Technical Appendix for “Unemployment Insurance Eligibility and Wage Dispersion” by Brennan Platt

This Mathematica file provides the algebraic manipulations used in solving the model in this paper, produce its graphs, and compute its comparative statics.  In some cases, the solutions from Mathematica are rearranged in the text for readability; the equivalence of these statements is verified with a FullSimplify command reporting “True”.

Base Model

Steady State Conditions

Worker population is normalized to one.

fn+fl+gn+gl+hn+hl==1

Flows through ineligible unemployment must remain steady.  To flow in, one must be employed and ineligible (gn+gl), and lose your job (δ).  To flow out, one must be unemployed and ineligible (fn), and receive an offer (ϕ).

(gn+gl) δ==ϕ fn

Flows through eligible unemployment must remain steady.  To flow in, one must be employed and eligible (hn+hl), and lose your job (δ).  To flow out, you must be unemployed and eligible (fl), and receive an offer (ϕ) that is at or above your reservation wage of Rl (a).

(hn+hl)δ ==fl ϕ a

Flows through ineligible employment at wn must remain steady.  Workers enter ineligible employment at wage wn only if they were unemployed and ineligible (fn) and were offered the wn wage (ϕ (1-a)).  They leave when they lose their job or gain eligibility (δ+γ).

ϕ (1-a)fn==gn(δ+γ)

Flows through ineligible employment at wl must remain steady.  Workers enter ineligible employment at wage wl if they were unemployed and either ineligible (fn) or lost their eligibility on being hired (λ fl), and were offered the wl wage (ϕ a).  They leave when they lose their job or gain eligibility (δ+γ).

ϕ a (fn+λ fl)==gl(δ+γ)

Flows through eligible employment at either wage must remain steady.  Workers enter eligible employment only if they were employed and ineligible (gn) and became eligible (γ).  They leave when they lose their job (δ).

γ gn==hn δ

γ gl==hl δ

Solution:

The system of equations is easily solved, omitting one of the colinear equations.

FullSimplify[Solve[{(gn+gl) δ==ϕ fn,(hn+hl)δ ==fl (ϕ a), ϕ (1-a)fn==gn(δ+γ),ϕ a (fn+λ fl)==gl(δ+γ),γ gn==hn δ,fn+fl+gn+gl+hn+hl==1},{fn,fl,gn,gl,hn,hl}]]

Bellman Equations

Let Vl[w] be the present value of the utility stream for someone employed at wage w who is already eligible for benefits.  Let Vn[w] be the same for someone who is not yet eligible.  Let Ul be the present value of the utility flow of someone who is unemployed was eligible, while Un is the same for anyone who is ineligible for benefits.  The discount rate is ρ.

ρ Vl[w]== w+δ(Ul-Vl[w])

ρ Vn[w]== w+δ(Un-Vn[w])+γ(Vl[w]-Vn[w])

ρ Ul==b+x+ϕ a((1 - λ) Vl[wl]+λ Vn[wl]-Ul)

ρ Un==x+ϕ(a(Vn[wl]-Un)+(1-a)(Vn[wn]-Un))

In addition, we know that reservation wages are chose for indifference:

(1-λ) Vl[wl]+λ Vn[wl]==Ul

Vn[wn]==Un

Solution:

First, simplify for the eligible employed:

Solve[ρ Vl[w]== w+δ( Ul-Vl[w]),Vl[w]]

Then for the ineligible employed:

FullSimplify[Solve[ρ Vn[w]== w+δ(Un-Vn[w])+γ(Vl[w]-Vn[w])/.Vl[w]->((w+Ul δ)/(δ+ρ)),Vn[w]]]

With the indifference conditions, the unemployed Bellman equations become:

ρ Ul==b+x

ρ Un==x+ϕ(a(Vn[wl]-Un))

which allows solving for the value functions:

FullSimplify[Solve[{ρ Ul==b+x,ρ Un==x+ϕ(a(Vn[wl]-Un))}/.Vn[wl]->((wl+Ul δ)/(δ+ρ))+(((-Ul+Un) δ)/(γ+δ+ρ)),{Ul,Un}]]

FullSimplify[((x ρ (δ+ρ) (γ+δ+ρ)+a (b γ δ+x γ δ+wl ρ (γ+δ+ρ)) ϕ)/(ρ (δ+ρ) (ρ (γ+δ+ρ)+a (γ+ρ) ϕ)))==((b+x)/ρ)-(((γ+δ+ρ) (b (δ+ρ)+a (b-wl+x) ϕ))/((δ+ρ) (ρ (γ+δ+ρ)+a (γ+ρ) ϕ)))]

Then substituting for Ul and Un into the indifference conditions, we obtain expressions for prices:

Simplify[FullSimplify[Solve[{(1-λ) ((wl+Ul δ)/(δ+ρ))+λ(((wl+Ul δ)/(δ+ρ))+(((-Ul+Un) δ)/(γ+δ+ρ)))==Ul,((wn+Ul δ)/(δ+ρ))+(((-Ul+Un) δ)/(γ+δ+ρ))==Un}/.{Ul->((b+x)/ρ),Un->((x ρ (δ+ρ) (γ+δ+ρ)+a (b γ δ+x γ δ+wl ρ (γ+δ+ρ)) ϕ)/(ρ (δ+ρ) (ρ (γ+δ+ρ)+a (γ+ρ) ϕ)))},{wl,wn}]]]

FullSimplify[((-b γ δ+x γ ρ+x δ ρ+x (ρ^2)+a (b+x) (γ+δ λ+ρ) ϕ)/(ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ))==b+x-((b (γ+ρ) (δ+ρ))/(ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ))]

Equal Profits Condition

The steady state profits must equate, defined as the realized profits times the average number of workers employed at w per firm offering that wage. The first term is the profit for firms offering wl, while the second is profit for firms offering wn.

FullSimplify[{(p-wl)((gl+hl)/a),(p-wn)((gn+hn)/(1-a))}/.{wl->b+x+((b δ λ (δ+ρ))/(ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ)),wn->((-b γ δ+x γ ρ+x δ ρ+x (ρ^2)+a (b+x) (γ+δ λ+ρ) ϕ)/(ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ)),gn->-(((-1+a) a (δ^2) λ ϕ)/((γ+δ) (a δ λ (δ+ϕ)+γ (δ+a ϕ)))),gl->((a δ (γ+a δ) λ ϕ)/((γ+δ) (a δ λ (δ+ϕ)+γ (δ+a ϕ)))),hn->-(((-1+a) a γ δ λ ϕ)/((γ+δ) (a δ λ (δ+ϕ)+γ (δ+a ϕ)))),hl->((a γ (γ+δ+(-1+a) δ λ) ϕ)/((γ+δ) (a δ λ (δ+ϕ)+γ (δ+a ϕ))))}]

FullSimplify[((a δ λ ϕ (b γ δ-a b (γ+δ λ+ρ) ϕ+(p-x) (ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ)))/((ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ) (a δ λ (δ+ϕ)+γ (δ+a ϕ))))== (p-x-b(1-((( γ δ+ρ (γ+δ+ρ)))/((ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ)))))((a δ λ ϕ)/q)/.q->a δ λ (δ+ϕ)+γ (δ+a ϕ)]

FullSimplify[(((γ+a δ λ) ϕ ((p-x) (ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ)-b (γ (ρ+a ϕ)+(δ λ+ρ) (δ+ρ+a ϕ))))/((ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ) (a δ λ (δ+ϕ)+γ (δ+a ϕ))))== (p-x-b(1+((δ λ (δ+ρ))/(ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ))))(((γ+a δ λ) ϕ)/q)/.q->a δ λ (δ+ϕ)+γ (δ+a ϕ)]

The following provide the Δ function used in proposition 2, which is simply the difference between the wl and wn profits.

FullSimplify[ (p-x-b(1-((( γ δ+ρ (γ+δ+ρ)))/((ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ)))))((a δ λ ϕ)/q)-(p-x-b(1+((δ λ (δ+ρ))/(ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ))))(((γ+a δ λ) ϕ)/q)]

Full Solution

wn degenerate equilibrium

Here we determine the condition required to sustain a wn degenerate equilibrium (the inequality) as well as the equilibrium solution.

FullSimplify[{(((γ+a δ λ) ϕ ((p-x) (ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ)-b (γ (ρ+a ϕ)+(δ λ+ρ) (δ+ρ+a ϕ))))/((ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ) (a δ λ (δ+ϕ)+γ (δ+a ϕ))))<((a δ λ ϕ (b γ δ-a b (γ+δ λ+ρ) ϕ+(p-x) (ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ)))/((ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ) (a δ λ (δ+ϕ)+γ (δ+a ϕ)))),{wl->b+x+((b δ λ (δ+ρ))/(ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ)),wn->((-b γ δ+x γ ρ+x δ ρ+x (ρ^2)+a (b+x) (γ+δ λ+ρ) ϕ)/(ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ)),fn->((a (δ^2) λ)/(a δ λ (δ+ϕ)+γ (δ+a ϕ))),fl->((γ δ)/(a δ λ (δ+ϕ)+γ (δ+a ϕ))),gn->-(((-1+a) a (δ^2) λ ϕ)/((γ+δ) (a δ λ (δ+ϕ)+γ (δ+a ϕ)))),gl->((a δ (γ+a δ) λ ϕ)/((γ+δ) (a δ λ (δ+ϕ)+γ (δ+a ϕ)))),hn->-(((-1+a) a γ δ λ ϕ)/((γ+δ) (a δ λ (δ+ϕ)+γ (δ+a ϕ)))),hl->((a γ (γ+δ+(-1+a) δ λ) ϕ)/((γ+δ) (a δ λ (δ+ϕ)+γ (δ+a ϕ))))}}/.a->0]

The condition rearranges to:

b (((γ ρ+(δ+ρ) (δ λ+ρ))/( ρ (γ+δ+ρ))))>p-x

wl degenerate equilibrium

Here we determine the condition required to sustain a wl degenerate equilibrium (the inequality) as well as the equilibrium solution.

