits of participation it seems natural to think that the participation in markets and hence the corresponding nature of competition will be derived endogenously from the model rather than being exogenously specified. We have provided one such model where agent behave competitively. Secondly, market failure at any finite positive price can arise from non-participation of agents in a market such that there is only demand for insurance but not supply
Chapter 5
113
or vice versa which requires the analysis of the degree of competition. In this chapter we have formalised a model of financial markets where agents use information and insurance as two tools for smoothing risk, and there by the number of participants varies across markets for endogenous reasons. This analysis has brought out a vital feature of financial (insurance) markets which is that if initial wealth distribution is asymmetric across agents and the ordering in terms of wealth levels (actual or expected as the case may be) is preserved over time then insurance markets may fail in the sense that not all agents can participate in all markets leading to deficient demand or supply. Appendix Proof : (of proposition) The saddle-point condition for the constrained maximisation problem (M) for (α0 , β 0 , γ 0 ) is written as : (SP) ∃x0 X and λ0 Rm , λ0 ≥ 0 s.t. φ(x, λ0 ; α0 , β,0 , γ 0 ) ≤ φ(x0 , λ0 ; α0 , β 0 , γ 0 ) ≤ φ(x0 , λ, α0 , β 0 , γ 0 ) ∀xX and λRm with λ ≥ 0, where φ(x, λ; α, β, γ) ≡ f (x, α) + λ.g(x, β, γ) As is well known if (SP) holds then x0 is a solution of (M) automatically.
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From the above saddle-point condition, we obtain f (x0 , α0 ) − f (x, α0 ) ≥ λ0 .g(x, β 0 , γ 0 ), ∀xX
(a)
Step I : Now, if f (x, α0 ) − f (x, α1 ) = fα (x, α1 )(α0 − α1 )
(b)
Then, adding (a) and (b) we have f (x0 , α0 ) − f (x1 , α1 ) ≥ fα (x1 , α1 )(α0 − α1 ) + λ01 g(x1 , β 0 , γ 0 )
(c)
Also, f (x1 , α1 ) − f (x0 , α0 ) ≥ fα (x0 , α0 )(α1 − α0 ) + λ11 g(x0 , β 1 , γ 1 )
(d)
Adding (c) and (d), we have
0 ≥ (α1 − α0 ){−fα (x1 , α1 ) + fα (x0 , α0 )} + λ01 g(x1 , β 0 , γ 0 ) +λ11 g(x0 , β 1 , γ 1 ) Conditions (i), (ii) and (c) and (d) imply,
f (x0 , α0 ) − f (x1 , α1 ) ≥ −fα (x1 , α1 )(α1 − α0 ) + λ01 (γ 1 − γ 0 )x1 f (x1 , α1 ) − f (x0 , α0 ) ≥ fα (x0 , α0 )(α1 − α0 ) + λ11 (γ 1 − γ 0 )x0 which implies,
(e)
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115
