Risk Sharing, Inequality, and Fertility

Transcription

1 Risk Sharing, Inequality, and Fertility Supplementary Appendix Roozbeh Hosseini ASU Larry E. Jones UMN Ali Shourideh UMN Contents Resetting Property for a General Class of Environments 2 2 Non-Homothetic Example 3 3 Full Information Efficient Allocation is not Incentive Compatible 5 4 Sufficiency of Downward Incentive Constraints 7 5 Spreading of Future Promises 0 6 Existence of Bounded Ergodic Set 0 7 A Uniqueness Result 6 8 Proofs 8 8. Proof of Lemma Proof of Corollary Proof of Remark A Numerical Example 20 0 Linear Utility of Leisure 23

2 Resetting Property for a General Class of Environments In this section, we show that the resetting property holds for a broader class of environments. Since, with private information, the resetting property at the top comes from no distortion at the top, we focus on the full information case and provide necessary and sufficient conditions for resetting to hold in a more general class of environments. Consider the model in section 2 of the paper where utility of an agent of type θ is given by Uc, y, n, θ+βn η uc 2,where y is income in the first period. The specification in section 2 is a special case where Uc, y, n, θ = uc + h y bn. Moreover, suppose that having children has an additional cost fn, θ θ in terms of period goods if the parent is of type θ. The planing problem, when types are public information, is given by max i=h,l π i [ y i c i fn i, θ i + R n ic 2i ] subject to π i [Uc i, y i, n i, θ i + βn η i uc 2i] w i=h,l θ i y i ; c i, n i, y i, c 2i 0. The following lemma can be proved about the solution of the problem : Lemma Suppose that the solution to the above problem is interior. Then η uc 2i u c 2i c 2i = Rf n n i, θ i + R U nc i, y i, n i, θ i U y c i, y i, n i, θ i. 2 Proof. The first order conditions for the above problem are given by λu c = + λu y = 0 λ = U y f n n i, θ i R c 2i + λ [ U n + βηn η i uc 2i ] = 0 3 R n i + λβn η i u c 2i = 0 λβn η i = R u c 2i where λ is the multiplier on promise keeping. By replacing terms in 3, we get the following: 2

3 which implies the lemma s claim. f n n i, θ i R c 2i U n uc 2i U y R u c 2i = 0 Note that for section 2 s specification, f = 0 and U n /U y = bθ, in which 2 becomes equation 4 in the paper. Given the above characterization for c 2i, one can state the following result: Remark 2 The resetting property, i.e., c 2i being independent of w, holds if and only if the function is only a function of θ and not of c, y, n. f n n i, θ i + U nc, y, n, θ U y c, y, n, θ The above remark implies that if we add linear goods cost of children to the model in the paper resetting still holds. Another environment that satisfies the above condition is when f is linear in n and U = θuc + h y bn. This analysis also shows that resetting property holds when there is no leisure cost of children and goods cost changes linearly with n for each type. 2 Non-Homothetic Example In this section, we will generalize the discussion from the two period example in section 2 of the paper to non-homthetic preferences. Suppose parents have the following, non-homothetic preferences: uc + h l bn + βgnuc 2 Assume that gnuc 2 is strictly increasing, strictly concave, differentiable and ng n/gn c 2 u c 2 /uc 2 < D < c 2, n, i.e., the negative of elasticity of substitution between C 2 and n is uniformly bounded above. In this case, the analog of equation 4 in the paper is: nw 0, θ H g nw 0, θ H uc 2 W 0, θ H = u c 2 W 0, θ H c 2 W 0, θ H + bθ H Ru c 2 W 0, θ H. gnw 0, θ H 3