FullSimplify[{(((γ+a δ λ) ϕ ((p-x) (ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ)-b (γ (ρ+a ϕ)+(δ λ+ρ) (δ+ρ+a ϕ))))/((ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ) (a δ λ (δ+ϕ)+γ (δ+a ϕ))))<((a δ λ ϕ (b γ δ-a b (γ+δ λ+ρ) ϕ+(p-x) (ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ)))/((ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ) (a δ λ (δ+ϕ)+γ (δ+a ϕ)))),{wl->b+x+((b δ λ (δ+ρ))/(ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ)),wn->((-b γ δ+x γ ρ+x δ ρ+x (ρ^2)+a (b+x) (γ+δ λ+ρ) ϕ)/(ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ)),fn->((a (δ^2) λ)/(a δ λ (δ+ϕ)+γ (δ+a ϕ))),fl->((γ δ)/(a δ λ (δ+ϕ)+γ (δ+a ϕ))),gn->-(((-1+a) a (δ^2) λ ϕ)/((γ+δ) (a δ λ (δ+ϕ)+γ (δ+a ϕ)))),gl->((a δ (γ+a δ) λ ϕ)/((γ+δ) (a δ λ (δ+ϕ)+γ (δ+a ϕ)))),hn->-(((-1+a) a γ δ λ ϕ)/((γ+δ) (a δ λ (δ+ϕ)+γ (δ+a ϕ)))),hl->((a γ (γ+δ+(-1+a) δ λ) ϕ)/((γ+δ) (a δ λ (δ+ϕ)+γ (δ+a ϕ))))}}/.a->1]

The condition rearranges to:

b (1+((δ λ (δ+ρ) (2 γ+δ λ+ρ))/(γ ρ (γ+δ+ρ)+γ (γ+δ λ+ρ) ϕ)))<p-x

because:

b (δ λ (δ+ρ) (δ λ+ρ)+γ (δ+ρ) (2 δ λ+ρ)+γ (δ λ+ρ) ϕ+(γ^2) (ρ+ϕ))<(p-x) γ (ρ (γ+δ+ρ)+(γ+δ λ+ρ) ϕ)

b FullSimplify[(δ λ (δ+ρ) (δ λ+ρ)+γ (δ+ρ) (2 δ λ+ρ)+γ (δ λ+ρ) ϕ+(γ^2) (ρ+ϕ))/(γ (ρ (γ+δ+ρ)+(γ+δ λ+ρ) ϕ))]<(p-x)

FullSimplify[((b (δ λ (δ+ρ) (δ λ+ρ)+γ (δ+ρ) (2 δ λ+ρ)+γ (δ λ+ρ) ϕ+(γ^2) (ρ+ϕ)))/(γ ρ (γ+δ+ρ)+γ (γ+δ λ+ρ) ϕ))==b (1+((δ λ (δ+ρ) (2 γ+δ λ+ρ))/(γ ρ (γ+δ+ρ)+γ (γ+δ λ+ρ) ϕ)))]

dispersed equilibrium

We now present the dispersed equilibrium solution.  First, we equate profits and solve for a:

FullSimplify[Solve[(((γ+a δ λ) ϕ ((p-x) (ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ)-b (γ (ρ+a ϕ)+(δ λ+ρ) (δ+ρ+a ϕ))))/((ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ) (a δ λ (δ+ϕ)+γ (δ+a ϕ))))==((a δ λ ϕ (b γ δ-a b (γ+δ λ+ρ) ϕ+(p-x) (ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ)))/((ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ) (a δ λ (δ+ϕ)+γ (δ+a ϕ)))),a]]

FullSimplify[((γ ((p-x) ρ (γ+δ+ρ)-b (γ ρ+(δ+ρ) (δ λ+ρ))))/((γ+δ λ+ρ) (b δ λ (δ+ρ)+(b-p+x) γ ϕ)))==((γ )/(γ+δ λ+ρ))(-((ρ (γ+δ+ρ))/(γ ϕ))+((b δ λ (δ+ρ) (ρ (γ+δ+ρ)-γ ϕ))/(γ ϕ (b δ λ (δ+ρ)+(b-p+x) γ ϕ))))]

Next we verify that it lies between 0 and 1.  For a > 0, this becomes:

FullSimplify[Solve[0==((γ )/(γ+δ λ+ρ))(-((ρ (γ+δ+ρ))/(γ ϕ))+((b δ λ (δ+ρ) (ρ (γ+δ+ρ)-γ ϕ))/(γ ϕ (b δ λ (δ+ρ)+(b-p+x) γ ϕ)))),p]]

FullSimplify[((b (γ ρ+(δ+ρ) (δ λ+ρ)))/(ρ (γ+δ+ρ)))==b(1+(((δ+ρ) δ λ)/(ρ (γ+δ+ρ))))]

The inequality goes the same direction as γ ϕ > ρ (γ+δ+ρ), since that needs to be divided in the rearrangement.

p-x>b(1+(((δ+ρ) δ λ)/(ρ (γ+δ+ρ))))

For a < 1, this becomes:

FullSimplify[Solve[1==((γ )/(γ+δ λ+ρ))(-((ρ (γ+δ+ρ))/(γ ϕ))+((b δ λ (δ+ρ) (ρ (γ+δ+ρ)-γ ϕ))/(γ ϕ (b δ λ (δ+ρ)+(b-p+x) γ ϕ)))),p]]

FullSimplify[((b (δ λ (δ+ρ) (δ λ+ρ)+γ (δ+ρ) (2 δ λ+ρ)+γ (δ λ+ρ) ϕ+(γ^2) (ρ+ϕ)))/(γ ρ (γ+δ+ρ)+γ (γ+δ λ+ρ) ϕ))==b(1+((δ λ (δ+ρ) (2 γ+δ λ+ρ))/(γ ρ (γ+δ+ρ)+γ (γ+δ λ+ρ) ϕ)))]

Again, the inequality goes the same direction as γ ϕ > ρ (γ+δ+ρ).

p-x>b(1+((δ λ (δ+ρ) (2 γ+δ λ+ρ))/(γ ρ (γ+δ+ρ)+γ (γ+δ λ+ρ) ϕ)))

Note that the conditions for 0 < a < 1 are the same as the conditions for the degenerate equilibria, with the sign agreeing when γ ϕ > ρ (γ+δ+ρ) and being opposite otherwise.  Thus we can either get uniqueness when this does not hold, or up to three equilibria when it does.  Also, if it is exactly equal and we have exactly the right parameters, then any a* will satisfy equilibrium.

FullSimplify[(((γ+a δ λ) ϕ ((p-x) (ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ)-b (γ (ρ+a ϕ)+(δ λ+ρ) (δ+ρ+a ϕ))))/((ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ) (a δ λ (δ+ϕ)+γ (δ+a ϕ))))-((a δ λ ϕ (b γ δ-a b (γ+δ λ+ρ) ϕ+(p-x) (ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ)))/((ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ) (a δ λ (δ+ϕ)+γ (δ+a ϕ))))/.p->x+b(1+((δ λ (δ+ρ))/(ρ (γ+δ+ρ))))/.ϕ-> ρ (γ+δ+ρ)/γ ]

Numeric Examples

Calibration

The average unemployment duration (2014-2017) for ineligible workers is 18.03 weeks

ϕ->52/18.03

The average unemployment duration for eligible workers is 21.48 weeks

a->(52/21.48)/ϕ/.%

Set γ = 1 (one year to get eligibility) and λ = 1 (not retained)

The unemployment rate averages 5.45%

Solve[((δ (γ+a δ λ))/(a δ λ (δ+ϕ)+γ (δ+a ϕ)))==0.0545/.ϕ->2.8840820854132003`/.a->0.8393854748603352`/.γ->1/.λ->1,δ]

We will normalize p = 1 and set the discount rate to 10%.

The replacement rate is 50%, and the equilibrium condition on a* must hold

FullSimplify[Solve[{b/(b+x+((b δ λ (δ+ρ) (-ρ+a (γ+δ λ+ρ)))/((γ+δ λ) (ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ))))==(1/2),((γ (-((ρ (γ+δ+ρ))/(γ ϕ))+((b δ λ (δ+ρ) (ρ (γ+δ+ρ)-γ ϕ))/(γ ϕ (b δ λ (δ+ρ)+(b-p+x) γ ϕ)))))/(γ+δ λ+ρ))==a},{x,b}]/.ϕ->2.8840820854132003`/.a->0.8393854748603352`/.γ->1/.λ->1/.δ->0.1423923515533254`/.p->1/.ρ->0.1]

Predicted statistics

This will yield a fraction of eligible workers as 89%

N[(γ/(γ+a δ λ))==L/.ϕ->2.8840820854132003`/.a->0.8393854748603352`/.γ->1/.λ->1/.δ->0.1423923515533254`]

This implies a wage range of 5% and a differential between eligible and ineligible of 0.8%:

N[{-1+((b+x+((b δ λ (δ+ρ))/(ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ)))/(b+x+((b (δ+ρ) (a δ λ+(-1+a) (γ+ρ)))/(ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ)))),-1+((b+x+((b δ λ (δ+ρ))/(ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ)))/((-b γ δ+x γ ρ+x δ ρ+x (ρ^2)+a (b+x) (γ+δ λ+ρ) ϕ)/(ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ)))}/.ϕ->2.8840820854132003`/.a->0.8393854748603352`/.γ->1/.λ->1/.δ->0.1423923515533254`/.p->1/.ρ->0.1/.x->0.49215053585919816`/.b->0.49666795430626337`]

Case 3 occurs when:

N[Solve[ ρ (γ+δ+ρ)==γ ϕ,γ]/.ϕ->2.8840820854132003`/.a->0.8393854748603352`/.λ->1/.δ->0.1423923515533254`/.p->1/.ρ->0.1/.x->0.49215053585919816`/.b->0.49666795430626337`]

Figures

The following generates Figure 1 for the paper at the calibrated parameters.