fα (x0 , α0 )4α1 + λ11 4γ1 x0 ≤ F (α1 , β 1 , γ 1 ) − F (α0 , β 0 , γ 0 ) ≤ fα (x1 , α1 )4α1 − λ01 4γ1 x1
(1)
(where4α1 = α1 − α0 , 4γ1 = γ 1 − γ 0 ) Conditions (i), (ii) and (e) imply, 0 ≥ 4α1 {−fα (x1 , α1 ) + fα (x0 , α0 )} + λ01 4γ1 x1 + λ1 4γ1 x0 ≥ 4α1 {−fα (x1 , α1 ) + fα (x0 , α0 )} + 4γ1 (λ01 x1 + λ11 x0 )
(2)
Step II : If f (x, α0 ) − f (x, α2 ) = fα (x, α2 )(α0 − α2 )
(b0 )
Then, adding (a) and (b0 ) we have f (x0 , α0 ) − f (x2 , α2 ) ≥ fα (x2 , α2 )(α0 − α2 ) + λ02 g(x2 , β, γ 0 )
(c0 )
also, f (x2 , α2 ) − f (x0 , α0 ) ≥ fα (x0 , α0 )(α2 − α0 ) + λ12 g(x0 , β, γ 2 )
(d0 )
Adding (c0 ) and (d0 ), we have
0 ≥ (α2 − α0 ){−fα (x2 , α2 ) + fα (x0 , α0 )} + λ02 g(x2 , β, γ 0 ) +λ21 g(x0 , β, γ 2 )
(e0 )
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Financial Economics
Conditions (iii), (iv) and (c0 ) and (d0 ) imply, f (x0 , α0 ) − f (x2 , α2 ) ≥ fα (x2 , α2 )(x0 − α2 ) + λ02 (γ 0 − γ 2 )x2 f (x2 , α2 ) − f (x0 , α0 ) ≥ fα (x0 , α0 )(α2 − α0 ) + λ12 (γ 0 − γ 2 )x0 which implies,
fα (x0 , α0 )4α2 + λ02 4γ2 x2 ≤ F (α2 , β 2 , γ 2 ) − F (α0 , β 0 , γ 0 ) ≤ fα (x2 , α2 )4α2 − λ21 4γ2 x0
(3)
(where,4α2 = α2 − α0 , 4γ2 = γ 2 − γ 0 ) Conditions (iii), (iv) and (e0 ) imply,
0 ≥ 4α2 {−fα (x2 , α2 ) + fα (x0 , α0 )} + λ02 4γ2 x2 + γ12 4γ2 x0 ≥ 4α2 {−fα (x2 , α2 ) + fα (x0 , α0 )} + 4γ2 (λ02 x2 + λ21 x0 )
(4)
Step III : If, f (x, α0 ) − f (x, α1 ) = fα (x, α1 )(α0 − α1 )
(b)
Then adding (a) and (b) 0
0
0
f (x0 , α0 ) − f (x1 , α1 ) ≥ fα (x1 , α1 )(α0 − α1 ) + λ03 g(x1 , β 0 , γ 0 )
(c00 )
Also, 0
0
f (x1 , α1 ) − f (x0 , α0 ) ≥ fα (x1 , α1 )(α1 , α0 ) + λ13 g(x0 , β 1 , γ 1 )
(d00 )
Chapter 5
117
Adding (c00 ) and (d00 ) we have,
0
0 ≥ (α1 − α0 ){−fα (x1 , α1 ) + fα (x0 , α0 ) + λ03 g(x1 , β 0 , γ 0 ) +λ13 g(x0 , β 1 , γ 1 )
(c00 )
Conditions (v), (vi), (c00 ) and (d00 ) imply, 0
0
f (x0 , α0 ) − f (x1 , α1 ) ≥ fα (x1 , α1 )(α0 − α1 ) +λ03 (β 0 − β 1 ) + λ03 (γ 1 − γ 0 )x1
0
0
f (x1 , α1 ) − f (x0 , α0 ) ≥ fα (x0 , α0 )(α1 − α0 ) +λ13 (β 1 − β 0 ) + λ13 (γ 1 − γ 0 )x0 which implies,3
fα (x0 , α0 )4)α1 + λ13 4β1 + λ13 4γ1 x0 ≤ F (α1 , β 1 , γ 1 ) − F (α0 , β 0 , γ 0 ) 0
≤ fα (x1 , α1 )4α1 + λ03 4β1 − λ03 4γ1 x1
3
0
The general iterative procedure discussed in this proof can be extended to an
algorithm for computing equilibria with integer constraints of course in such a case the model has to be closed by suitable redistribution. For a generalisation see chapter 6.