4 This can be rewritten as: nw 0, θ H g nw 0, θ H /gnw 0, θ H c 2 W 0, θ H u c 2 W 0, θ H /uc 2 W 0, θ H = + bθ HR c 2 W 0, θ H. Since the elasticity on the left is assumed bounded, it follows that c 2 W 0, θ H must be bounded away from zero for all values of W 0. It follows immediately that per capita utility uc 2 W 0, θ H is also bounded below. In other words, including fertility in the model will give rise to a level of per capita continuation utility that is bounded below as long as these elasticities are bounded. That is homothetic utility is not required for c 2 to be bounded away from zero. Rather what is required is that income expansion paths in C 2, n space should have slope that is bounded away from zero. The example given below illustrates this point. Example.Suppose gn = n η + An η 2 section 2 becomes the following subject to with 0 > η > η 2. In this case, problem 8 in min c 2,n bθ Hn + R nc 2 n η + An η 2 uc 2 = W W 0, θ H. If we equate the marginal rate of transformation between c 2 and n to the associated marginal rate of substitution, we get η n η + η 2 An η 2 n η + An η 2 uc 2 u c 2 c 2 = brθ H. Now suppose that W 0 and therefore, W W 0, θ H converges to. In this case, one can argue that n has to converge to zero. If not, by promise keeping c 2 has to converge to 0 violating the above equation. Note that η n η + η 2 An η 2 lim = η n 0 n η + An η 2. 2 This means that as W 0 converges to, the above equation becomes: η 2 uc 2 u c 2 c 2 = brθ H which implies that c 2 is bounded away from 0. Income exapnsion paths for this example are giveun in Figure. Note that at W 0 =, c 2 W 0, θ H is the slope of the income expansion path at the origin which is positive. Morevoer, C 2 n is bounded away from zero for 4

5 all points on the curve. 2 Income expansion path for fertility and aggregate consumption of children.8.6 Aggregate consumption of children, C Fertility, n Figure : Income expansion path in an example with non-homothetic formulation. The slope of the income expantion path is per capita consumption. Example for gn = n η + An η 2. 3 Full Information Efficient Allocation is not Incentive Compatible In this section, we show that the efficient allocations when there is no private information does not satisfy the incentive compatibility constraints for the maximization problem in section 3 of the paper. The efficient allocation with full information solves the following recursive problem: vw = min θ [ πθ cθ θlθ + ] R nθvw θ Pfi subject to πθ[ucθ + h lθ bnθ + βnθ η w θ] w. θ 5

6 This implies the following first order and Envelope conditions: u cθ = u cθ 4 θu cθ = h lθ bnθ 5 u cθv w θ = βrnθ η 6 ηv w θw θ vw θ = brθ. 7 Intuitively, from intra-family risk sharing, equation 4, we know that per capita consumption among siblings is equal. Moreover, efficiency requires that leisure is decreasing in productivity, equation 5. It is therefore sufficient to show that future utility for a low productivity agent is higher than a high productivity agent. This is shown below. One intuition for this comes from the curvature properties of the cost function, vw. In the unconstrained efficient allocation, the planner equates per capita marginal cost nθ η v w θ across various types. Notice that convexity of the above problem implies that both v w and ηv ww vw are increasing functions of w. This implies that v has a curvature higher than η, i.e., v w w η v w η η w. Therefore, for a given relative fertility nθ nθ = >, equating per capita marginal cost implies that relative promised utility w θ w θ is at most η. This implies that nθ η w θ > nθ η w θ. Hence, overall promised value, nθ η w θ, is higher for agents with a higher number of children agents with lower productivity. Hence, we can state the following lemma: Lemma 3 The solution to the problem Pfi, is not incentive compatible, i.e., it violates the incentive constraint 9 in the paper. Proof. Consider the solution to Pfi. From 4 and 6, we have that nθ η v w θ = nθ η v w θ, θ, θ. 8 Moreover, since ηv ww vw is increasing in w, we know that wv w v w > η η v w v w > η η w. If we assume that θ > θ, then w θ > w θ and we can integrate the above equation to obtain that v w θ w θ v w log = v w θ w θ v w dw > η w θ log η w θ 6

7 and therefore v w θ w η v w θ > θ η. w θ Combining the above with 8, we get the following inequality nθ η = v w θ nθ v w θ > w η θ η w θ nθ η w θ > nθ η w θ. 9 Moreover, since cθ does not depend on θ and leisure is decreasing in θ. Therefore lθ bnθ < lθ bnθ < θ lθ /θ bnθ, when θ > θ. These properties together with 9 gives us the following inequality ucθ + h lθ bnθ + βnθ η w θ < ucθ + h θ lθ bnθ + βnθ η w θ, θ > θ θ which means that under the efficient allocation, agents with higher productivity would like to pretend to be low productivity. So the unconstrained efficient allocation is not incentive compatible. 4 Sufficiency of Downward Incentive Constraints In this section we show, if the an allocation satisfies certain properties then downward incentive constraints are sufficient. Hence, in any solution, we can check whether the any solution of the model satisfies these conditions. These conditions are easy to check and they hold in our numerical examples that are done with two types. We provide our suffieicnt conditions in the following lemma: Lemma 4 Suppose an allocation cθ, lθ, nθ, w θ satisfies the following:. lθθ is increasing in θ, 2. lθ bnθ θ lθ θ bnθ, for all θ > θ 3. Local downward incentive constraints are binding: ucθ i + h lθ i bnθ i + βnθ i η w θ i = ucθ i + h θ i lθ i θ i bnθ i + βnθ i η w θ i. Then, incentive compatibility holds for any θ, θ. 7