Clear[p,x,δ,γ,λ,ρ,ϕ]

p=1;
x=0.49215053585919816`;
λ=1;
δ=0.1423923515533254`;
ρ=0.1;
ϕ=2.8840820854132003`;
Grid[{{"γ=0.004","γ=0.0087","γ=1"},{γ=0.004;Plot[{If[ (b γ ρ-(p-x) ρ (γ+δ+ρ)+b (δ+ρ) (δ λ+ρ)) ϕ>0,0],If[(-(p-x) γ (ρ (γ+δ+ρ)+(γ+δ λ+ρ) ϕ)+b (δ λ (δ+ρ) (δ λ+ρ)+γ (δ+ρ) (2 δ λ+ρ)+γ (δ λ+ρ) ϕ+(γ^2) (ρ+ϕ)))<0,1],If[0≤((γ (-((ρ (γ+δ+ρ))/(γ ϕ))+((b δ λ (δ+ρ) (ρ (γ+δ+ρ)-γ ϕ))/(γ ϕ (b δ λ (δ+ρ)+(b-p+x) γ ϕ)))))/(γ+δ λ+ρ))≤1,((γ (-((ρ (γ+δ+ρ))/(γ ϕ))+((b δ λ (δ+ρ) (ρ (γ+δ+ρ)-γ ϕ))/(γ ϕ (b δ λ (δ+ρ)+(b-p+x) γ ϕ)))))/(γ+δ λ+ρ))]},{b,0,0.3},PlotRange->Full,AxesLabel->{"b","a"},AxesOrigin->{0,-0.02},PlotStyle->{Directive[Darker[Red],Thick,Dashing[{.05,.03}]],Directive[Darker[Blue],Thick,Dotted],Directive[Darker[Green],Thick]}],γ=0.0087;
Show[Plot[{If[ (b γ ρ-(p-x) ρ (γ+δ+ρ)+b (δ+ρ) (δ λ+ρ)) ϕ>0,0],If[(-(p-x) γ (ρ (γ+δ+ρ)+(γ+δ λ+ρ) ϕ)+b (δ λ (δ+ρ) (δ λ+ρ)+γ (δ+ρ) (2 δ λ+ρ)+γ (δ λ+ρ) ϕ+(γ^2) (ρ+ϕ)))<0,1]},{b,0,0.3},PlotRange->Full,AxesLabel->{"b","a"},AxesOrigin->{0,-0.02},PlotStyle->{Directive[Darker[Red],Thick,Dashing[{.05,.03}]],Directive[Darker[Blue],Thick,Dotted],Directive[Darker[Green],Thick]}],ListPlot[{{(p-x)/(((γ ρ+(δ+ρ) (δ λ+ρ))/( ρ (γ+δ+ρ)))),1},{(p-x)/(((γ ρ+(δ+ρ) (δ λ+ρ))/( ρ (γ+δ+ρ)))),0}},Joined->True,PlotStyle->Directive[Darker[Green],Thick]]],γ=1;
Plot[{If[ (b γ ρ-(p-x) ρ (γ+δ+ρ)+b (δ+ρ) (δ λ+ρ)) ϕ>0,0],If[(-(p-x) γ (ρ (γ+δ+ρ)+(γ+δ λ+ρ) ϕ)+b (δ λ (δ+ρ) (δ λ+ρ)+γ (δ+ρ) (2 δ λ+ρ)+γ (δ λ+ρ) ϕ+(γ^2) (ρ+ϕ)))<0,1],If[0≤((γ (-((ρ (γ+δ+ρ))/(γ ϕ))+((b δ λ (δ+ρ) (ρ (γ+δ+ρ)-γ ϕ))/(γ ϕ (b δ λ (δ+ρ)+(b-p+x) γ ϕ)))))/(γ+δ λ+ρ))≤1,((γ (-((ρ (γ+δ+ρ))/(γ ϕ))+((b δ λ (δ+ρ) (ρ (γ+δ+ρ)-γ ϕ))/(γ ϕ (b δ λ (δ+ρ)+(b-p+x) γ ϕ)))))/(γ+δ λ+ρ))]},{b,0.30,0.6},PlotRange->Full,AxesOrigin->{0.30,-0.02},AxesLabel->{"b","a"},PlotStyle->{Directive[Darker[Red],Thick,Dashing[{.05,.03}]],Directive[Darker[Blue],Thick,Dotted],Directive[Darker[Green],Thick]}]},{γ=0.004;Plot[{If[ (b γ ρ-(p-x) ρ (γ+δ+ρ)+b (δ+ρ) (δ λ+ρ)) ϕ>0,{b+x+((b δ λ (δ+ρ))/(ρ (γ+δ+ρ))),x-((b γ δ)/(ρ (γ+δ+ρ)))}],If[(-(p-x) γ (ρ (γ+δ+ρ)+(γ+δ λ+ρ) ϕ)+b (δ λ (δ+ρ) (δ λ+ρ)+γ (δ+ρ) (2 δ λ+ρ)+γ (δ λ+ρ) ϕ+(γ^2) (ρ+ϕ)))<0,{b+x+((b δ λ (δ+ρ))/(ρ (γ+δ+ρ)+(γ+δ λ+ρ) ϕ)),((-b γ δ+x γ ρ+x δ ρ+x (ρ^2)+(b+x) (γ+δ λ+ρ) ϕ)/(ρ (γ+δ+ρ)+(γ+δ λ+ρ) ϕ))}],If[0≤((γ (-((ρ (γ+δ+ρ))/(γ ϕ))+((b δ λ (δ+ρ) (ρ (γ+δ+ρ)-γ ϕ))/(γ ϕ (b δ λ (δ+ρ)+(b-p+x) γ ϕ)))))/(γ+δ λ+ρ))≤1,{((b γ ρ+x ρ (γ+δ+ρ)+b (δ+ρ) (δ λ+ρ)-p γ ϕ)/(ρ (γ+δ+ρ)-γ ϕ)),((δ λ (-b γ δ+x ρ (γ+δ+ρ))-γ (-p (γ+ρ)+b (γ+δ λ+ρ)+x (γ+δ λ+ρ)) ϕ)/(δ λ (ρ (γ+δ+ρ)-γ ϕ)))}]},{b,0,0.3},PlotRange->Full,AxesLabel->{"b","w"},PlotStyle->{Directive[Darker[Red],Thick,Dashing[{.05,.03}]],Directive[Darker[Blue],Thick,Dotted],Directive[Darker[Green],Thick]}],γ=0.0087;
Show[Plot[{If[ (b γ ρ-(p-x) ρ (γ+δ+ρ)+b (δ+ρ) (δ λ+ρ)) ϕ>0,{b+x+((b δ λ (δ+ρ))/(ρ (γ+δ+ρ))),x-((b γ δ)/(ρ (γ+δ+ρ)))}],If[(-(p-x) γ (ρ (γ+δ+ρ)+(γ+δ λ+ρ) ϕ)+b (δ λ (δ+ρ) (δ λ+ρ)+γ (δ+ρ) (2 δ λ+ρ)+γ (δ λ+ρ) ϕ+(γ^2) (ρ+ϕ)))<0,{b+x+((b δ λ (δ+ρ))/(ρ (γ+δ+ρ)+(γ+δ λ+ρ) ϕ)),((-b γ δ+x γ ρ+x δ ρ+x (ρ^2)+(b+x) (γ+δ λ+ρ) ϕ)/(ρ (γ+δ+ρ)+(γ+δ λ+ρ) ϕ))}]},{b,0,0.3},PlotRange->Full,AxesLabel->{"b","w"},PlotStyle->{Directive[Darker[Red],Thick,Dashing[{.05,.03}]],Directive[Darker[Blue],Thick,Dotted],Directive[Darker[Green],Thick]}],ListPlot[{{(p-x)/(((γ ρ+(δ+ρ) (δ λ+ρ))/( ρ (γ+δ+ρ)))),x+(((p-x) ρ (γ+δ+ρ))/(γ ρ+(δ+ρ) (δ λ+ρ)))+(((p-x) δ λ ρ (δ+ρ) (γ+δ+ρ))/((γ ρ+(δ+ρ) (δ λ+ρ)) (ρ (γ+δ+ρ)+(γ+δ λ+ρ) ϕ)))},{(p-x)/(((γ ρ+(δ+ρ) (δ λ+ρ))/( ρ (γ+δ+ρ)))),p}},Joined->True,PlotStyle->Directive[Darker[Green],Thick]],ListPlot[{{(p-x)/(((γ ρ+(δ+ρ) (δ λ+ρ))/( ρ (γ+δ+ρ)))),x+(((-p+x) γ δ)/(γ ρ+(δ+ρ) (δ λ+ρ)))},{(p-x)/(((γ ρ+(δ+ρ) (δ λ+ρ))/( ρ (γ+δ+ρ)))),((x (δ+ρ) (γ+δ λ+ρ) (ρ (γ+δ+ρ)+δ λ ϕ)+p ρ (γ+δ+ρ) (-γ δ+(γ+δ λ+ρ) ϕ))/((γ ρ+(δ+ρ) (δ λ+ρ)) (ρ (γ+δ+ρ)+(γ+δ λ+ρ) ϕ)))}},Joined->True,PlotStyle->Directive[Darker[Green],Thick]]],γ=1;
Plot[{If[ (b γ ρ-(p-x) ρ (γ+δ+ρ)+b (δ+ρ) (δ λ+ρ)) ϕ>0,{b+x+((b δ λ (δ+ρ))/(ρ (γ+δ+ρ))),x-((b γ δ)/(ρ (γ+δ+ρ)))}],If[(-(p-x) γ (ρ (γ+δ+ρ)+(γ+δ λ+ρ) ϕ)+b (δ λ (δ+ρ) (δ λ+ρ)+γ (δ+ρ) (2 δ λ+ρ)+γ (δ λ+ρ) ϕ+(γ^2) (ρ+ϕ)))<0,{b+x+((b δ λ (δ+ρ))/(ρ (γ+δ+ρ)+(γ+δ λ+ρ) ϕ)),((-b γ δ+x γ ρ+x δ ρ+x (ρ^2)+(b+x) (γ+δ λ+ρ) ϕ)/(ρ (γ+δ+ρ)+(γ+δ λ+ρ) ϕ))}],If[0≤((γ (-((ρ (γ+δ+ρ))/(γ ϕ))+((b δ λ (δ+ρ) (ρ (γ+δ+ρ)-γ ϕ))/(γ ϕ (b δ λ (δ+ρ)+(b-p+x) γ ϕ)))))/(γ+δ λ+ρ))≤1,{((b γ ρ+x ρ (γ+δ+ρ)+b (δ+ρ) (δ λ+ρ)-p γ ϕ)/(ρ (γ+δ+ρ)-γ ϕ)),((δ λ (-b γ δ+x ρ (γ+δ+ρ))-γ (-p (γ+ρ)+b (γ+δ λ+ρ)+x (γ+δ λ+ρ)) ϕ)/(δ λ (ρ (γ+δ+ρ)-γ ϕ)))}]},{b,0.35,0.55},PlotRange->Full,AxesLabel->{"b","w"},PlotStyle->{Directive[Darker[Red],Thick,Dashing[{.05,.03}]],Directive[Darker[Blue],Thick,Dotted],Directive[Darker[Green],Thick]}]}}]

Comparative Statics

The reported signs in the dispersed equilibria all assume ρ (γ+δ+ρ)<γ ϕ, and will reverse otherwise (unless noted differently).