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Financial Economics
0
≤ fα (x1 , α1 )4α1 + λ03 4β1 − λ03 4γ1 x1
0
(5)
Conditions (v), (vi) and (e00 ) imply, 0
0 ≥ 4α1 {−fα (x1 , α1 ) + f (x0 , α0 )} 0
−λ03 4β1 + λ03 4γ1 x1 + λ13 4β1 + λ13 4γ1 x0 0
≥ 4α1 {−fα (x1 , α1 ) + fα (x0 , α0 )} 0
+4β1 (λ13 − λ03 ) + 4γ 1 (λ03 x1 + λ13 x0 ) (where 4β1 = β1 − β0 )
(6) Proved
Proof : (of Lemma 1)
1 L−1 (Li ) : {I+ /I}U {∞} → R+
by (3b)
Also, L−1 (.)exists ∀Li I+ by (3a) and (3d) L−1 (.) is monotonic : by (3c) if l0 i > li then L(l0 1 ) ≤ L(li ) 00
now, suppose Li 000 > Li > L0 i and, min{x : L(x) = Li 00 } < min{x : L(x) = Li 0 } but, min{x : L(x) = Li 000 } < min{x : L(x) = Li 00 } Also, 0
Suppose, min{x : L(x) = Li } = x
0
Chapter 5
119
00
00
000
000
min{x : L(x) = Li } = x
min{x : L(x) = Li } = x
x00 < x0 Then, x000 > x00 · · ·
Either x000 < x0 but x000 > x00 or x000 = x0 or x000 > x0 000
0
· 00 · · Li
> Li
000
00
now, if x000 < x0 , then by 3(c) Li ≥ Li but Li ≤ Li - a contradiction
0
0
000
If, x000 = x0 , then Li = Li - again a contradiction · 000 · · Li
0
00
000
0
> L i > L i ⇒ Li > L i 000
0
if, x000 > x0 , then by (3c) Li ≤ Li , again a contradiction. Hence, L−1 is monotonically decreasing. L−1 (.) is bounded : · −1 · · L (.)
is monotonically decreasing, and I+ is ordered therefore
the maximum and minimum are attained at the minimum and maximum of I+ Now, By (3a), L−1 (∞) = 0 and by (3d), L−1 (2) is bounded.
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Financial Economics
Hence, L−1 (.) is bounded over I+ − {1} L−1 is continuous : L(.) : R → {I+ /I}U (∞} by 3(a) and 3(b) Now, both R and I+ U {∞} are metric spaces. · · ·
If L(.) is a continuous 1-1 mapping, then it can be shown that
L−1 (.) is continuous. (Theorem 4.17 pg. 90 Rudin (1976)). Now, pick any point x in R Pick any > 0 Suppose, 0 < < 1 Then, by (3c) ∃∞ > δ > 0
s.t. ∀yR+ :
| y − x |< δ, | L(y) − L(x) |< Suppose, we pick any ∞ > ≥ 1 then by 3(c) and 3(d) ∃∞ > δ > 0 s.t. ∀yR+ :
| y − x |< δ, | L(y) − L(x) |< This holds ∀xR+ (by 3(b)).
Chapter 5
121
Hence, by definition L(li ) : R+ → {I+ /1}U {∞} is continuous, L(L−1 (.)) is a 1-1 mapping. Hence L−1 (.) is continuous. Proved. Proof (or Lemma 2) : V (L−1 (Li ), {p(s)}, wi0,1 ) is the indirect utility for agent i given the choice of investment in information L−1 (Li ). Now, given a particular choice of Li and therefore L−1 (Li ) (see 3(a)) the value of optimal consumption at the given price vector {p(s))} is given by,
cs∗ i (1, {p(s)}) =
χ(L−1 (Li )) p(s)Li
−1 c1∗ i (0, {p(s)}) = χ(L (Li ))
Replacing in the utility function gives the result
Proved
Proof : (of Lemma 3) First we shall prove that if ({c∗i }, li∗ ) is a local maximum of u˜({c∗i }, .) then it is necessary that li∗ = L−1 (θi∗ ) for some θi∗ I+ . −1 ) + (1 − Suppose, li∗ 6= L−1 (θi ) for any θi I+ Then, li∗ = τ ∗ (li∗