8 Proof. By part 3 of the assumption, we have ucθ i + βnθ i η w θ i ucθ i βnθ i η w θ i = = h lθ i bnθ i h θ i lθ i θ i bnθ i. By part 2 and 3 of the assumption we have θ i θ i lθ i θ i lθ i θ i θ i lθ i θ i lθ i bnθ i nθ i. Hence, for any x [/θ i, /θ i ] xθ i lθ i θ i lθ i bnθ i nθ i xθ i lθ i bnθ i xθ i lθ i bnθ i. Therefore, using part and concavity of h, h xθ i lθ i bnθ i θ i lθ i h xθ i lθ i bnθ i θ i lθ i. Integrating both sides from /θ i to /θ i, we get h lθ i bnθ i h θ i lθ i θ i bnθ i h θ ilθ i θ i bnθ i h lθ i bnθ i. Therefore, ucθ i + βnθ i η w θ i ucθ i βnθ i η w θ i h θ ilθ i θ i bnθ i h lθ i bnθ i. Hence, the local upward incentive constraints are satisfied. Now, we will show that other upward incentive constraints are satisfied. To illustrate we show the argument for i and i + 2 and a similar inductive argument works for higher differences. By condition 2, we know that: θ i+2 θ i+2 lθ i+2 θ i+ lθ i+ bnθ i+ nθ i+2 8

9 and therefore, θ i+2 lθ i+2 θ i+ lθ i+ θ i+2 lθ i+2 θ i+ lθ i+ bnθ i+ nθ i+2. θ i θ i+ Hence, for any x [/θ i+, /θ i ], xθ i+ lθ i+ bnθ i+ xθ i+2 lθ i+2 bnθ i+2 and we have, h xθ i+ lθ i+ bnθ i+ h xθ i+2 lθ i+2 bnθ i+2 h xθ i+ lθ i+ bnθ i+ θ i+ lθ i+ h xθ i+2 lθ i+2 bnθ i+2 θ i+2 lθ i+2. So, h θ i+lθ i+ θ i bnθ i+ h lθ i+ bnθ i+ 0 h θ i+2lθ i+2 θ i bnθ i+2 h θ i+2lθ i+2 θ i+ bnθ i+2. Rewriting local IC s for i, i + and i +, i + 2: ucθ i + βnθ i η w θ i ucθ i+ βnθ i+ η w θ i+ h θ i+lθ i+ θ i bnθ i+ h lθ i bnθ i ucθ i + βnθ i+ η w θ i+ ucθ i+2 βnθ i+2 η w θ i+2 2 h θ i+2lθ i+2 θ i+ bnθ i+2 h lθ i+ bnθ i+. Summing over inequalities 0-2, we get: ucθ i + βnθ i η w θ i ucθ i+2 βnθ i+2 η w θ i+2 h θ i+2lθ i+2 θ i bnθ i+2 h lθ i bnθ i which is the upward incentive constraint for i, i + 2. The rest of the upward and downward incentive constraints can be proved in a similar way. The conditions provided are very intuitive. The first condition asserts that income has to be increasing in type. A similar condition arises in most Mirrleesian environments. The second assumption implies that leisure from lying downward is higher than leisure under telling the truth. Notice that, in environments without fertility, conditions and 2 are 9

10 equivalent. However, since fertility is endogenous and potentially different for different types condition 2 is needed. Condition 3 is very common in the literature can sometimes shown to be binding. This lemma is similar to a result in mechanism design that provides sufficiency of local incentive constraints, see Matthews and Moore 987, and Pavan et al Spreading of Future Promises In this section, we show that when βr =, then it is optimal for the planner to spread continuation utility for the highest and lowest values of shock. This is in fact very similar to lemma 5 in Thomas and Worrall 990. Therefore, it suggests that the same proof as theirs would work to show that total continuation utility, N η t w t converges to its lowest bound, when βr =. Lemma 5 Suppose that βr =. Then nw, θ η v w w, θ v w nw, θ I η v w w, θ I, for all w. Moreover, if µi, jµj, is positive for some j, v w < nw, θ I η v w w, θ I v w > nw, θ η v w w, θ. Proof. The FOC w.r.t w θ and w θ I are given by: βnθ η [λπθ ] µi, i=2 ] I βnθ I [λπθ η I + µi, i i= = R πθ v w θ = R πθ Iv w θ I where µi, j 0 is the Lagrange multiplier on the incentive constraint i, j with i > j and λ > 0 is the multiplier associated with the promise-keeping constraint. Since βr =, the above equalities imply that nθ η v w θ λ nθ I η v w θ I with the inequalities being strict if and only if one of the multipliers µi, i and µi, are positive. Envelope theorem, we know that λ = v w. This proves the claim. By the 6 Existence of Bounded Ergodic Set Assumption 6 For any given w, if lw, θ i = 0 for some i I, then lw, θ j = 0 for all j < i. Assumption 7 The policy function w w, θ is continuous for all θ. 0

