Risk! " #$%&'() *!'+,'''## -. / # $
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24 6 - K CD/@- ( (9 /C/CCC? 7 (( % ( %" 9 % ; &''&K/0 (% %&''& 0 7 Q AR (;. (Q% RG+ + D ( >( > > +/#'/% '"+(;(%-&''6&6,&6D C ( > ( - K ( - 4<.=>?;9"@!"( F/, &''6 /,//6D /' ( >A 7! 0%% /'!0?$F6@@.&''6D@C& // ( > A 7 F - #- $ #$ /% ;(%>'&''?! /& ( #( + ; + / ( A&''' /, ( 7! K > 7A&'', /6 ( 7Q A,(R 7 EF/&&''/ /? (Q9KA R./0&'', /0 (+ /@ - 4 > Q R. :% 5 % 7A/CC&/6@ /D - -Q R + &+'/,C@/&/0//'''. &''6 /C -A; (A '"+(;" '--:A +@ + "; (= &!:7 + TU;;%&''& 0. Ehrlich, I., and G. S. ecker, Market Insurance, Self-Insurance, and Self-Protection Journal of Political Economy, Vol. 80,No. 4,97, pg &/! - K V V VVV V V!&?&''6 && 4" 4 &'', K % #':% /% ; ( &, 4-74%A3%%(E. 7Q( K (A>- A %% R ++ &6 4;;A%!0? /% ;(%; (F?6&''&6,D6?@ 4
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26 " 3 A%( A/CDD 6D F >Q! R &''& 6C W - K( % WA F6/,6'/6/D/CC? Appendix Proof. aseline scenario: Obective based on the expected utility function of the firm is max ( z U ( W + [ π ( z ] I z + ( z U N ( W π ( z I z z, I The first order condition for IT security investment is π ( z ( z[ U U N ] + I ( z U + { ( z } U N = 0 z. (A First order condition for insurance is ( z[ π ( z ] U π ( z ( z U N = 0. (A Since ( z ( z U = U and I=. That is, the marginal utilities in both π =, from the latter condition we see that N states are equal. From the first condition, ( z = (A3 Proof. IT Security Spending with No Insurance Market Utility function of the firm is z U W z + z U W z max ( ( ( ( N ( z The first order condition for IT security investment is z U U z U + z U = 0 { } ( [ ] ( ( ( N N For W large enough, first order Taylor series approximation gives U U + U (A4 since ( z ( ( z N { } ( ( ( ( z U z U + z U = 0 N U N ( z + ( ( z U ( z = U N + <, IT security investment when insurance is available at fair market price is lower U than IT security investment when there is no insurance market available. Proof 3. Interdependent case: Utility function of the firm is max ( z, z U ( W + [ π ( z : z ] I z + ( z, z UN ( W π ( z : z I z z, I where ( z, z = ( p ( z ( qp ( z The first order condition for IT security investment is ( z, z π ( z (, z U U N + I { ( z, z U + ( z, z U N } = 0 z z First order condition for insurance is ( z, z ( π ( z, z U ( ( z, z π ( z, z U =0 If =0, then I= and ( z z N, = p ( z( qp( z = z (A5 6
27 If >0, for W large enough, using first order Taylor series approximation U U U I U U + U I + ( ; ( N N Substituting in A5, dividing byu and using first order condition for insurance (, ( [ + λ] (, [ λ] ( (, U z z z z N = U + z z we get ( z, z = z + λ [ ] and assuming the CARA utility function we get from the FOC for insurance λ r[ I] = [ + λ] p ( z qp ( z U where r = is a constant and greater than 0. Using identical firms, U p ( z qp ( z = (A6 + λ [ ] λ I = (A7 r [ + λ] p ( z qp ( z Proof 4: Condition for Unique Equilibrium From the first order condition of IT security investment (A6, Π ( R ( z, z = p ( z qp ( z + = 0 The slope of reaction function for Firm