τ ∗)(li∗+1 ) for some θi∗ I+ , 0 < τ ∗ < 1. Hence, from (9),
u˜({ci∗ }, li∗ ) = τ ∗ u({c∗i }, L−1 (θi∗ − 1)) + (1 − τ ∗ )u({c∗i }, L−1 (θi∗ )) Now, either u({c∗i }, L−1 (θi∗ )) ≥ u({c∗i }, L−1 (θi∗ − 1))
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Financial Economics
Hence, either u({c∗i }, L−1 (θi∗ )) ≥ u˜(c∗i }, li∗ ) or u({c∗i }, L−1 (θi∗ − 1)) ≥ u˜({c∗i }, li∗ ) In every case the maximum is attained at li∗ = L−1 (θi∗ ) for some θi∗ I+ . Hence, it is not possible that li∗ 6= L−1 (θi ) for any θi I+ . 0
Now, from the budget constraint in (600 ) if li0 = L−1 (θi ) and , li1 < li : L(li∗ ) = θi , then li∗ cannot be an optimal choice of li due to the fact that u(., li ) and hence u˜(., li ) is concave in {csi }. Hence, the optimal choice of li can only be such that li∗ = L−1 (θi ) for some θi ∈ I+ Now, since u˜({csi }, li , πi ) = u({csi }, li , πi ) ∀li
li = L−1 (θi ) for θi ∈ I+ . Hence, if li∗ = Arg maxli u({c∗i }, li ) s.t. the b.c. in (600 ), Then, li∗ = Arg maxli u({c∗i }, L−1 (θi ∗)) s.t. the b.c. in (60 ), and li∗ = L−1 (θi ∗), θi∗ I+ To prove the converse Suppose, ({c∗i }, L−1 (θi∗ )) solves (60 ) But, ({c∗i }, L−1 (θi∗ )) does not solve (600 ) · · ci }, ˜li ), ˜li · ∃({˜
6= L−1 (θi∗ )
Chapter 5
123
s.t. u({˜ ci }, ˜li , πi (˜li )) >
u({c∗i }, li∗ , πi (li∗ )) and ˜li is feasible where, πi (˜li ) =
1 L(˜li )
πi (li∗ ) =
1 L(li∗ )
Now by 3(b) L(˜li ) = θ˜i I+ Now L(˜li ) 6= θi∗ , for by the concavity of u(., .) with respect to {ci }, if L(˜li ) = θi∗ , then ˜li > L−1 (θi∗ ), from the budget constraint this implies that u({c∗i }, li∗ , πi (li∗ )) cannot be an optimum of (60 ). L(˜li ) = θ˜i < θi∗ by3(c) But, u˜({˜ ci }, L−1 (θ˜i ), πi (˜li )) > u˜({c∗i }, L−1 (θi∗ ), πi (li∗ ))
⇒ u({˜ ci }, L−1 (θ˜i ), πi (˜li )) > u(c∗i }, L−1 (θi∗ ), πi (li∗ )) which contradicts the fact that (c{c∗i }, L−1 (θi∗ )) solve (60 ) Hence the Lemma. Proved
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Financial Economics
Proof : (of Lemma 4) We pick any set of prices {p(s)} > 0. For notational simplification we drop prices from the argument. From definition 2 of subgradient, we have to first check whether u({csi }, li ) is convex or concave. Now, u({csi }, li ) is convex in li if for any ∞ > l0 i > li
u({csi }, αli + (1 − α)li0 )) ≤ αu({csi }, li ) + (1 − α)u({csi }, li0 ) where, L(li ) = Li , (the inequality is reversed if u(., .) is concave) and 0 < α < 1. Let, li0 be close enough s.t. L(li0 ) = L(li ) or L(li0 ) = L(li )−1. If li0 is greater the convexity or concavity can be derived by induction from the fact that L−1 (.) is monotonic and continuous (Lemma 1 conds. 3(a) - 3(d) hold).