Summary of equilibrium variables:

Dispersed a*

MixedPercentOffering=((γ ((p-x) ρ (γ+δ+ρ)-b (γ ρ+(δ+ρ) (δ λ+ρ))))/((γ+δ λ+ρ) (b δ λ (δ+ρ)+(b-p+x) γ ϕ)))

Eligible Equilibrium Wage

EligWage=b+x+((b δ λ (δ+ρ))/(ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ))

Non-eligible Equilibrium Wage

IneligWage=((-b γ δ+x γ ρ+x δ ρ+x (ρ^2)+a (b+x) (γ+δ λ+ρ) ϕ)/(ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ))

Unemployment Rate

UnemploymentRate=((δ (γ+a δ λ))/(a δ λ (δ+ϕ)+γ (δ+a ϕ)))

Fraction of Unemployed Who Are Eligible

FracEligUnemployed=(γ/(γ+a δ λ))

Expected Wage Offer

ExpectedWageOffer=((x ρ (γ+δ+ρ)+a x (γ+δ λ+ρ) ϕ+b ((-1+a) γ δ+a γ (ρ+ϕ)+a (δ λ+ρ) (δ+ρ+ϕ)))/(ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ))

Substitutions for a*

It is useful to substitute for p to determine the sign unambiguously.

FullSimplify[Solve[a==((γ (-((ρ (γ+δ+ρ))/(γ ϕ))+((b δ λ (δ+ρ) (ρ (γ+δ+ρ)-γ ϕ))/(γ ϕ (b δ λ (δ+ρ)+(b-p+x) γ ϕ)))))/(γ+δ λ+ρ)),p]]

FullSimplify[x+((b (a δ λ (δ+ρ) (δ λ+ρ)+γ (δ+ρ) ((1+a) δ λ+ρ)+a γ (δ λ+ρ) ϕ+(γ^2) (ρ+a ϕ)))/(γ ρ (γ+δ+ρ)+a γ (γ+δ λ+ρ) ϕ))==b+x+((b δ λ (δ+ρ) (γ+a γ+a δ λ+a ρ))/(γ ρ (γ+δ+ρ)+a γ (γ+δ λ+ρ) ϕ))]

Effect of x

Degenerate - Eligible

Eligible Wage = +

FullSimplify[D[EligWage/.a->1,x]]

Non-eligible Wage = +

FullSimplify[D[IneligWage/.a->1,x]]

Expected Wage Offer = +

FullSimplify[D[ExpectedWageOffer/.a->1,x]]

Unemployment Rate = 0

FullSimplify[D[UnemploymentRate/.a->1,x]]

Fraction of Unemployed Who Are Eligible = 0

FullSimplify[D[FracEligUnemployed/.a->1,x]]

Degenerate - Ineligible

Eligible Wage = +

FullSimplify[D[EligWage/.a->0,x]]

Non-eligible Wage = +

FullSimplify[D[IneligWage/.a->0,x]]

Expected Wage Offer = +

FullSimplify[D[ExpectedWageOffer/.a->0,x]]

Unemployment Rate = 0

FullSimplify[D[UnemploymentRate/.a->0,x]]

Fraction of Unemployed Who Are Eligible = 0

FullSimplify[D[FracEligUnemployed/.a->0,x]]

Dispersed

Fraction offering high wage = +

FullSimplify[D[MixedPercentOffering,x]/.p->b+x+((b δ λ (δ+ρ) (γ+a γ+a δ λ+a ρ))/(γ ρ (γ+δ+ρ)+a γ (γ+δ λ+ρ) ϕ))]

Eligible Wage = -

FullSimplify[D[EligWage/.a->((γ ((p-x) ρ (γ+δ+ρ)-b (γ ρ+(δ+ρ) (δ λ+ρ))))/((γ+δ λ+ρ) (b δ λ (δ+ρ)+(b-p+x) γ ϕ))),x]]

Non-eligible Wage = + (requires lower γ to reverse)

FullSimplify[D[IneligWage/.a->((γ ((p-x) ρ (γ+δ+ρ)-b (γ ρ+(δ+ρ) (δ λ+ρ))))/((γ+δ λ+ρ) (b δ λ (δ+ρ)+(b-p+x) γ ϕ))),x]]

1-((γ (γ+ρ) ϕ)/(δ λ (ρ (γ+δ+ρ)-γ ϕ)))

Expected Wage Offer = + (requires lower γ to reverse)

FullSimplify[D[ExpectedWageOffer/.a->((γ ((p-x) ρ (γ+δ+ρ)-b (γ ρ+(δ+ρ) (δ λ+ρ))))/((γ+δ λ+ρ) (b δ λ (δ+ρ)+(b-p+x) γ ϕ))),x]]

1-((γ (ρ (γ+δ+ρ)+(γ+ρ) ϕ))/(δ λ (ρ (γ+δ+ρ)-γ ϕ)))

Unemployment Rate  = -

FullSimplify[D[UnemploymentRate/.a->((γ ((p-x) ρ (γ+δ+ρ)-b (γ ρ+(δ+ρ) (δ λ+ρ))))/((γ+δ λ+ρ) (b δ λ (δ+ρ)+(b-p+x) γ ϕ))),x]/.p->b+x+((b δ λ (δ+ρ) (γ+a γ+a δ λ+a ρ))/(γ ρ (γ+δ+ρ)+a γ (γ+δ λ+ρ) ϕ))]

Fraction of Unemployed Who Are Eligible = + under condition 1; - under condition 2 inequality; 0 under condition 2 equality

FullSimplify[D[FracEligUnemployed/.a->((γ ((p-x) ρ (γ+δ+ρ)-b (γ ρ+(δ+ρ) (δ λ+ρ))))/((γ+δ λ+ρ) (b δ λ (δ+ρ)+(b-p+x) γ ϕ))),x]/.p->b+x+((b δ λ (δ+ρ) (γ+a γ+a δ λ+a ρ))/(γ ρ (γ+δ+ρ)+a γ (γ+δ λ+ρ) ϕ))]

Effect of b

Degenerate - Eligible

Eligible Wage = +

FullSimplify[D[EligWage/.a->1,b]]

Non-eligible Wage = depends on additional condition.

FullSimplify[D[IneligWage/.a->1,b]]

Unemployment Rate = 0

FullSimplify[D[UnemploymentRate/.a->1,b]]

Fraction of Unemployed Who Are Eligible = 0

FullSimplify[D[FracEligUnemployed/.a->1,b]]

Expected Wage Offer = +

FullSimplify[D[ExpectedWageOffer/.a->1,b]]

Degenerate - Ineligible

Eligible Wage = +

FullSimplify[D[EligWage/.a->0,b]]

Non-eligible Wage = -

FullSimplify[D[IneligWage/.a->0,b]]

Unemployment Rate = 0

FullSimplify[D[UnemploymentRate/.a->0,b]]

Fraction of Unemployed Who Are Eligible = 0

FullSimplify[D[FracEligUnemployed/.a->0,b]]

Expected Wage Offer = -

FullSimplify[D[ExpectedWageOffer/.a->0,b]]

Dispersed

% Offering Eligible Wage = +

FullSimplify[D[MixedPercentOffering,b]/.p->b+x+((b δ λ (δ+ρ) (γ+a γ+a δ λ+a ρ))/(γ ρ (γ+δ+ρ)+a γ (γ+δ λ+ρ) ϕ))]

Eligible Wage = -

FullSimplify[D[EligWage/.a->((γ (-((ρ (γ+δ+ρ))/(γ ϕ))+((b δ λ (δ+ρ) (ρ (γ+δ+ρ)-γ ϕ))/(γ ϕ (b δ λ (δ+ρ)+(b-p+x) γ ϕ)))))/(γ+δ λ+ρ)),b]]

Non-eligible Wage = +

FullSimplify[D[IneligWage/.a->((γ (-((ρ (γ+δ+ρ))/(γ ϕ))+((b δ λ (δ+ρ) (ρ (γ+δ+ρ)-γ ϕ))/(γ ϕ (b δ λ (δ+ρ)+(b-p+x) γ ϕ)))))/(γ+δ λ+ρ)),b]]

Expected Wage Offer = +

FullSimplify[D[ExpectedWageOffer/.a->((γ (-((ρ (γ+δ+ρ))/(γ ϕ))+((b δ λ (δ+ρ) (ρ (γ+δ+ρ)-γ ϕ))/(γ ϕ (b δ λ (δ+ρ)+(b-p+x) γ ϕ)))))/(γ+δ λ+ρ)),b]]

Unemployment Rate = -

FullSimplify[D[UnemploymentRate/.a->((γ (-((ρ (γ+δ+ρ))/(γ ϕ))+((b δ λ (δ+ρ) (ρ (γ+δ+ρ)-γ ϕ))/(γ ϕ (b δ λ (δ+ρ)+(b-p+x) γ ϕ)))))/(γ+δ λ+ρ)),b]/.p->b+x+((b δ λ (δ+ρ) (γ+a γ+a δ λ+a ρ))/(γ ρ (γ+δ+ρ)+a γ (γ+δ λ+ρ) ϕ))]

FullSimplify[(γ (γ+δ λ) ϕ (ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ) (a δ λ (δ+ρ) (δ λ+ρ)+γ (δ+ρ) ((1+a) δ λ+ρ)+a γ (δ λ+ρ) ϕ+(γ^2) (ρ+a ϕ)))/(b λ (δ+ρ) (γ+δ λ+ρ) (ρ (γ+δ+ρ)-γ ϕ) (((a δ λ (δ+ϕ)+γ (δ+a ϕ)))^2))==((γ (γ+δ λ) ϕ (ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ))/((ρ (γ+δ+ρ)-γ ϕ))) ((δ λ (δ+ρ) (γ+a( γ+δ λ+ρ))+ (γ ρ (γ+δ+ρ)+a γ (γ+δ λ+ρ) ϕ))/(b λ (δ+ρ) (γ+δ λ+ρ) (((a δ λ (δ+ϕ)+γ (δ+a ϕ)))^2) ))]