11 Proposition 8 Suppose V N, W is convex and continuously differentiable and assumptions 6 and 7 hold. The, there exist w < 0 and w > such that for every w [w, w], w w, θ belongs to [w, w] for all θ Θ. Suppose there are I types Θ = {θ,..., θ I } and θ i+ > θ i for all < i I. We break the proof into few lemmas. Consider the following problem vw = min cθ,lθ,nθ s.t. θ Θ πθ cθ θlθ + R nθvw θ πθ ucθ + h lθ bnθ + βnθ η w θ w θ Θ ucθ i + h lθ i bnθ i + βnθ i η w θ i ucθ j + h θ jlθ j bnθ j + βnθ j η w θ j ; i, j < i. θ i We have shown in Lemma in the paper that convexity and differentiability of V N, W implies that vw is differentiable and convex. It also implies that ηwv w vw is an increasing function of w. Lemma 9 For any w such that lw, θ I > 0 we have w w, θ i < w w, θ I for all i < I. Proof. Let λ and µi, j be multipliers on promise keeping and incentive constraint of type θ i who wants to pretend to be of type θ j. For now suppose lw, θ i > 0 for all i. First order conditions are we suppress the dependence of the allocation on w, it plays no role in the following arguments: θ I πθ I = θ i πθ i = I λπθ I + µi, j h lθ I bnθ I 3 j= i λπθ i + µi, j h lθ i bnθ i 4 k=i+ j= µk, i θ i θ k h θ ilθ i bnθ i θ k

12 πθ I v w θ I = πθ I v w θ i = I λπθ I + µi, j βrnθ I η 5 j= i λπθ i + µi, j j= k=i+ µk, i βrnθ i η 6 πθ I vw θ I = πθ i vw θ i = I ηβrnθi πθ I λ + µi, j η w θ I Rbh lθ I bnθ I j= i ηβrnθi λπθ i + µi, j η w θ i Rbh lθ i bnθ i k=i+ j= µk, i ηβrnθ i η w θ i Rbh θ ilθ i bnθ i θ k 7 for i < I. Combining these equations we can get the following ηw θ I v w θ I vw θ I = Rbθ I 8 ηw θ i v w θ i vw θ i = Rb λ + i µi, j h lθ i bnθ i πθ i j= Rb µk, ih θ ilθ i bnθ i. 9 πθ i θ k k=i+ Since ηwv w vw is increasing in w, to establish the claim of the lemma it is enough to show that the right hand side of the equation 9 is smaller that Rbθ I. But notice that Rb πθ i Rb πθ i k=i+ k=i+ µk, ih θ ilθ i bnθ i θ k θ i µk, ih θ ilθ i bnθ i θ k θ k < Rb Rb λ + πθ i λ + πθ i = Rbθ i < Rbθ I. i µi, j h lθ i bnθ i j= i µi, j h lθ i bnθ i j= 2

13 The first inequality follows from the fact that θ i θ k < for k > i. The rest follows from 4. This finishes the proof for the case in which lw, θ i > 0. Now consider the case in which the non-negativity constraint on hours is binding for some types i < I. Then by assumption 6 for all types θ j, j < i we have lw, θ j = 0. Then all types θ i, i < j receive the same allocations and therefore µi, j = 0 for j < i. The equations 4 and 9 for type θ i change to and θ i < λ πθ i k=i+ ηw θ i v w θ i vw θ i = Rb Suppose w θ i > w θ I, then Rbθ I < ηw θ i v w θ i vw θ i = Rb and therefore µk, i θ i θ k λ πθ i h bnθ i k=i+ λ πθ i µk, i k=i+ µk, i h bnθ i h bnθ i h bnθ i > > θ I λ I πθ i k=i+ µk, i λ + πθ I θ I I j= µi, j = h bnθ I lθ I. Hence bnθ i < lθ I bnθ I. On the other hand w θ i > w θ I implies λ πθ i k=i+ µk, i nθ i η = v w θ i βr > v w θ I βr = λ + πθ I I µi, j nθ I η j= and therefore nθ I > nθ i. These implies that lθ I has to be negative which is a contradiction. Therefore, we must have w w, θ i < w w, θ I for all w such that lw, θ I > 0. By assumption 6 lw, θ j = 0 for 3