and is Π ( R ( z, z p z qp z R ( z = = Π R z, z p z qp ( z ( ( ( ( ( ; R ( z ( [ + λ] ( (, ( R ( z, z ( ( ( ( ( Π R z z p z qp z = = Π p z qp z In order for reaction curve to intersect, the slope of R should be higher than the slope of R. So p ( z ( qp ( z p ( z qp ( z >. p ( z qp ( z p ( z ( qp( z cross-multiplying and rearranging { p ( z ( qp( z p ( z qp ( z }{ p ( z ( qp( z + p ( z qp ( z } > 0 Note that the second term in the HS multiplicand is positive. Hence for an unique equilibrium, p ( z ( qp( z p ( z qp ( z > 0 Assuming symmetric firms, the condition for unique equilibrium can be written as ( ( ( p z qp( z q p z > 0 Proof 5: Denote the level of IT security investment and insurance coverage taken in independent firm and dependent firms I D I D as z and z respectively and I and I. I I λ p ( z = ; r [ + λ] I = I [ + λ] p ( z (A8(a,b D D D λ p ( z qp ( z = ; r [ + λ] I = D D [ + λ] p ( z qp ( z (A9(a,b Dividing A8a and A9a, p z I = p z D qp z D p z I > p z D z I > z D If >0, dividing A8b and A9b, ( ( ( ; ( ( 7
28 Proof 6: Proof 7: Proof 8: D D ( qp ( z I ( I I p z = D I p z I I D I D z > z p ( z > p ( z ; I D < or I I p ( z p ( z ( ( ( ( z = < 0 q p z qp z p z qp z (i λ p ( z + q qp ( z [ λ] ( ( I D > I (if =0, I I D = I. since denominator is less than zero. I = > 0 z + r p z qp z λ ( + ( [ + λ] r p ( z qp ( z ( ( ( ( ( ( I z p z q qp z p z p z = 0 z q { p z qp z p z qp z } z (ii = > 0 [ + λ] p ( z ( qp ( z p ( z qp ( z λ p ( z + q qp ( z [ + λ] r p ( z qp ( z [ + λ] ( ( ( ( ( I I z + = > 0 z p z qp z p z qp z z I λ (iii = 0, = r r r 0 + λ p z qp z π z ( ( ( ( ( = + < 0 and = ( + λ q z q, ( z = qp ( z p ( z, p ( z q qp ( z z z π < 0 from proposition (i: As a result, > 0 q q z = > 0 { } λ [ + λ] p ( z ( qp ( z p ( z qp ( z λ p ( z + q qp ( z [ + λ] r p ( z qp ( z [ + ] ( ( ( ( ( I I z I = + = λ z λ λ λ p z qp z p z qp z p ( z qp ( z r[ + λ] { } λ p ( z + q qp ( z ( + λ p ( z qp ( z p ( z qp ( z p ( z qp ( z = p ( z qp ( z r[ + λ] { } ( ( p z + q qp z Denote λ = > 0 p ( z qp ( z { p ( z qp ( z p ( z qp ( z } I λλ =. If λ > λ p ( z qp ( z r[ + λ] ( + λ λ, I > λ 0 I, else < 0. λ Proof 9: I = λ ( ( ( λ ( [ ( ] λ r + λ p z qp z = r λ z = + r ( + λ π ( z 8
29 ( π z ( z ( λ = + + z λ z λ λ π z ( ( z π ( z I λ π + λ + = = λ λ λ λ π z ( ( π I We know that insured amount will greater than premium paid.(i.e. ( π ( z > 0 As a result, if > 0, < 0. λ λ If loading factor increases, insurance coverage will decreases if price of insurance ((z increases as well. z π ( z ( z ( λ = + + z > 0 (if λ z λ π ( z > = λ, > 0. λ z λ z λ Proof 0: In region ; p ( z ( z p 3 = ( I r = 4 =, I ( qp ( z3, I3 =. In region ; p ( z = ( + λ p ( z qp ( z ( + λ ( =, r ( I =. In region 4, p ( z4 = + λ qp z λ 4 4 ( ( ( 4, ( λ ( + λ p ( z ( z. In region 3; z z > z3, z > z4, z > z and since > 0, z4 > z3. λ As a result, z > z4, z > z3. For the insurance amount, since z > z4, I = I3 > I > I4. Traditional Insurance Market (q=0, =0 versus Current cyber insurance market. We will compare IT security investment level in region 4 and in region. For