0 )=L(l
Case a If, L(li
i)
0 )=u({cs },l ) i i
then, u({csi }, li
u({csi }, αli +(1−α)li0 ) = u({csi }, li ) (L−1 (.)is
continuous from Lemma 1)
u({csi }, li ) is both concave and convex. Case b: If, L(li0 ) = L(li ) − 1
Chapter 5
125
Then, u({csi }, αli )
+ (1 −
α)li0 )
−
ln c1i (0)
+
LX i −1 s=1
δ ln csi (1) Li − 1
now, 0)
αu({csi }, li ) + (1 − α)u({csi }, li = α{ln c1i (0) +
Li X δ
Li LX i −1 s=1
= ln c1i (0) + α
Li X δ
ln csi (1)} + (1 − α){ln c1i (0)+
δ ln csi (1) Li − 1
ln csi (1) + (1 − α)
Li
LX i −1
δ ln csi (1) Li − 1
u(., li ) is convex in li iff : ln c1i (0)
+α
Li X δ
Li
ln csi (1)
+ (1 − α)
LX i −1
LX i −1
δ ln csi (1) ≤ ln c1i (0)+ Li − 1
δ ln csi (1) Li − 1
u(., .) is convex in li iff : Li i −1 δ X δ LX ln csi (1) ≤ ln csi (1) Li Li − 1
or,
Li X
1
ln(csi (1)) Li ≤
LX i −1
1
ln(csi (1)) Li −1 [ δ > 0]
∞ > wi1,s > 0 f ors ≤ si < ∞ = 0 f or s > si then {csi (1)} is bounded ∀{p(s)} > 0
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Financial Economics
csi (1) is bounded below by 0 csi (1) is bounded above by (wi0,1 + P∞
s=1
p(s)wi1,s )/p(s) = Qn
csi (1)
then, cn+1 =
Qn+1
Denote, cn =
(wi0,1 +
PS i
s=1
p(s)wi1,s )
p(s)
(f rom(6))
csi (1)
if, {csi (1)} > 0, then
lim sup(cn )1/n ≤ lim sup
n→∞
⇒ n→∞ lim
n→∞
cn+1 cn
n Y
lim sup(cn+1 (1)) (csi (1))1/n ≤ n→∞ i
lim inf(cn+1 (1)) ≤ n→∞ lim inf i n→∞
n Y
(csi (1))1/n
[Theorem 3.37 Rudin (1976), pg.68] Now,
· · ·
the sequence {csi (1)}∞ s=1 is real and bounded (see above)
by Weirstrass Theorem (Rudin (1976), pg.68)) lim cn (1) n→∞ i exists. Hence, limn→∞ sup cn+1 (1) =limn→∞ inf cn+1 (1) i i = n→∞ lim cn+1 (1) i
Chapter 5
127 = lim sup
n Y
(csi (1))1/n
n→∞
= n→∞ lim inf = lim
n→∞
Thus, limn→∞
Qn
n Y
(csi (1))1/n
n Y
(csi (1))1/n
(csi (1))1/n exists in R
Hence, ∃ a sequence of (csi (1))s.t. n−1 Y
For such a sequence,
ln
(csi (1))1/n−1 ≥
· · ·
n−1 Y
n Y
(csi (1))1/n
ln is a monotonically increasing function.
(csi (1))1/n−1 ≥ ln
n Y
(csi (1))1/n
n X 1 n−1 1X s ⇒ ln(ci (1)) ≥ ln(csi (1)) n−1 n
Hence, u(., .) is convex in li for such a sequence of {csi (1)}. Now, Let us denote
ui ({csi }, li ) = [ln c1i (0) +
PLi
=[
δ Li
ln csi (1) − ln c1i (0) − Li − (Li − 1)
Li X δ
Li
ln csi (1) −
LX i −1
PLi −1
δ Li −1
ln csi (1)] 4Li 4li
δ 4Li ln csi (1)] Li − 1 4li
[This is one of the sub-gradients, see Note (4)]
ui (., .)(li1 − li0 ) = (
Li i −1 δ X δ LX ln csi (1) − ln csi (1))(−1) Li Li − 1
128
Financial Economics Let, li0 = L−1 (L0i ), li1 = L−1 (L0i − 1) Now, u({csi }, li0 ) = ln c11 (0) + 0 )=ln c1 (0)+
u({csi }, li
δ L0 −1 i
Li δ X ln csi (1) L0i
PLi −1
ln csi (1)
u({csi }, li1 ) − u({csi }, li0 ) 0
0
Li −1 Li X δ X δ ln csi (1) − 0 ln csi (1) ≥ 0 = 0 Li − 1 Li
u({csi }, .)is convex in li u({csi }, li1 ) − u({csi }, li0 ) = ul (., li0 )(li1 − li0 ) Similar arguments show,
u({csi }, li2 ) − u({csi }, li1 ) = ul (., li1 )(li2 − li1 ) Hence, u({csi }, li2 ) − u({csi }, li0 ) ≥ ul (., li0 )(li2 − li0 ) Hence, by definition 2 u({csi }, li ) is subdifferentiable in li and the subgradient is given by
ul ({csi }, li )
Li i −1 δ LX δ X 4Li s ln ci (1) − ln csi (1)] =[ Li − 1 Li 4li
Here, 4Li = Li − (Li − 1) and 4li = L−1 (Li ) − L−1 (Li − 1), where Li = L(li ) This holds for any prices ∞ > {p(s)} > 0.