14 all j < i, and we know that w w, θ j = w w, θ i < w w, θ I for all j < i since all types θ j receive the same allocations for < j i. Next we will find the promised utility at which the non-negativity for type θ I just binds. At this point, all types work zero hours and therefore receive the same allocation assumption 6. Let ĉ, ˆn and w I solve the following equations θ I u ĉ = h bˆn v w I u ĉ = βrˆn η ηw I v w I vw I = Rbθ I. Define ŵ = uĉ + h bˆn + βˆn η w I. Note that w ŵ, θ I = w ŵ, θ i = w I for all i. Our goal is to show that for w > ŵ, it is optimal for type θ I to work zero hours for all θ. After we establish that, we can prove the claim of the proposition for two cases on w I > ŵ and w I < ŵ. Lemma 0 If w > ŵ, then lw, θ I = 0. Proof. Suppose otherwise and consider the following equations w = uc + hm + βn η w I and θ H u c = h m v w I u c = βrn η in which m = l bn. Also, the first order conditions at ŵ are θ I u ĉ = h ˆm v w I u ĉ = βrˆn η 4

15 ŵ = uĉ + h bˆn + βˆn η w I in which ˆm = bˆn. Subtract the first order conditions at w and ŵ to obtain: w ŵ = uc uĉ + hm h ˆm + βw I ˆn η n η 20 θ H u c u ĉ = h m h ˆm v w I u c u ĉ = βrn η ˆn η. Then, by concavity of uc, hm and n η we know that c ĉ,m ˆm and n ˆn must all have the same signs. Also, from 20 we have u ĉc ĉ + hm + h ˆmm ˆm + βw I ηˆn η n ˆn > w ŵ > 0. This implies that c ĉ,m ˆm and n ˆn must be all positive. If m > ˆm and n > ˆn, then we must have l = m bn < ˆm bˆn = 0. This is a contradiction, therefore, the non-negativity constraint on hours must be binding at ŵ. Lemma If w I > ŵ, there exist ŵ w 0 such that w w, θ = w. Proof. Recall that that since lw, θ 0 is binding, by assumption 6 all types work zero hours and receive the same allocations. Therefore, the incentive constraint is slack. The first order conditions for type θ I are λh bnθ I > θ I and Therefore vw θ I + Rbλh bnθ I = ληβrnθ I η w θ I. ηw θ I v w θ I vw θ I = Rbλh bnθ I > Rbθ I. This implies w w, θ I > w I > ŵ. Define the function w ɛ, θ : [ŵ, ɛ] [ŵ, ɛ] as w, θ if w ɛ, θ ɛ w ɛ, θ =. ɛ if w ɛ, θ > ɛ 5

16 This function must have a fixed point wɛ [ŵ, ɛ]. We know that w, θ = lim ɛ 0 w ɛ, θ. Then, either a ŵ < w < 0 exists such that w w, θ = w or lim w 0 w w, θ = 0. Note that because no one works all types receive the same allocations. So far we have established that if w I > ŵ, then we can choose w = w = w and the proposition is proved. Now suppose w I ŵ. By lemma 9, we have ww, θ w I ŵ for all w. Let w = ŵ and notice that proposition 3 together with continuity of w w, θ i assumption 7 implies that there exist a w small enough so that w w, θ i >. By assumption 7, w w, θ i is continuous in w and hence has a minimum w in [ w, w]. This concludes the proof of the proposition. 7 A Uniqueness Result In this example, we assume that resetting property at the top holds for every w < w 0. For this case, using a direct constructive proof, we show that the model implies a long-run distribution for per capita consumption and characterize its properties. What is convenient about this example is that we can show that if resetting holds at all w and Assumption 8 holds with w = w 0, then there is a unique stationary distribution within a certain class. We know that w 0 = w w, θ H is independent of w for all w w 0. Hence, we can define the following set of promised values: W = {w n w n+ = w w n, θ L, n 0} By Corollary 7 in the paper, there is a lower bound w such that w w w 0, for all w W. Assumption 2 Assume that w j w i if j i. Consider a distribution over W, Ψ = ψ 0, ψ, with i=0 ψ i =. For Ψ to be a stationary distribution, there must exist a γ such that the following conditions hold: γψ 0 = π H i=0 Iterating on equation 22 implies the following: ψ m = πl γ nw i, θ H ψ i 2 γψ j = π L nw j, θ L ψ j, j 22 m nw m, θ L nw m 2, θ L nw 0, θ L ψ 0 6

