Health insurance without single crossing: web appendix

Introduction

Appendix to section 2.2.1: Perfect competition and Miyazaki-Wilson contracts

• Lemma A solution to the MW problem exists. In such a solution, the resource constraints (RC) is binding and at least one type has full coverage. If the no-cross-subsidy constraint (NCS) is binding in the solution, then the solution is a RS equilibrium.
  
• Proof
  
  • Existence:
    

    The menu \((0,0,0,0)\) satisfies all constraints. Hence, the domain is non-empty. As it is also compact and the objective is continuous, a solution exists. Let \((p^h,q^h,p^l,q^l)\) be such a solution menu.

  • (RC) is binding.
    

    We do this by contradiction: Suppose that (RC) was slack. Note that (NCS) and (RC) imply that in this case \(p^l-c(q^l,u(q^l,p^l,\theta ^l),\theta ^l)>0\), i.e. positive profits are made on the \(\theta ^l\) types. If the incentive compatibility constraint for the h-type (IC_h) was slack, decreasing \(p^l\) and thereby increasing the objective would be feasible. As this would contradict optimality, we conclude that (IC_h) binds. If (IC_l) is slack, then decreasing \(p^l\) by \(\varepsilon > 0\) and decreasing \(p^h\) by \(\varepsilon '>0\) such that (IC_h) still holds is feasible and increases the objective value. As this would contradict optimality, we conclude that (IC_h) and (IC_l) would have to bind. Now we distinguish two cases:

    • If \(p(1,u(q^l,p^l,\theta ^l),\theta ^l)\geq c(1,u(q^h,p^h,\theta^h),\theta ^h)\), then pooling both types on the contract \((c(1,u(q^h,p^h,\theta ^h),\theta ^h)-\varepsilon ,1)\) is feasible for \(\varepsilon >0\) small enough and would increase (MW).
      
    • If \(p(1,u(q^l,p^l,\theta ^l),\theta ^l)< c(1,u(q^h,p^h,\theta ^h),\theta^h)\), then consider the menu \((min\{c(1,u^h(q^h,p^h,\theta^h),\theta^h),p(1,u(q^h,p^h,\theta ^h),\theta ^h)\}-\varepsilon ,1,p^l-\varepsilon' ,q^l)\) where \(\varepsilon\geq0,\,\varepsilon'>0\) are chosen (i) small enough to satisfy (RC) and (ii) \(\varepsilon'\) small enough to satisfy (IC_h). This menu is feasible and leads to a higher objective value than \((p^h,q^h,p^l,q^l)\).
      
  • In both cases the optimality of \((p^h,q^h,p^l,q^l)\) is contradicted and therefore (RC) has to bind.
    
  • At least one type has to have full coverage.
    

    This follows almost directly from assumption 3. Suppose to the contrary that \(q^l<1\) and \(q^h<1\) and define \(\iota=\arg\max_{k\in\{h,l\}} p(1,u(q^k,p^k,\theta^k),\theta^k)\). Again we distinguish two cases:

    • If \(\iota=l\), then the menu \((p^h,q^h,p(1,u(q^l,p^l,\theta^l),\theta^l)-\varepsilon ,1)\) is feasible by assumption 3. This contradicts the optimality of \((p^h,q^h,p^l,q^l)\).
      
    • If \(\iota=h\), we have to distinguish two subcases:
      
      • First, \(c(1,u(q^h,p^h,\theta ^h),\theta ^h)\leq p(1,u(q^l,p^l,\theta ^l),\theta^l)\). Then pooling both types on the contract \((c(1,u(q^h,p^h,\theta ^h),\theta ^h)-\varepsilon ,1)\) is feasible and leads to higher (MW).
        
      • Second, \(c(1,u(q^h,p^h,\theta^h),\theta ^h)> p(1,u(q^l,p^l,\theta ^l),\theta^l)\). But then the menu \((min\{c(1,u^h(q^h,p^h,\theta ^h),\theta^h),p(1,u(q^h,p^h,\theta ^h),\theta^h)\}-\varepsilon ,1,p^l-\varepsilon' ,q^l)\) (where again \(\varepsilon \geq 0,\,\varepsilon '>0\) are chosen such that the incentive constraints hold) is feasible and leads to a higher objective value.
        

        Hence, in both cases optimality of \((p^h,q^h,p^l,q^l)\) with \(q^h,q^l<1\) is contradicted and therefore we can conclude that at least one type has full coverage.

  • The solution of the MW problem is a RS equilibrium whenever (NCS) binds.
    