Fraction of Unemployed Who Are Eligible = -

FullSimplify[D[FracEligUnemployed/.a->((γ (-((ρ (γ+δ+ρ))/(γ ϕ))+((b δ λ (δ+ρ) (ρ (γ+δ+ρ)-γ ϕ))/(γ ϕ (b δ λ (δ+ρ)+(b-p+x) γ ϕ)))))/(γ+δ λ+ρ)),b]/.p->b+x+((b δ λ (δ+ρ) (γ+a γ+a δ λ+a ρ))/(γ ρ (γ+δ+ρ)+a γ (γ+δ λ+ρ) ϕ))]

FullSimplify[((γ (ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ) (a δ λ (δ+ρ) (δ λ+ρ)+γ (δ+ρ) ((1+a) δ λ+ρ)+a γ (δ λ+ρ) ϕ+(γ^2) (ρ+a ϕ)))/(b (((γ+a δ λ))^2) (δ+ρ) (γ+δ λ+ρ) (ρ (γ+δ+ρ)-γ ϕ)))==((γ(ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ))/( (ρ (γ+δ+ρ)-γ ϕ))) ((δ λ (δ+ρ) (γ+a γ+a δ λ+a ρ)+γ  (ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ))/(b (((γ+a δ λ))^2) (δ+ρ) (γ+δ λ+ρ) ))]

Effect of λ

Degenerate - Eligible

Eligible Wage = +

FullSimplify[D[EligWage/.a->1,λ]]

Non-eligible Wage = +

FullSimplify[D[IneligWage/.a->1,λ]]

Expected Wage Offer = +

FullSimplify[D[ExpectedWageOffer/.a->1,λ]]

Unemployment Rate = 0

FullSimplify[D[UnemploymentRate/.a->1,λ]]

Fraction of Unemployed Who Are Eligible = -

FullSimplify[D[FracEligUnemployed/.a->1,λ]]

Degenerate - Ineligible

Eligible Wage = +

FullSimplify[D[EligWage/.a->0,λ]]

Non-eligible Wage = +

FullSimplify[D[IneligWage/.a->0,λ]]

Expected Wage Offer = +

FullSimplify[D[ExpectedWageOffer/.a->0,λ]]

Unemployment Rate = -

FullSimplify[D[UnemploymentRate/.a->0,λ]]

Fraction of Unemployed Who Are Eligible = -

FullSimplify[D[FracEligUnemployed/.a->0,λ]]

Dispersed

% Offering Eligible Wage = +

FullSimplify[D[MixedPercentOffering,λ]/.p->b+x+((b δ λ (δ+ρ) (γ+a γ+a δ λ+a ρ))/(γ ρ (γ+δ+ρ)+a γ (γ+δ λ+ρ) ϕ))]

Eligible Wage = -

FullSimplify[D[EligWage/.a->((γ (-((ρ (γ+δ+ρ))/(γ ϕ))+((b δ λ (δ+ρ) (ρ (γ+δ+ρ)-γ ϕ))/(γ ϕ (b δ λ (δ+ρ)+(b-p+x) γ ϕ)))))/(γ+δ λ+ρ)),λ]]

Non-eligible Wage = +

FullSimplify[D[IneligWage/.a->((γ (-((ρ (γ+δ+ρ))/(γ ϕ))+((b δ λ (δ+ρ) (ρ (γ+δ+ρ)-γ ϕ))/(γ ϕ (b δ λ (δ+ρ)+(b-p+x) γ ϕ)))))/(γ+δ λ+ρ)),λ]/.p->b+x+((b δ λ (δ+ρ) (γ+a γ+a δ λ+a ρ))/(γ ρ (γ+δ+ρ)+a γ (γ+δ λ+ρ) ϕ))]

Expected Wage Offer = +

FullSimplify[D[ExpectedWageOffer/.a->((γ (-((ρ (γ+δ+ρ))/(γ ϕ))+((b δ λ (δ+ρ) (ρ (γ+δ+ρ)-γ ϕ))/(γ ϕ (b δ λ (δ+ρ)+(b-p+x) γ ϕ)))))/(γ+δ λ+ρ)),λ]/.p->b+x+((b δ λ (δ+ρ) (γ+a γ+a δ λ+a ρ))/(γ ρ (γ+δ+ρ)+a γ (γ+δ λ+ρ) ϕ))]

Unemployment Rate = -

FullSimplify[D[UnemploymentRate/.a->((γ (-((ρ (γ+δ+ρ))/(γ ϕ))+((b δ λ (δ+ρ) (ρ (γ+δ+ρ)-γ ϕ))/(γ ϕ (b δ λ (δ+ρ)+(b-p+x) γ ϕ)))))/(γ+δ λ+ρ)),λ]/.p->b+x+((b δ λ (δ+ρ) (γ+a γ+a δ λ+a ρ))/(γ ρ (γ+δ+ρ)+a γ (γ+δ λ+ρ) ϕ))]

FullSimplify[γ δ ϕ((a δ λ ((γ ρ+a (δ λ+ρ) (γ+δ λ+ρ)) ϕ)+((γ+δ λ) (1+a)+a ρ )(a δ λ ρ (γ+δ+ρ)+γ( ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ)))/(λ (γ+δ λ+ρ) (ρ (γ+δ+ρ)-γ ϕ) (((a δ λ (δ+ϕ)+γ (δ+a ϕ)))^2)))==(γ δ ϕ (γ ρ ((1+a) (δ^2) λ (1+a+a λ)+δ (a+λ+2 a (1+a) λ) ρ+a (ρ^2))+a γ ((1+2 a) (δ^2) (λ^2)+(2+3 a) δ λ ρ+a (ρ^2)) ϕ+(1+a) (γ^3) (ρ+a ϕ)+a δ λ (ρ (δ+ρ) ((1+a) δ λ+a ρ)+a (((δ λ+ρ))^2) ϕ)+(γ^2) ((1+2 a) ρ (ρ+a ϕ)+(1+a) δ (ρ+λ ρ+a λ ρ+2 a λ ϕ))))/(λ (γ+δ λ+ρ) (ρ (γ+δ+ρ)-γ ϕ) (((a δ λ (δ+ϕ)+γ (δ+a ϕ)))^2))]

Fraction of Unemployed Who Are Eligible = -

FullSimplify[D[FracEligUnemployed/.a->((γ (-((ρ (γ+δ+ρ))/(γ ϕ))+((b δ λ (δ+ρ) (ρ (γ+δ+ρ)-γ ϕ))/(γ ϕ (b δ λ (δ+ρ)+(b-p+x) γ ϕ)))))/(γ+δ λ+ρ)),λ]/.p->b+x+((b δ λ (δ+ρ) (γ+a γ+a δ λ+a ρ))/(γ ρ (γ+δ+ρ)+a γ (γ+δ λ+ρ) ϕ))]

FullSimplify[((γ δ ((γ+a δ λ) ρ (γ+δ+ρ)+a ((2+a) (γ^2)+(1+2 a) γ δ λ+2 (1+a) γ ρ+a (((δ λ+ρ))^2)) ϕ))/((((γ+a δ λ))^2) (γ+δ λ+ρ) (ρ (γ+δ+ρ)-γ ϕ)))==((γ δ (a (γ+ρ) (γ+a (γ+δ λ+ρ)) ϕ+(γ+a δ λ)( ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ)))/((((γ+a δ λ))^2) (γ+δ λ+ρ) (ρ (γ+δ+ρ)-γ ϕ)))]

Effect of ϕ

Degenerate - Eligible

Eligible Wage = -

FullSimplify[D[EligWage/.a->1,ϕ]]

Non-eligible Wage = +

FullSimplify[D[IneligWage/.a->1,ϕ]]

Expected Wage Offer = -

FullSimplify[D[ExpectedWageOffer/.a->1,ϕ]]

Unemployment Rate = -

FullSimplify[D[UnemploymentRate/.a->1,ϕ]]

Fraction of Unemployed Who Are Eligible = 0

FullSimplify[D[FracEligUnemployed/.a->1,ϕ]]

Degenerate - Ineligible

Eligible Wage = 0

FullSimplify[D[EligWage/.a->0,ϕ]]

Non-eligible Wage = 0

FullSimplify[D[IneligWage/.a->0,ϕ]]

Unemployment Rate = 0

FullSimplify[D[UnemploymentRate/.a->0,ϕ]]

Fraction of Unemployed Who Are Eligible = 0

FullSimplify[D[FracEligUnemployed/.a->0,ϕ]]

Expected Wage Offer = 0

FullSimplify[D[ExpectedWageOffer/.a->0,ϕ]]

Average Employed Worker Wage = 0

FullSimplify[D[AverageEmployedWage/.a->0,ϕ]]

Dispersed

% Offering Eligible Wage = -

FullSimplify[D[MixedPercentOffering,ϕ]/.p->b+x+((b δ λ (δ+ρ) (γ+a γ+a δ λ+a ρ))/(γ ρ (γ+δ+ρ)+a γ (γ+δ λ+ρ) ϕ))]

Eligible Wage = +

FullSimplify[D[EligWage/.a->((γ (-((ρ (γ+δ+ρ))/(γ ϕ))+((b δ λ (δ+ρ) (ρ (γ+δ+ρ)-γ ϕ))/(γ ϕ (b δ λ (δ+ρ)+(b-p+x) γ ϕ)))))/(γ+δ λ+ρ)),ϕ]/.p->b+x+((b δ λ (δ+ρ) (γ+a γ+a δ λ+a ρ))/(γ ρ (γ+δ+ρ)+a γ (γ+δ λ+ρ) ϕ))]

Non-eligible Wage = -

FullSimplify[D[IneligWage/.a->((γ (-((ρ (γ+δ+ρ))/(γ ϕ))+((b δ λ (δ+ρ) (ρ (γ+δ+ρ)-γ ϕ))/(γ ϕ (b δ λ (δ+ρ)+(b-p+x) γ ϕ)))))/(γ+δ λ+ρ)),ϕ]/.p->b+x+((b δ λ (δ+ρ) (γ+a γ+a δ λ+a ρ))/(γ ρ (γ+δ+ρ)+a γ (γ+δ λ+ρ) ϕ))]