cyber insurance coverage taken = I > I4. For the IT security investment, If [ λ] qp ( z + = 4, then 4 z = z. If [ λ ] ( ( ( p p ( z ( z + qp z < p z > p z z > z If [ λ ] ( ( ( 4 [ λ] qp ( z = qp z4 > p z < p z4 z < z4 Proof : Joint decision-making solution Suppose that there are two firms; firm and firm. Social planner will maximize the following U ( W + ( π I z + ( U N ( W πi z + U ( W + ( π I z + ( U N ( W π I z where = ( p ( z ( qp ( z, ( ( ( ( = p z qp z and π ( = + λ FOC for self protection with respect to z ; π π ( U U N U + ( U N + I+ ( U U N U + ( U N I= 0 z z z z FOC for insurance is; π U π U = 0 ( ( N Taylor st order approximation yields as before U N π U U U N π ( I + ( + I+ ( I + ( I= 0 z U z z U U U z For identical agent, U = U = U and U N = U N = U N and since ( ( + λ UN = ( [ + λ] U 9
30 ( ( ( p z + q qp z = The above equation can be written as ( ( ( p z qp z + M = ( + λ ( + λ Comparing with individual firm s first order condition z p ( z ( qp ( z + M = where M=0 and since < 0. + λ M ( Proof : Utility of firm will be qp ( z p ( z U W + I π z, z I z + ; M = p ( z ( q qp ( z 0 ( A ( ( ( ( ( ( ( + π (, + ( ( C ( π (, p z qp z p z U W I z z I z p z U W z z I z First order condition with respect to z qp ( z ( p ( z U A + p ( z qp ( z ( p ( z U p ( z UC π ( z, z ( + I qp ( z ( p ( z U A + p ( z qp ( z ( p ( z U + ( p( z U C = 0 z First order condition with respect to I ( π ( z, z qp ( z ( p ( z U A + ( π ( z, z p ( z qp ( z ( p ( z U ( p( z π ( z, z U C = 0 Following first order Taylor series approximation, U U ( A + U A I, UC U A + U A( I and U U A + U A( I. From first order condition with respect to I and replacing ( p( z U C in first order condition, ( ( ( ( A ( p z qp z p z U ( I p z U A( I π ( z ( ( ( ( ( ( (, z qp z p z p z qp z p z ( + I U A + U = 0 z π ( z, z π ( z, z Substituting the Taylor approximation, dividing by U, and since, and for symmetric firms, p ( z + qp ( z qp ( z p ( z = ( + λ The equation above can be written as ( ( ( A ( ( ( ( ( π z, z p z qp z p z ( + I [ r ( I ] z π ( z, z + K, where K = > 0 p z qp z p z K = [ + λ] ', where K K qp ( z = > 0 ' z For the individual choice of z earlier K =0.From the previous equation = > 0 K p ( z qp ( z qp ( z p ( z As a result IT security investment with liability is greater than without liability. The equation above can be written as p ( z + q qp ( z K = where K = K qp ( z p ( z > 0. + λ [ ] For the oint choice of z earlier, K =0. Thus, from the last equation z = > 0 K p ( z + q qp ( z qp ( z p ( z IT security investment level with liability is higher than social optimum level of IT security investment without liability. Proof 3: Generalization to Several Interdependent Firms Proof is omitted due to space limitation. However, proof is available from authors upon request. 30
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Online Appendix I Appendix D Additional Existence Proofs Denote B q = {q : A E A S [0, +r ]}, Bψ = {ψ : Y E A S S}, B W = {W : I E A S R}. I slightly abuse the notation by defining B q (L q ) the subset