Unemployment insurance with limited commitment wage contracts and savings

A.1 Competitive equilibrium

The (stationary) competitive equilibrium consists of a set of value functions {U(a, j), W(a, J)} for unemployed and employed workers respectively, a set of decision rules on asset holdings {a′_e(a, J), a′_u(a, j)}, the firm’s continuation value J′(a, J), and the search intensity rule s(a, j). It also consists of a level of taxes τ and an invariant measure μ of agents across assets, employment status and J such that:

1) Agents optimize: {U(a, j), W(a, J)} solve functional equations (2) and (6) and optimal policies derive.

2) Taxes and benefits are consistent with budget balance: eτ = ∑_{j < m}u_j b_j

3) The measure μ is consistent: In particular the law of motion of μ can be represented as:

where 𝒜 and 𝒥 are subsets of the relevant state space and μ_{(u, 𝒜, j)}, μ_{(e, 𝒜, 𝒥)} are the probability distributions conditional on employment status. ℐ_(x) is an indicator variable which takes the value one if x is true and zero otherwise.^[27]

Proof of Proposition 2 Suppose that we have ϕ₀ = 0 and ϕ_T > 0 for some T. From (7) and (8) we have that: i) c0e=c1e=...=cTe≡c¯e and ctectu=1+(1β−r)rλ≡κ for t = 1, 2,… T.

Note that if the firm’s constraint first binds in period T we have that J₁, J₂, … J_T > 0 and J_{T + 1} = 0. From the optimality conditions we can show that J₁ = J₂ = … = J_T > 0. Suppose the converse; note that from ct−1e=cte and ct−1ect−1u=ctectu we have that consumption and assets are constant. From the agent’s budget constraint it must also be that wages are constant. From the promise keeping constraints it is trivial to show that the Js are constant.

Given this property we can write (J₀, J₁, …, J_T, J_{T + 1}) = (0, J¯, …, J¯, 0). J starts at zero, subsequently remains to some constant value J¯ > 0 for T periods and then drops to 0. The sequence of wages between period 0 and T satisfies: (w0,w1,...,wT−1,wT)=(y+β(1−λ)J¯,y−(1−β(1−λ))J¯,...,y−(1−β(1−λ))J¯,y−J¯).

To simplify assume wlog that r = 1 and τ = 0. We have that c¯e=y−(1−β(1−λ))J¯. Moreover, a₁ = a₂ = … = a_T = a₀ + J¯ ≡ a¯. Let a~ denote the level of assets in T + 1; we obtain: a~=a¯−β(1−λ)J¯. In T + 1 assets drop since wages drop in T.

To show that the worker cannot optimize and choose (a_{T + 1}, J_{T + 1}) = (a~, 0) I proceed in two steps. First, I assume that there is a unique optimum in terms of the policy rule which solves the value function. Otherwise, if the agent is indifferent between (a¯, J¯) and (a~, 0) in period T, she will also be indifferent in period 0 as will become evident subsequently.^[28] We can now rule out T > 1 through claiming that (a~, 0) is obviously inconsistent with this worker’s optimization. Notice that in T − 1 we have:

since the vector (c¯e, a~, 0) is in the constraint set in T − 1. However, in T we have that

In other words, the agent solved the same program in T − 1 as in T and with exactly the same state variables. Unless she is indifferent between the two solutions we have a contradiction.

This contradiction is not obvious in the case where T = 1 (since now W(a_{T − 1}, J_{T − 1}) = W(a₀, 0)). To show the problem in this case note that since ϕ₀ = 0 we have that:

Again (a~, 0) is affordable in period zero since a~ = a₀ + y − c¯e. Now consider the agent’s program in period 1. The agent has two choices: i) to remain in the unconstrained optimum (i.e. (a₂, J₂) = (a¯, J¯)) ^[29], ii) to choose (a₂, J₂) = (a~, 0). In either case consumption in period one must equal c¯e because of (7). Assume that ii) is preferred over i). Then we have:

which contradicts (11). Hence, i) is preferable to ii). ■.

Since the program is recursive it is trivial to generalize the above to have that if ϕ_t = 0 then ϕ_{t + j} = 0 for j > 0.

Proof of Proposition 4 To prove that the optimal contract is a flat contract insofar as the firm’s constraint is binding, I show here that when the constraint binds it binds forever. In other words, the only possibility is to have T = ∞ (where T is the last period that the constraint binds in proposition 4). Then, it is trivial to show that w_t = y ∀t. This can be immediately seen from the firm’s promise keeping constraint and the fact that J_t = 0 ∀t.

Assume that T < ∞. We have that ϕ_t > 0 for t = 0, 1, 2, …, T − 1 but then ϕ_T = 0 and the firm’s constraint never binds again after T + 1.^[30] From the promise keeping constraint of the firm we can show that w₀ = w₁ = … = w_{T − 1} = y. From Proposition 1 we know that subsequently we have w_T ≥ y and w_t = w¯ ≤ y, t = T + 1, T + 2, ….

From the proof of Proposition 1 we know that if the constraint ceases to bind the worker’s wealth follows a_T ≤ a_{T + 1} = a_{T + 2} = … = a~ (a_T < a_{T + 1} if w_T > y) and J_{T + 1} = J_{T + 2} = … = J¯ ≥ 0. Before period T, since the firm’s constraint binds, assets and consumption drop over time (e.g. (7)). Therefore a₀ > a₁ > … > a_T and c0e>c1e>...>cTe=cT+1e=.... Consumption drops over time and remains constant from period T onwards.

From equation (8) for t ≤ T and t > 1 we have that

Note that according to the above equation, it holds that cTecTu<cT+1ecT+1u. Since cTe=cT+1e this implies that cTu>cT+1u. This contradicts the assumption that a_T ≤ a_{T + 1} from the fact that rctu=Ua(at,0). This path is therefore not consistent with the worker’s optimization. There cannot be any T < ∞ such that the firm’s constraint ceases to bind. ■.

The optimality conditions hold only when T = ∞ (put differently, when the firm’s constraint always binds). The fact that the constraint always binds should not be surprising. In standard incomplete market models which feature a buffer stock savings, the level of assets which defines the buffer stock is reached in the limit. Here the optimality conditions suggest that the contract features an analogous property. Assets are reduced by smaller and smaller amounts, and in the limit the firm’s constraint becomes slack.