Proved
Chapter 5
129
Proof : (of Lemma 5) Let us denote,
Φ(ci , λi , Li , wi0,1 , Li ) ≡ ln c1i (0) + +λi [wi0,1 +
Li X
Li X 1
Li
ln csi (1)
p(s)(wi1.s − csi (1)) − c1i (0) − L−1 (Li )]
where, φ(ci , λi , Li , wi0,1 , Li ) is as given in the proof of the proposition and the maximisation problem (M) corresponds to our (60 ) with Li held as a parameter, where ci is the vector (c1i (0), (csi (1)L1 i ) Now, if ∃ a c∗i , λ∗i at prices {p(s)}, such that Φ(ci , λ∗i , Li , wi0,1 , Li ) ≤ Φ(c∗i , λ∗i , Li , wi0,1 , Li ) ≤ [Φ(c∗i , λi , Li , wi0,1 , Li ) then, {c∗i } is necessarily a solution to (70 ) given Li , wi0,1 and {p(s)}. Now, for any λi ≥ 0, c∗i as derived in Lemma 2 satisfies, Φ(ci , λ∗i , Li , wi0,1 , Li ) ≤ Φ(c∗i λ∗i , Li , wi0,1 , Li ) ≤ Φ(c∗i , λi , Li , wi0,1 , Li )(SP ∗) [
· · ·
f (x, α) is concave in x and g(.) is linear in x-see foot note 3
of Anderson et.al.(1979)] Now, denote in terms of the Proposition :
130
Financial Economics 0
L0
0,1 0,1 0 0 s∗ 0 0 i as, x0i RLi +1 the vector (c1∗ i (0; Li , wi , Li ), (ci (1; Li , wi , Li ))s=1 ) 0
0,1 0,1 0 0 s∗ 0 0 as, x1i RLi the vector (c1∗ i (0; Li −1, wi , Li −1), (ci (1; Li −1, wi , Li − L0 −1
i 1))s=1 ) 0
0,1 0,1 0 0 s∗ 0 0 as, x2i RLi +2 the vector (c1∗ i (0; Li +1, wi , Li +1), (ci (1; Li +1, wi , Li + L0 +1
i 1))s=1 )
as, αi0 the scalar L0i as, αi1 the scalar L0i − 1 as, αi2 the scalar L0i + 1 as, βi the scalar wi0,1 as, γi0 the scalar L0i as, γi1 the scalar L0i − 1 as, γi2 the scalar L0i + 1 as, f (x, α) the maxima nd and as, g(x, β, γ) the budget constraint in (60 ). From Lemma 4,f (x, α0 ) − f (x, α1 ) = fα (x, α0 )(α0 − α1 ), by virtue of assumptions (i), (ii) and (iii). Hence, condition (b) in the proof of the proposition is satisfied. Also, by virtue of (SP*), condition (a) is satisfied. Also, it can be easily verified by writing down the budget constraint that conditions (i) and (ii) of the proposition are satisfied, if (iv) holds. Similarly, by virtue of assumptions (iv) and (v) it can be shown that conditions (iii) and(iv) of the proposition hold.