    In this case, (RC) and (NCS) imply that both contracts make zero profits (recall that (RC) is binding). We have to show that there is no profitable deviation contract \((p^d,q^d)\) given that the MW menu \((p^h,q^h,p^l,q^l)\) is offered. There cannot be a profitable deviation contract which attracts the \(\theta ^l\) types: This would contradict the definition of the MW menu as solution to the MW maximization problem. Hence, we only have to check whether there is a profitable contract \((p^d,q^d)\) which attracts only the \(\theta ^h\) types. From lemma 2, we know that \(q^h=1\) and therefore \(p^h=c(1,u,\theta ^h)\) as (NCS) binds. Because of assumption 3, there cannot be a profitable contract that high types prefer to \((c(1,u,\theta ^h),1)\). Q.E.D.

• Lemma The solution of the MW problem is a Wilson equilibrium.
  
• Proof
  

  Say the solution to the MW problem is \((p^h,q^h,p^l,q^l)\). This implies that this menu satisfies the constraints of optimization problem (MW). In particular, we have that the profits per low type satisfy \(\pi^l \geq 0\), profits per high type \(\pi^h \leq 0\) and \(\phi \pi^h +(1-\phi) \pi^l \geq 0\). Further, we know from lemma 2 in the paper that \(q^h=1\) and hence \(p^h \leq c(1,.,\theta^h)\) because of (NCS). What we have to show is that given this menu there is no deviation menu that would be profitable after unprofitable contracts –contracts that became unprofitable as a consequence of the introduction of the deviation menu– are retracted. We will show this by contradiction and divide the proof in three cases depending on which types are attracted by the deviation menu.

  • case where after deviation the l-type chooses the deviation contract \((p^d,q^d)\) while the h-type chooses an existing contract.
    

    That is, the h-type chooses \((\tilde p,\tilde q)\) which is either \((p^h,q^h),(p^l,q^l)\) or \((0,0)\). This implies that \(u(q^d,p^d,\theta^l)>u(q^l,p^l,\theta^l), \pi^d>0\) and \(\tilde \pi \geq 0\) (otherwise the contract \((\tilde p,\tilde q)\) would be retracted). We consider two cases:

    • \(\theta^h\) chooses \((\tilde p,\tilde q)\) with \(\tilde \pi =0\).
      

      But then \((\tilde p,\tilde q,p^d,q^d)\) leads to higher utility for the l-type, contradicting that \((p^h,q^h,p^l,q^l)\) solves (MW).

    • \(\theta^h\) chooses \((\tilde p,\tilde q)\) with \(\tilde \pi >0\).
      

      This implies that (i) \(\pi^h <0\) (hence \((p^h,q^h)\) is retracted) and (ii) \((\tilde p,\tilde q)=(p^l,q^l)\) (as contract \((0,0)\) yields zero profits). Now one of the two following menus is feasible in the MW problem and gives \(\theta^l\) a higher utility (thereby contradicting the MW solution). The first menu is \((p^d,q^d, c(1,\cdot,\theta^h),1)\). By assumption 3 and \(\tilde \pi >0\), the contract \((c(1,\cdot,\theta^h),1)\) is preferred by \(\theta^h\) to the contract \((p^l,q^l)\).¹ The menu \(( p^d,q^d, c(1,\cdot,\theta^h),1)\) is feasible if the \(\theta^l\) type prefers \((p^d,q^d)\) to \((c(1,\cdot,\theta^h),1)\). If this is not the case, then pooling both types on \((c(1,\cdot,\theta^h),1)\) is feasible in the MW problem and leads to a higher utility for \(\theta^l\) than the alleged solution of the MW problem, i.e. we get the desired contradiction.

  • case where after the deviation the h-type chooses the deviation contract \((p^d,q^d)\) with \(\pi^d >0\) while the l-type chooses an existing contract.
    

    This case cannot occur. As \(p^h \leq c(1,.,\theta^h), q^h=1\), by assumption 3 there does not exist a deviation contract that is both profitable and preferred by the h-type above \((p^h,q^h)\).

  • third case where deviation menu \((p^{hd},q^{hd},p^{ld},q^{ld})\) with \(\phi \pi^{hd}+(1-\phi)\pi^{ld}>0\) attracts both types.
    

    Note that the contract \((p^l,q^l)\) will not be retracted because \(\pi^l \geq 0\) and the h-types choose the \((p^{hd},q^{hd})\) contract. Therefore, the deviation menu must lead to strictly higher utility for the l-type. We consider two cases:

    • \(\pi^{hd} \leq 0\):
      

      Then \((p^{hd},q^{hd},p^{ld},q^{ld})\) leads to higher utility of the l-type and satisfies all constraints of the (MW) problem. This contradicts that \((p^h,q^h,p^l,q^l)\) solves (MW).

    • \(\pi^{hd} > 0\):
      

      This case cannot happen. As \(p^h \leq c(1,.,\theta^h), q^h=1\), by assumption 3 there does not exist a deviation contract that is both profitable and preferred by the h-type above \((p^h,q^h)\). Q.E.D.