17 Replacing in 2 implies the following equation: γ = π H nw 0, θ H + m= πl γ m nw m, θ L nw m 2, θ L nw 0, θ L nw m, θ H 23 Given that in the original problem, we must have nw, θ /b. This means that the right hand side of the above equation is lower than m=0 π L/bγ m. Therefore, if we let γ, the right hand side converges to a finite number, π H nw 0, θ H. Notice that the left hand side is strictly increasing and the right hand side is strictly decreasing in γ. Moreover, at γ = 0 RHS is higher than LHS and at γ =, RHS is lower than LHS. Because of this, if we knew that RHS was continuous, this would be sufficient to say that there is a γ satisfying equation 23 and that it is unique. To handle this last technical detail, we proceed as follows Define γ K as follows: γ K = π H nw 0, θ H + K m= πl γ K m nw m, θ L nw m 2, θ L nw 0, θ L nw m, θ H By definition, γ K < γ K+. We know that nw, θ < b, therefore γ K < π H b + K m= πl γ K m m+. b Suppose that π L bγ K < or π L b < γ K. Then, the above inequality implies that. bγ K < π H π L bγ K γ K < π H + π L b = b. This shows that γ K is a bounded increasing sequence. Hence, there exists γ such that γ K γ with γ > γ K. It needs to be shown that at γ, RHS of 23 exists. Suppose not and that the sum is infinity. Define F K γ to be the RHS of 23 up to K-th term. F K γ is a continuous and decreasing function. Therefore, γ K = F K γ K > F K γ. Moreover, F K γ F γ and hence F γ γ. This means that RHS of 23 cannot be infinity and 23 is satisfied at γ. Now by Corollary 3, we know that A > 0 ; nw, θ H Anw, θ L w [w, w]. 24 7

18 Therefore, using 23, we will have Hence γ = π H nw 0, θ H + π H A m=0 Now define, ψ 0 as πl m=0 γ πl m= πl γ m nw m, θ L nw m 2, θ L nw 0, θ L nw m, θ H m nw m, θ L nw m, θ L nw 0, θ L. γ m+ nw m, θ L nw m, θ L nw 0, θ L π L Aπ H. ψ 0 = + m+ π Lγ m=0 nwm, θ L nw m, θ L nw 0, θ L By the above inequality, we know that ψ 0 exists and it is greater than zero. Moreover, we can automatically define ψ i s using 22. Hence, the definition of γ, being the solution to 23 together with the definition of ψ 0, makes sure that Ψ satisfies 2-22 and hence, it is a stationary distribution. As it appears in the proof, in some sense, bounded relative fertility together with the resetting property at the top are the key elements of having a long-run stationary distribution. First, every time any one receives a high shock, her promised value is reset. Secondly, relatively, there are enough children being born by high types so that we get stationarity. Moreover, the above proof shows that when the set of w s is restricted to W, the stationary distribution is unique. 2 8 Proofs In this section we provide various proofs omitted in the paper. 8. Proof of Lemma 2 We show that when V is continuously differentiable and strictly convex, vw is continuously differentiable, and strictly convex and ηv ww vw is strictly increasing. By definition, we have V N, W = NvN η W A high type s number of kids relative to the low type s 2 A similar argument shows that there is a unique stationary distribution on W even if Assumption 2 does not hold. In this case, it follows that W is necessarily finite and the same logic applies. 8