Chapter 5
131
equations (1) - (4) of the proposition hold. Now, x0i is the local optimal choice for the set of parameters (αi0 , βi , γi0 ) Iff, F (αi1 , βi , γi1 ) − F (αi0 , βi , γi0 ) ≤ 0(A) and F (αi2 , βi , γi2 ) − F (αi0 , βi , γi0 ) ≤ 0(B) [see defn(1)] which using the upper bounds of the differences in equations (1) and (3) can be written down as :
fα (x1i , αi1 )4αi,1 − λ0i 4γi,1 x1i ≤ 0(A0 ) fα (x2i , αi2 )4αi,2 − λ2i 4γi,2 x0i ≤ 0(B 0 ) and from equations (2) and (4) :
0 ≥ 4αi,1 {−fα (x1i , αi1 ) + fα (x0i , αi0 )} + 4γi,1 (λ0i x1i + λ1i x0i )(C) 0 ≥ 4αi,2 {−fα (x2i , αi2 ) + fα (x0i , αi0 )} + 4γi,2 (λ0i x2i + λ1i x0i )(D) Using Lemma 2 to replace x0i and Lemma 4 for fα , we get the following four sufficient conditions. L0i −1
ln 0,1 0 wi +λi {
+
Y
δ
L0
i Y 1 L0 −1 1 Lδ0 { } i − ln { } i p(s) p(s)
PL0 −1 i
p(s)wi1,s − L−1 (L0i − 1) } ≤ 0(A∗) 2p(s)
132
Financial Economics L0i +1
Y
ln
w0,1 −λ2i { i
PL0
+
i
L0i −2
ln
Y
0,1 0 wi +λi { 0,1 1 wi +λi {
ln
Y
L0
δ
+
PL0 −1 i
PL0 i
p(s)wi1,s − L−1 (L0i − 1) 2(L0i − 1)
p(s)wi1,s − L−1 (L0i ) } ≥ 0(C∗) 2(L0i ) 0
Li +1 δ Y 1 L0 −1 1 L0δ+1 { } 1 − ln { } i p(s) p(s)
w0,1 +λ0i { i w0,1 +λ2i { i
p(s)wi1,s − L−1 (L0i ) } ≤ 0(B∗) 2L0i
i Y 1 Lδ0 1 L0 −2 { } i − ln { } i p(s) p(s)
+
L0i −1
L0
δ
i Y 1 L0 +1 1 Lδ0 { } i − ln { } i p(s) p(s)
+
PL0 +1 i
PL0
+
i
p(s)wi1,s − L−1 (L0i + 1) } 2p(s)
p(s)wi1,s − L−1 (L0i ) } ≤ 0(D∗) 2p(s)
where, (A∗) holds ∀s ≤ L0i and (D∗) holds ∀s ≤ L0i Using the fact that λ0 i s ≥ 0 in (B∗) gives the condition 0
L0i
s.t.
Li Y
0
{p(s)}δ ≤ (p(L0i + 1))δLi
and using (B*) and (D*) to solve for λ0i (treating them as holding with equality) and then using (A*) we have L0i −1
L0i
≥ 2 s.t.
Y
0
{p(s)}δ ≥ (p(L0i ))δ(Li −1)
Thus, if ∃L0i ≥ 2, s.t. 0
Li Y
0
{p(s)}δ ≤ (p(L0i + 1))δLi
Chapter 5
133
and, L0i −1
Y
0
{p(s)}δ > (p(L0i ))δ(Li −1)
and conditions (i) -(v) hold, then L0i is an interior optimum for agent i.
Proved Proof : (of Theorem) from (7a) - (8) the insurance market at t = 0 is imperfect at prices
{p(s)} > 0 if ∃ at least one agent i∗, s.t.