Expected Wage Offer = -

FullSimplify[D[ExpectedWageOffer/.a->((γ (-((ρ (γ+δ+ρ))/(γ ϕ))+((b δ λ (δ+ρ) (ρ (γ+δ+ρ)-γ ϕ))/(γ ϕ (b δ λ (δ+ρ)+(b-p+x) γ ϕ)))))/(γ+δ λ+ρ)),ϕ]/.p->b+x+((b δ λ (δ+ρ) (γ+a γ+a δ λ+a ρ))/(γ ρ (γ+δ+ρ)+a γ (γ+δ λ+ρ) ϕ))]

Unemployment Rate = +

FullSimplify[D[UnemploymentRate/.a-> ((γ (-((ρ (γ+δ+ρ))/(γ ϕ))+((b δ λ (δ+ρ) (ρ (γ+δ+ρ)-γ ϕ))/(γ ϕ (b δ λ (δ+ρ)+(b-p+x) γ ϕ)))))/(γ+δ λ+ρ)),ϕ]/.p->b+x+((b δ λ (δ+ρ) (γ+a γ+a δ λ+a ρ))/(γ ρ (γ+δ+ρ)+a γ (γ+δ λ+ρ) ϕ))]

Fraction of Unemployed Who Are Eligible = +

FullSimplify[D[FracEligUnemployed/.a->((γ (-((ρ (γ+δ+ρ))/(γ ϕ))+((b δ λ (δ+ρ) (ρ (γ+δ+ρ)-γ ϕ))/(γ ϕ (b δ λ (δ+ρ)+(b-p+x) γ ϕ)))))/(γ+δ λ+ρ)),ϕ]/.p->b+x+((b δ λ (δ+ρ) (γ+a γ+a δ λ+a ρ))/(γ ρ (γ+δ+ρ)+a γ (γ+δ λ+ρ) ϕ))]

Effect of γ

Degenerate - Eligible

Eligible Wage = -

FullSimplify[D[EligWage/.a->1,γ]]

Non-eligible Wage = -

FullSimplify[D[IneligWage/.a->1,γ]]

Expected Wage Offer = -

FullSimplify[D[ExpectedWageOffer/.a->1,γ]]

Unemployment Rate = 0

FullSimplify[D[UnemploymentRate/.a->1,γ]]

Fraction of Unemployed Who Are Eligible = -

FullSimplify[D[FracEligUnemployed/.a->1,γ]]

Degenerate - Ineligible

Eligible Wage = -

FullSimplify[D[EligWage/.a->0,γ]]

Non-eligible Wage = -

FullSimplify[D[IneligWage/.a->0,γ]]

Expected Wage Offer = -

FullSimplify[D[ExpectedWageOffer/.a->0,γ]]

Unemployment Rate = 0

FullSimplify[D[UnemploymentRate/.a->0,γ]]

Fraction of Unemployed Who Are Eligible = 0

FullSimplify[D[FracEligUnemployed/.a->0,γ]]

Dispersed

% Offering Eligible Wage = -

FullSimplify[D[MixedPercentOffering,γ]/.p->b+x+((b δ λ (δ+ρ) (γ+a γ+a δ λ+a ρ))/(γ ρ (γ+δ+ρ)+a γ (γ+δ λ+ρ) ϕ))]

FullSimplify[(((1+a) (γ^2) (ρ+a ϕ)+2 a γ (δ λ+ρ) (ρ+a ϕ)+a (δ λ+ρ) (ρ (δ+ρ)+a (δ λ+ρ) ϕ))/(γ (γ+δ λ+ρ) (ρ (γ+δ+ρ)-γ ϕ)))==((( γ+ a  (γ+δ λ+ρ) )γ(ρ+a ϕ)+a (δ λ+ρ) (ρ (γ +δ+ρ)+a (γ+δ λ+ρ) ϕ))/(γ (γ+δ λ+ρ) (ρ (γ+δ+ρ)-γ ϕ)))]

Eligible Wage = +

FullSimplify[D[EligWage/.a->((γ (-((ρ (γ+δ+ρ))/(γ ϕ))+((b δ λ (δ+ρ) (ρ (γ+δ+ρ)-γ ϕ))/(γ ϕ (b δ λ (δ+ρ)+(b-p+x) γ ϕ)))))/(γ+δ λ+ρ)),γ]/.p->b+x+((b δ λ (δ+ρ) (γ+a γ+a δ λ+a ρ))/(γ ρ (γ+δ+ρ)+a γ (γ+δ λ+ρ) ϕ))]

Non-eligible Wage = -

FullSimplify[D[IneligWage/.a->((γ (-((ρ (γ+δ+ρ))/(γ ϕ))+((b δ λ (δ+ρ) (ρ (γ+δ+ρ)-γ ϕ))/(γ ϕ (b δ λ (δ+ρ)+(b-p+x) γ ϕ)))))/(γ+δ λ+ρ)),γ]/.p->b+x+((b δ λ (δ+ρ) (γ+a γ+a δ λ+a ρ))/(γ ρ (γ+δ+ρ)+a γ (γ+δ λ+ρ) ϕ))]

FullSimplify[-((b (δ+ρ) )/(γ (ρ (γ+δ+ρ)+ (γ+δ λ+ρ) ϕ a)))(γ+(((γ+ρ) (γ ρ+ (γ+δ λ+ρ) ϕ a))/(γ ϕ-ρ (γ+δ+ρ))))==((b (δ+ρ) (-γ δ ρ+((1+a) (γ^2)+a ρ (δ λ+ρ)+a γ (δ λ+2 ρ)) ϕ))/(γ (ρ (γ+δ+ρ)-γ ϕ) (ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ)))]

Expected Wage Offer = -

FullSimplify[D[ExpectedWageOffer/.a->((γ (-((ρ (γ+δ+ρ))/(γ ϕ))+((b δ λ (δ+ρ) (ρ (γ+δ+ρ)-γ ϕ))/(γ ϕ (b δ λ (δ+ρ)+(b-p+x) γ ϕ)))))/(γ+δ λ+ρ)),γ]/.p->b+x+((b δ λ (δ+ρ) (γ+a γ+a δ λ+a ρ))/(γ ρ (γ+δ+ρ)+a γ (γ+δ λ+ρ) ϕ))]

FullSimplify[((b (δ+ρ) (γ ρ (δ (-1+a+a λ)+2 a ρ)+a γ (δ λ+2 ρ) ϕ+(1+a) (γ^2) (ρ+ϕ)+a ρ (δ λ+ρ) (δ+ρ+ϕ)))/(γ (ρ (γ+δ+ρ)-γ ϕ) (ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ)))==-((b (δ+ρ) )/(γ (ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ)))(((a (γ+δ λ+ρ) (γ (ρ+ϕ)+ρ (δ+ρ+ϕ))+γ ρ (2 γ+ρ))/((γ ϕ-ρ (γ+δ+ρ)) ))+γ)]

Unemployment Rate =  +

FullSimplify[D[UnemploymentRate/.a->((γ (-((ρ (γ+δ+ρ))/(γ ϕ))+((b δ λ (δ+ρ) (ρ (γ+δ+ρ)-γ ϕ))/(γ ϕ (b δ λ (δ+ρ)+(b-p+x) γ ϕ)))))/(γ+δ λ+ρ)),γ]/.p->b+x+((b δ λ (δ+ρ) (γ+a γ+a δ λ+a ρ))/(γ ρ (γ+δ+ρ)+a γ (γ+δ λ+ρ) ϕ))]

((δ ϕ (ρ ((γ+δ λ) ((1+a) (γ^2)+a δ (γ+a γ+a δ) λ)+a (2 γ+δ+δ λ) (γ+a δ λ) ρ+a (γ+a δ λ) (ρ^2))+a ((1+a) (γ^3)+a δ λ (((δ λ+ρ))^2)+2 (γ^2) ((1+a) δ λ+a ρ)+γ (δ λ+ρ) (δ (λ+2 a λ)+a ρ)) ϕ))/((γ+δ λ+ρ) (γ ϕ-ρ (γ+δ+ρ)) (((a δ λ (δ+ϕ)+γ (δ+a ϕ)))^2)))

FullSimplify[(γ^2) δ (γ+δ λ) ρ ϕ]+FullSimplify[a δ ϕ ((γ^2) (γ+δ λ) ρ+γ δ λ (γ+δ λ) ρ+γ (2 γ+δ+δ λ) (ρ^2)+γ (ρ^3)+(γ^3) ϕ+2 (γ^2) δ λ ϕ+γ δ λ (δ λ+ρ) ϕ)]+FullSimplify[(a^2) δ ϕ (γ δ λ (γ+δ λ) ρ+(δ^2) λ (γ+δ λ) ρ+δ λ (2 γ+δ+δ λ) (ρ^2)+δ λ (ρ^3)+(γ^3) ϕ+2 (γ^2) δ λ ϕ+2 (γ^2) ρ ϕ+2 γ δ λ (δ λ+ρ) ϕ+γ ρ (δ λ+ρ) ϕ+δ λ (((δ λ+ρ))^2) ϕ)]

(γ^2) δ (γ+δ λ) ρ ϕ+(a^2) δ (γ+δ λ+ρ) ϕ (δ λ ρ (γ+δ+ρ)+((γ^2)+(γ+δ λ) (δ λ+ρ)) ϕ)+a γ δ ϕ ((ρ+δ+2 γ) (ρ^2)+((((γ+δ λ))^2)+δ λ ρ )(ρ+ϕ))