19 Therefore, if V is continuously differentiable, v has the same property. Moreover, strict convexity and differentiability of V imply that V N V W is strictly increasing in NW. Notice that V W W, N = N η v N η W V N W, N = vn η W N η W vn η W Hence, V W being strictly increasing in W implies that v w is strictly increasing and hence v is strictly convex. Moreover, V N being strictly increasing in N implies that vw ηwv w is strictly decreasing in w. 8.2 Proof of Corollary 7 By Proposition 6, we know that This implies that there exists a w ɛ such that lim w w w, θ i = w i w w ɛ, w w, θ i w i < ɛ By assumption 6, w i < w. Now define, { } w = min w ɛ, w ɛ, w 2 ɛ,, w n ɛ, inf w [w w w, θ i ɛ,w],i Notice that since w is a continuous function that is always in R and the infimum is taken over a compact set, w is well-defined. Pick ɛ > 0 small enough so that inf w [wɛ,w],i w w, θ i < w j ɛ for all j. Then by definition of w ɛ we must have w w, θ i [w, w], w [w, w] Since utility is unbounded below and η is negative, nw, θ i must be positive. Hence we can have the following corollary: Corollary 3 For all w [w, w], we must have nw, θ i n and nw,θn nw,θ i i {,, n} and for some n, A > 0. A, for all 9

20 8.3 Proof of Remark 0 Since lw, θ n > 0 for all w [w, w 0 ], resetting property at the top holds. Therefore, by definition γψ Ψ {w 0 } = π n nw, θ n dψw γψ = S π n S n π i nw, θ i dψw i= nw, θ n dψw S + π n A nw, θ n dψw S = π n + π n A nw, θ n dψw S Therefore, Ψ {w 0 } π n A π n + π n A 9 A Numerical Example In this section, we provide a numerical solution for the model provided in section 3 of the paper. In light of this example, we are able to provide some more intuitive properties of the model that we have not proved in the paper. We assume that individuals have CRRA preferences over consumption and leisure uc = c σ, and hm = φm σ σ γ in which m = l bn is leisure, l is hours worked and n is number of kids for each parent. For this example we assume the following values for parameters: β = 0.3, R = 4, σ =.5, φ = 0.5, b = 0.4 and η = 2. We assume two levels of productivity shocks {θ L, θ H } = {2, 6}. Shocks are i.i.d across generations and dynasties and the probability of the high shock is π H = 0.. Figure 2, presents the optimal policy functions in the recursive problem given above specification. It can be seen that whenever hours worked are positive, the per capita continuation value of parents with high productivity shock is constant, i.e., the Resetting Property. Moreover, as it is established in proposition 6 in the paper, the per capita continuation value 20

21 for a parent with a low productivity shock converges to a finite number as parent s promised utility tends to. One observation that cannot be shown analytically is that fertility for less productive agents is lower than that of more productive agents. When there is no private information, full risk sharing and cost minimization deliver this result. Full risk sharing implies that per capita marginal utility from having children must be equated across types. This implies that consumption per child is negatively correlated with the number of children. On the other hand, kids are more expensive for more productive types in terms of their time. Therefore, it is cost effective to deliver utility from having children to more productive types by giving more per capita consumption to each of their children and have them have fewer children. With private information, however, full risk sharing is violated and the argument above does not work. However, it is observed in our numerical example that more productive types have a lower fertility rate. Our calculations, also show that the spread in fertility is lower under private information compared to full information High shock Low shock Consumption Function, cw,! 0.8 Hours, lw,! cw,!.5 lw,! !4!2!0!8!6!4!2 Current promised utility, w 0!4!2!0!8!6!4!2 Current promised utility, w 2.5 Fertility, nw,!!2 Future utility promised to each kid, w " w,! 2!4 nw,!.5 w " w,!!6!8!0 0.5!2 0!4!2!0!8!6!4!2 Current promised utility, w!4!4!2!0!8!6!4!2 Current promised utility, w Figure 2: Optimal consumption, hours, fertility and promised utility allocations for the numerical example It is important to note here that incentives are provided both by the level of per capita promised utility to the children and the number of children. In other words the future 2

22 utility that is promised to a parent is nw, θ η w w, θ. This promised utility is always monotone increasing in the current utility promised to the parent. This property is similar to benchmark dynamic Mirrleesian models with no fertility choice. Figure 3 illustrates this. Future utility promised to each parent, nw,! " w # w,!!2 High shock Low shock!4!6 nw,! " w # w,!!8!0!2!4!6!8!4!2!0!8!6!4!2 Current promised utility, w Figure 3: Future promised utility to the parents. We can use the construction procedure in section 7 to calculate the stationary distribution of per capita promised utilities and the growth rate of population, γ Ψ. The stationary distribution is shown in Figure 4. For this economy, the growth rate of population is γ Ψ = As it is mentioned in remark 0 in the paper, a positive fraction of agents are always at the resetting value w 0 = Stationary distribution of promised utility 0.8! * w !4!2!0!8!6!4!2 Current promised utility, w Figure 4: Stationary distribution 22