L0i∗ ({p(s)}) > L0i∗ ({p(s)}) ∀i 6= i∗ and, csi∗ (1, {p(s)}) − w1,s > 0, s = L0i∗ ({p(s)}) Since, N is a finite set of integers and (11) holds the agents can be ordered in terms of decreasing wealth by decreasing order of initial wealth levels. Let w.l.o.g., 0,1 0,1 0,1 0,1 0,1 wN < wN −1 < wN −2 0,
L0N ({p(s)}) > L0i ({p(s)})∀i 6= N and, csN (1, {p(s)}) − w1,s > 0, s = L0N ({p(s)})(∗)
134
Financial Economics
Now, by the conditions of the theorem, wj0,1 − wi0,1 > 0 ∀j < i∀1 < i ≤ N, in terms of the above ordering of agents. We shall show that this is sufficient for (*) to hold, if the conditions of Lemma 5 are satisfied, mini si ≤ N and (11) holds ∀i, j ≤ N Now, L0N ({p(s)}) > L0i ({p(s)}), i 6= N if, 0,1 φ(c∗N , λN , L0N , wN , L0N ) > (c∗i , λN , L0i , wi0,1 , L0i )
∀λN ≥ 0 [Notice that at c∗i for any Li the budget constraint in (7’) is fulfilled with equality due to the strictly monotonically increasing nature of the utility function and the perfect divisibility of the consumption good, hence, φ(c∗i , λi , L0i ({p(s)}), wi0,1 , L0i ) ≡ V (L−1 (L0i ), {p(s)}i , wi0,1 ) of Lemma 2 ∀λi ≥ 0] for any agent i < N and i + 1,
0,1 φ(c∗i+1 , λi , L0i − 1, wi+1 , L0i − 1) − φ(c∗i , λi , L0i , wi0,1 , L0i ) > 0
Chapter 5
135
at prices {p(s)}. (where as before {c∗i } is the optimal choice of consumption when choice of Li is L0i by agent i). iff, from equation (5) of the proposition,
o,1 ul (ci ∗, L0i )(L0i − 1 − L0i ) + λi (wi+1 − wi0,1 ) + λi (L0i − 1 − L0i ) > 0
[where, ul (., .) is given by Lemma 4] or iff, 0,1 −ul (c∗i , L0i ) + λi (wi+1 − wi0,1 ) + λi (−1) > 0
or iff, 0,1 wi+1 − wi0,1 >
ul (c∗i , L0i ) ul (c∗i , L0i ) + λi = + 1(A) λi λi
if, ul (., .) ≤ 0 condition (A) is satisfied if 0,1 wi+1 − wi0,1 > 1.
Now, ul (c∗i , L0i ) ≤ 0 ui ({c∗i }, .) is convex in li (see lemma 4) for {c∗i } as derived in Lemma 2 for Li = L0i . L0i −1
(A) becomes
Y
which is fulfilled from Lemma 5, i.
0
(p(s))δ/Li −1 > (p(L0i ))δ · · ·
L0i is the optimal choice of Li by
136
Financial Economics
Now since this is true for any agent i, this is true for agents N and N − 1. Thus, at any prices {p(s)}
L0N ({p(s)}) > L0i ({p(s)})∀i 6= N (by induction) Now, from assumptions (3a), (6d) and condition (ii) of Lemma 5, L0i ≥ 2∀i. hence, L0N ({p(s)}) > N . Also, N : sN = mini si Hence, 0
w1,LN ({p(s)}) = 0 for agent N, sN ≤ N. ∗
Now, csN (1, {p(s)}) = 0 for s = L0N ({p(s)}) is not possible. 0,1 ) = −∞ if cs∗ since, V (L−1 (L0N ), {p(s)}, wN N (1, {p(s)}) = 0 and, 0,1 −∞ < V (L−1 (sN ), {p(s)}, wN ), since by virtue of assumption (iv) −1 Lemma 5, χ(L−1 (sN )) > 0 for agent N, hence from Lemma 2, cs∗ N (L 0,1 0 (sN ), {p(s)}, wN ) > 0. Hence, cs∗ N (1, {p(s)}) > 0 for s = LN ({p(s)}) 1.s This implies, cs∗ > 0 for s = L0N ({p(s)}). N (1, {p(s)}) − w
Thus, the two parts of condition (*) of this Theorem are proved for agent N. Hence the theorem.
Proved
Chapter 5
137 References
1. Anderson, R.K. and A. Takayama : Comparative Statics with Discrete Jumps in Shift Parameters, or, How to do Economics on the Saddle (-Point), Journal of Economic Theory, 21, 491-509, 1979. 2. Hahn. F. : Excess Capacity and Imperfect Competition, ch. 19 in F. Hahn : Money, Growth and Stability, Cambridge University Press, Cambridge, 1985. 3. Rockfeller, R.T. : Convex Analysis, Princeton University Press, Princeton, N.J., 1970. 4. Rudin, W. : Principles of Mathematical Analysis, Mc. Graw Hill Book Co., New York, 3rd edn., 1976. 5. Varian, H. : Microeconomic Analysis, W.W. Norton and Co., New York and London, 3rd edn., 1992. 6. Mallick, S : Bounded Rationality and Arrow-Debreu economies (wealth distribution), Ph.D. dissertation, Dept. of Economics, New York University, 1993.