FullSimplify[-((δ ϕ (ρ ((γ+δ λ) ((1+a) (γ^2)+a δ (γ+a γ+a δ) λ)+a (2 γ+δ+δ λ) (γ+a δ λ) ρ+a (γ+a δ λ) (ρ^2))+a ((1+a) (γ^3)+a δ λ (((δ λ+ρ))^2)+2 (γ^2) ((1+a) δ λ+a ρ)+γ (δ λ+ρ) (δ (λ+2 a λ)+a ρ)) ϕ))/((γ+δ λ+ρ) (ρ (γ+δ+ρ)-γ ϕ) (((a δ λ (δ+ϕ)+γ (δ+a ϕ)))^2)))==((γ^2) δ (γ+δ λ) ρ ϕ+(a^2) δ (γ+δ λ+ρ) ϕ (δ λ ρ (γ+δ+ρ)+((γ^2)+(γ+δ λ) (δ λ+ρ)) ϕ)+a γ δ ϕ ((ρ+δ+2 γ) (ρ^2)+((((γ+δ λ))^2)+δ λ ρ )(ρ+ϕ)))/((γ+δ λ+ρ) (γ ϕ-ρ (γ+δ+ρ)) (((a δ λ (δ+ϕ)+γ (δ+a ϕ)))^2))]

Fraction of Unemployed Who Are Eligible = +

FullSimplify[D[FracEligUnemployed/.a->((γ (-((ρ (γ+δ+ρ))/(γ ϕ))+((b δ λ (δ+ρ) (ρ (γ+δ+ρ)-γ ϕ))/(γ ϕ (b δ λ (δ+ρ)+(b-p+x) γ ϕ)))))/(γ+δ λ+ρ)),γ]/.p->b+x+((b δ λ (δ+ρ) (γ+a γ+a δ λ+a ρ))/(γ ρ (γ+δ+ρ)+a γ (γ+δ λ+ρ) ϕ))]

FullSimplify[((δ λ (-γ (γ+a δ (-1+λ)) ρ-a ((2+a) (γ^2)+(1+2 a) γ (δ λ+ρ)+a (((δ λ+ρ))^2)) ϕ))/((((γ+a δ λ))^2) (γ+δ λ+ρ) (ρ (γ+δ+ρ)-γ ϕ)))==((δ λ )/((((γ+δ λ a))^2) (γ+δ λ+ρ)))((((γ+a γ+a δ λ+a ρ) (γ ρ+a (γ+δ λ+ρ) ϕ))/((γ ϕ-ρ (γ+δ+ρ))))+γ a)]

Numeric size:

Manipulate[Grid[{{"% scale","","da","dwl","dwn","du"},{"1% of x","dx",-((γ (((ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ))^2))/(b δ λ (δ+ρ) (γ+δ λ+ρ) (ρ (γ+δ+ρ)-γ ϕ)))(0.01x),((ρ (γ+δ+ρ))/(ρ (γ+δ+ρ)-γ ϕ))(0.01x),(1-((γ (γ+ρ) ϕ)/(δ λ (ρ (γ+δ+ρ)-γ ϕ))))(0.01x),(((γ+δ λ) ϕ (((γ ρ (γ+δ+ρ)+a γ (γ+δ λ+ρ) ϕ))^2))/(b λ (δ+ρ) (γ+δ λ+ρ) (ρ (γ+δ+ρ)-γ ϕ) (((a δ λ (δ+ϕ)+γ (δ+a ϕ)))^2)))(0.01x)},{"1% of b","db",-(((ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ) (a δ λ (δ+ρ) (δ λ+ρ)+γ (δ+ρ) ((1+a) δ λ+ρ)+a γ (δ λ+ρ) ϕ+(γ^2) (ρ+a ϕ)))/(b δ λ (δ+ρ) (γ+δ λ+ρ) (ρ (γ+δ+ρ)-γ ϕ)))(0.01b),((γ ρ+(δ+ρ) (δ λ+ρ))/(ρ (γ+δ+ρ)-γ ϕ))(0.01b),-((γ ((δ^2) λ+(γ+δ λ+ρ) ϕ))/(δ λ (ρ (γ+δ+ρ)-γ ϕ)))(0.01b),((γ (γ+δ λ) ϕ (ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ))/((ρ (γ+δ+ρ)-γ ϕ))) ((δ λ (δ+ρ) (γ+a( γ+δ λ+ρ))+ (γ ρ (γ+δ+ρ)+a γ (γ+δ λ+ρ) ϕ))/(b λ (δ+ρ) (γ+δ λ+ρ) (((a δ λ (δ+ϕ)+γ (δ+a ϕ)))^2) ))(0.01b)},{"10% of λ","dλ",-((ρ (γ+δ+ρ) (γ+a γ+2 a δ λ+a ρ)+a ((1+a) (γ^2)+a (((δ λ+ρ))^2)+γ (ρ+2 a (δ λ+ρ))) ϕ)/(λ (γ+δ λ+ρ) (ρ (γ+δ+ρ)-γ ϕ)))(0.1λ),((b δ (δ+ρ))/(ρ (γ+δ+ρ)-γ ϕ))(0.1λ),-((b (γ+ρ) (δ+ρ) (γ+a γ+a δ λ+a ρ) ϕ)/(λ (ρ (γ+δ+ρ)-γ ϕ) (ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ)))(0.1λ),(0.1λ)γ δ ϕ((a δ λ ((γ ρ+a (δ λ+ρ) (γ+δ λ+ρ)) ϕ)+((γ+δ λ) (1+a)+a ρ )(a δ λ ρ (γ+δ+ρ)+γ( ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ)))/(λ (γ+δ λ+ρ) (ρ (γ+δ+ρ)-γ ϕ) (((a δ λ (δ+ϕ)+γ (δ+a ϕ)))^2)))},{"10% of ϕ","dϕ",((a (γ+a γ+a δ λ+a ρ))/(ρ (γ+δ+ρ)-γ ϕ))(0.1ϕ),-((a b δ λ (δ+ρ) (γ+δ λ+ρ))/((ρ (γ+δ+ρ)-γ ϕ) (ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ)))(0.1ϕ),((a b (γ+ρ) (δ+ρ) (γ+δ λ+ρ))/((ρ (γ+δ+ρ)-γ ϕ) (ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ)))(0.1ϕ),((a δ (γ+δ λ) (-(γ+a δ λ) ρ (γ+δ+ρ)-a γ (γ+ρ) ϕ))/((ρ (γ+δ+ρ)-γ ϕ) (((a δ λ (δ+ϕ)+γ (δ+a ϕ)))^2)))(0.1ϕ)},{"10% of γ","dγ",((( γ+ a  (γ+δ λ+ρ) )γ(ρ+a ϕ)+a (δ λ+ρ) (ρ (γ +δ+ρ)+a (γ+δ λ+ρ) ϕ))/(γ (γ+δ λ+ρ) (ρ (γ+δ+ρ)-γ ϕ)))(0.1γ),-((b δ λ (δ+ρ) (γ ρ+a (γ+δ λ+ρ) ϕ))/(γ (ρ (γ+δ+ρ)-γ ϕ) (ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ)))(0.1γ),-((b (δ+ρ) )/(γ (ρ (γ+δ+ρ)+ (γ+δ λ+ρ) ϕ a)))(γ+(((γ+ρ) (γ ρ+ (γ+δ λ+ρ) ϕ a))/(γ ϕ-ρ (γ+δ+ρ))))(0.1γ),-(0.1γ)((δ ϕ (ρ ((γ+δ λ) ((1+a) (γ^2)+a δ (γ+a γ+a δ) λ)+a (2 γ+δ+δ λ) (γ+a δ λ) ρ+a (γ+a δ λ) (ρ^2))+a ((1+a) (γ^3)+a δ λ (((δ λ+ρ))^2)+2 (γ^2) ((1+a) δ λ+a ρ)+γ (δ λ+ρ) (δ (λ+2 a λ)+a ρ)) ϕ))/((γ+δ λ+ρ) (ρ (γ+δ+ρ)-γ ϕ) (((a δ λ (δ+ϕ)+γ (δ+a ϕ)))^2)))}}/.a->((γ ((p-x) ρ (γ+δ+ρ)-b (γ ρ+(δ+ρ) (δ λ+ρ))))/((γ+δ λ+ρ) (b δ λ (δ+ρ)+(b-p+x) γ ϕ)))],{{p,1},0,1},{{x,0.49215053585919816`},0,1},{{b,0.49666795430626337`},0,1},{{γ,1},0,1},{{δ,0.1423923515533254`},0.001,0.4},{{ϕ,2.8840820854132003`},0,4},{{ρ,0.1},0.001,0.2},{{λ,1},0,1},{{b,0.49666795430626337`},0,1}]

Endogenous Arrival

Bellman Equations

Employment duration does not depend on the type of worker hired, so the value of employing someone at wage w is:

Solve[ρ Jl==p-wl+δ(Kl-Jl),Jl]

Solve[ρ Jn==p-wn+δ(Kn-Jn),Jn]

The expected profit from a vacancy among firms targeting eligible workers is:

FullSimplify[Solve[ρ Kl==-k+α (Jl-Kl)/.Jl->((p-wl+Kl δ)/(δ+ρ)),Kl]]

The expected profit from a vacancy among firms targeting ineligible workers is:

FullSimplify[Solve[ρ Kn==-k+α(fn/(fl+fn)) (Jn-Kn)/.Jn->((p-wn+Kn δ)/(δ+ρ)),Kn]]

In a dispersed equilibrium, both must equal zero:

FullSimplify[Solve[((p α-wl α-k (δ+ρ))/(ρ (α+δ+ρ)))==0,α]]

FullSimplify[Solve[((-fl k (δ+ρ)+fn (p α-wn α-k (δ+ρ)))/(ρ (fl (δ+ρ)+fn (α+δ+ρ))))==0,α]]

And therefore:

FullSimplify[Solve[(((fl+fn) k (δ+ρ))/(fn (p-wn)))==((k (δ+ρ))/(p-wl)),wl]]

Which is equivalent to the equal profit condition of the base model:

FullSimplify[p-wl==(fn/(fn+fl))(p-wn)/.wl->((fl p+fn wn)/(fl+fn))]

Matching Rates

To endogenize α and ϕ, we now use the matching function M(u,v) = A min[u,v].  Thus, if u < v, then ϕ==A and α==A ((u/v)).  We proceed as if u < v, then verify whether this holds in equilibrium.

Dispersed, and u < v

If u < v, then α->((k (δ+ρ))/(p-wl)) and  α==A ((u/v)) and ϕ = A.  First,

((k (δ+ρ))/(p-wl))==A ((u/v))<A

Using the equilibrium value of wl, we obtain:

FullSimplify[((k (δ+ρ))/(p-wl))<A/.wl->((b γ ρ+x ρ (γ+δ+ρ)+b (δ+ρ) (δ λ+ρ)-p γ ϕ)/(ρ (γ+δ+ρ)-γ ϕ))/.ϕ->A]

which rearranges as:

k (δ+ρ)<A(((b (δ+ρ) δ λ-(p-x-b) ρ (γ+δ+ρ))/(γ A-ρ (γ+δ+ρ))))

Thus, the base model solution holds if this condition holds.