23 0 Linear Utility of Leisure In this section we focus on the case with linear utility from leisure. This allows to prove many of the unproved assumptions in the main body of the paper for the general case. To do so, we consider the problem P in the paper with the additional assumption that hm = ψm: V N, W = min C i,l i,n i,w i s.t i= i= N η u [ πθ i C i θ i L i + ] R V N i, W i πθ i [N η u Ci N N η u i, j. P 5 Ci + ψ L i N N bn i N + ψ L i N bn i + βw i N + ψ θ jl j θ i N bn j + βw j N Cj N ] + βw i W Notice that the set of reports is not restricted to lower reports since we can prove that general incentive compatibility is equivalent to local downward constraints being binding and output being increasing, a similar approach to Thomas and Worrall 990. Notice that adding the IC constraint where j pretends to be i and the reverse implies that: θ i L i θ j N + θ jl j θ i N L i N + L j N. Therefore, if θ i > θ j, then θ i L i θ j L j which means output is increasing. Moreover, if we assume that local downward IC constraints are binding and output is increasing, it can be easily shown that the local upward constraints are satisfied. Thus, summing over local incentive constraints gives the general ones. We also assume that output being increasing is not binding so we can neglect it. Therefore the functional equation becomes 23

24 the following: V N, W = min C i,l i,n i,w i s.t i= i= N η u [ πθ i C i θ i L i + ] R V N i, W i πθ i [N η u Ci N N η u Ci + ψ L i N N bn i N + ψ L i N bn i + βw i N + ψ θ i L i N θ i N bn i N Ci ] + βw i W + βw i. Let λn η be the lagrange multiplier on promise-keeping constraint and µ i N η be the multiplier for i-th IC constraint. following: Then the first order condition for hours worked is the π I θ I = λπ I + µ I ψ π i θ i = λπ i + µ i θ iµ i+ θ i+ ψ, i = 2,, n 25 π θ = λπ θ µ 2 θ 2 ψ. 26 We can define µ = µ I+ = 0 and 25 holds for i =,..., I. If we divide the i-th equation by θ i and sum over all i s, the µ i s will cancel and we have λ = ψ i π i = θ i ψe θ 27 Therefore, µ i = ψ θ i j i π j θ i ψ j π j j i θ j π j θ j Since θ i s increasing, all the µ i s are positive. = θ i π j j The first order conditions with respect to consumption are: π i = λπ i + µ i µ i+ u Ci. N θ j j i π j j i ψ j π j θ j π j. θ j Obviously, we need consumption to be increasing and marginal utility to be positive. This gives us a condition on distribution of θ i. Moreover, we can see that consumption is independent of the state variable N, W. 24

25 The first order conditions with respect to N i, W i are: π i R V NN i, W i = bλπ i + µ i µ i+ ψ 28 π i R V W N i, W i = N η λπ i + µ i µ i+ β. 29 Now for every i, define after-tax-productivity as follows: θ i = ψ λπ i + µ i µ i+ π i. Notice that we have u C i /N θ i = ψ and θ i does not depend on the state variables. From before, we know that there exists a function v such that V N, W = NvN η W. Therefore, V N N, W = vw ηwv w, V W N, W = N η v w where w = N η W. Hence, from 29 we have that: ηw iv w i vw i = br θ i N η i v w i = βrn η θi. The above, implies that n i = N i/n, w i are also independent of the state. Moreover, from the Envelope condition we have that: V W N, W = λn η = N η ψ i π i θ i = N η v w. Therefore, v is a linear function and we have: w vw = A + ψ i π i θ i V N, W = AN + W N η ψ i π i. θ i Notice that for the problem to be concave, we need N η h M to be concave and therefore, N to have N η M be weakly concave, we must have η =. In this case V N, W is linear in N, W and therefore weakly convex. References Matthews, S. and J. Moore 987: Monopoly Provision of Quality and Warranties: An Exploration in the Theory of Multidimensional Screening, Econometrica, 55,

26 Pavan, A., I. Segal, and J. Toikka 2009: Dynamic Mechanism Design: Incentive Compatibility, Profit Maximization and Information Disclosure, Working Paper, Northwestern University. Thomas, J. and T. Worrall 990: Income Fluctuation and Asymmetric Information: An Example of a Repeated Principal-agent Problem, Journal of Economic Theory, 5,