Dispersed, and u > v

If u > v, then  α->((k (δ+ρ))/(p-wl)) and  α==A  and ϕ = A ((v/u)).  Again, we use the equilibrium value of wl:

FullSimplify[Solve[((k (δ+ρ))/(p-wl))== A/.wl->((b γ ρ+x ρ (γ+δ+ρ)+b (δ+ρ) (δ λ+ρ)-p γ ϕ)/(ρ (γ+δ+ρ)-γ ϕ))/.ϕ->A(v/u),v]]

u>v==((u (((ρ (γ+δ+ρ))/A)+((b γ ρ-(p-x) ρ (γ+δ+ρ)+b (δ+ρ) (δ λ+ρ))/(k (δ+ρ)))))/γ)

which rearranges to

A(p-x-b+((A (b-p+x) γ+b δ λ (δ+ρ))/(A γ-ρ(γ+δ+ρ))))<k (δ+ρ)

So if this condition applies, then the base model solution holds, except with:

ϕ ==A ((v/u))==  ((ρ (γ+δ+ρ))/(γ A))+(A/(γ k ))(b δ λ-(((p-x-b) ρ (γ+δ+ρ))/(δ+ρ)))

Eligible degenerate, and u < v

As before,

((k (δ+ρ))/(p-wl))==A ((u/v))<A

All that changes is wl:

FullSimplify[((k (δ+ρ))/(p-wl))<A/.wl->b+x+((b δ λ (δ+ρ))/(ρ (γ+δ+ρ)+(γ+δ λ+ρ) ϕ))/.ϕ->A]

k (δ+ρ)<A(p-x-b-((b δ λ (δ+ρ))/(ρ (γ+δ+ρ)+A (γ+δ λ+ρ))))

If this condition holds, then the previous solution applies.

Ineligible degenerate

In this setting, the match rate is poorly defined because no one accepts any offers made in equilibrium, so effectively there is no reason to make any offers of either type.  If so, then v->0 while u->1.  Thus u > v for sure if such an equilibrium occurs.  Then α = A and ϕ = 0.

v->0

Here, we must use Kn = 0 to define α->(((fl+fn) k (δ+ρ))/(fn (p-wn)))and α==A ((u/v)).

(((fl+fn) k (δ+ρ))/(fn (p-wn)))==A ((u/v))<A

We then insert the wn-degenerate solution:

FullSimplify[(fl+fn) k (δ+ρ)<A fn (p-wn)/.wn->((-b γ δ+x γ ρ+x δ ρ+x (ρ^2)+a (b+x) (γ+δ λ+ρ) ϕ)/(ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ))/.fn->0/.fl->1/.ϕ->A]

So that can’t happen; profits would always be negative.  In other words, no firms will want to offer this.  But the also don’t want to offer wl.

FullSimplify[((p α-wl α-k (δ+ρ))/(ρ (α+δ+ρ)))<0/.α->A/.wl->b+x+((b δ λ (δ+ρ))/(ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ))/.a->0]

k (δ+ρ)>A (p-b-x-((b δ λ (δ+ρ))/(ρ (γ+δ+ρ))))

If this condition holds, then the ineligible degenerate solution can arise in which no one offers anything.

Deterministic Eligibility

Steady State Conditions

Worker population is normalized to one.

fn+fl+gn+gl+hn+hl==1

No change to flows through unemployment.

(gn+gl) δ==ϕ fn

(hn+hl)δ ==fl ϕ a

Flows from unemployment to ineligible employment at wn are:

ϕ (1-a)fn==zn[0]

Flows from unemployment to ineligible employment at wl are:

ϕ a (fn+λ fl)==zl[0]

Flows through eligible employment are:

zn[T]==hn δ

zl[T]==hl δ

In between these two, people lose their job at rate δ

DSolve[{zn'[t]==-δ zn[t],zn[T]==hn δ},zn[t],t]

So

ϕ (1-a)fn (E^(-T δ))==hn δ

ϕ a (fn+λ fl)(E^(-T δ))==hl δ

Flows through eligible employment at either wage must remain steady.  Workers enter eligible employment only if they were employed and ineligible (gn) and became eligible (γ).  They leave when they lose their job (δ).

γ gn==hn δ

γ gl==hl δ

Integrating across all these ineligible workers people, we can get gn:

gn==(∫_0^T)(E^(-t δ+T δ)) hn δ\[DifferentialD]t

gl==(∫_0^T)(E^(-t δ+T δ)) hl δ\[DifferentialD]t

Solution

FullSimplify[Solve[{(gl+gn) δ==ϕ fn,(hn+hl)δ ==fl (ϕ a), ϕ (1-a)fn (E^(-T δ))==hn δ, ϕ a (fn+λ fl)(E^(-T δ))==hl δ,gn==(-1+(E^(T δ))) hn,fn+fl+gl+gn+hn+hl==1},{fn,fl,gl,gn,hn,hl}]]

Note that these are equivalent to the base model solution if γ->((δ λ)/((e^(T δ))-λ)).

FullSimplify[Solve[(δ/(δ+ϕ+((δ+a ϕ)/(a (E^(T δ))-a λ))))==((a (δ^2) λ)/(a δ λ (δ+ϕ)+γ (δ+a ϕ))),γ]]

Bellman Equations

The Bellman equations are only changed for the ineligible:

ρ Vn[w,t]== w+δ(U0-Vn[w,t])+Derivative[0,1][Vn][w,t]

This solves as:

FullSimplify[DSolve[{ρ Vn[w,t]== w+δ(Un-Vn[w,t])+Derivative[0,1][Vn][w,t],Vn[w,T]==((w+Ul δ)/(δ+ρ))},Vn[w,t],t]]

At time t = 0 (when the worker is newly hired), this means:

Vn[w,0]->((w+Ul δ)/(δ+ρ))+(((1-(E^(-T (δ+ρ))))(Un-Ul) δ)/(δ+ρ))

So the Bellman equations would be identical to those of the base model if γ->((δ+ρ)/(-1+(e^(T (δ+ρ))))), so if ρ = 0 and λ = 1 this holds for both models.

Solve[(((1-(E^(-T (δ+ρ))))(Un-Ul) δ)/(δ+ρ))==(((-Ul+Un) δ)/(γ+δ+ρ)),γ]

Solution:

From the indifference conditions, we get:

FullSimplify[Solve[{ρ Ul==b+x,ρ Un==x+ϕ(a(Vn[wl,0]-Un))}/.Vn[wl,0]->((wl+Ul δ)/(δ+ρ))+(((1-(E^(-T (δ+ρ))))(Un-Ul) δ)/(δ+ρ)),{Ul,Un}]]

Then substituting for Ul and Un, we obtain expressions for prices:

Simplify[FullSimplify[Solve[{(1-λ) ((wl+Ul δ)/(δ+ρ))+λ(((wl+Ul δ)/(δ+ρ))+(((1-(E^(-T (δ+ρ))))(Un-Ul) δ)/(δ+ρ)))==Ul,((wn+Ul δ)/(δ+ρ))+(((1-(E^(-T (δ+ρ))))(Un-Ul) δ)/(δ+ρ))==Un}/.{Ul->((b+x)/ρ),Un->((a (b+x) δ ϕ+(E^(T (δ+ρ))) ρ (x (δ+ρ)+a wl ϕ))/(ρ (a δ ϕ+(E^(T (δ+ρ))) ρ (δ+ρ+a ϕ))))},{wl,wn}]]]

Equal Profits Condition

FullSimplify[{((gl+hl)/a)(p-wl),((gn+hn)/(1-a))(p-wn)}/.{wl->((-b δ λ (δ+ρ)-a (b+x) δ (-1+λ) ϕ+(E^(T (δ+ρ))) ((δ+ρ) (b δ λ+(b+x) ρ)+a (b+x) (δ λ+ρ) ϕ))/(-a δ (-1+λ) ϕ+(E^(T (δ+ρ))) (ρ (δ+ρ)+a (δ λ+ρ) ϕ))),wn->((-b δ (δ+ρ)-a (b+x) δ (-1+λ) ϕ+(E^(T (δ+ρ))) (x ρ (δ+ρ)+a (b+x) (δ λ+ρ) ϕ))/(-a δ (-1+λ) ϕ+(E^(T (δ+ρ))) (ρ (δ+ρ)+a (δ λ+ρ) ϕ))),gl->((a (E^(-T δ)) (1+a (-1+(E^(T δ)))) ((E^(T δ))-λ) ϕ)/(δ-a δ λ+a ϕ-a λ ϕ+a (E^(T δ)) (δ+ϕ))),gn->-(((-1+a) a (E^(-T δ)) (-1+(E^(T δ))) ((E^(T δ))-λ) ϕ)/(δ-a δ λ+a ϕ-a λ ϕ+a (E^(T δ)) (δ+ϕ))),hn->-(((-1+a) a (E^(-T δ)) ((E^(T δ))-λ) ϕ)/(δ-a δ λ+a ϕ-a λ ϕ+a (E^(T δ)) (δ+ϕ))),hl->((a (E^(-T δ)) (a ((E^(T δ))-λ)+λ) ϕ)/(δ-a δ λ+a ϕ-a λ ϕ+a (E^(T δ)) (δ+ϕ)))}]

Full Solution

Finding the dispersed solution

FullSimplify[Solve[(((-1+a (-(E^(T δ))+λ)) ϕ (-b δ λ (δ+ρ)-a (b-p+x) δ (-1+λ) ϕ+(E^(T (δ+ρ))) ((δ+ρ) (b δ λ+(b-p+x) ρ)+a (b-p+x) (δ λ+ρ) ϕ)))/((δ-a δ λ+a ϕ-a λ ϕ+a (E^(T δ)) (δ+ϕ)) (-a δ (-1+λ) ϕ+(E^(T (δ+ρ))) (ρ (δ+ρ)+a (δ λ+ρ) ϕ))))==((a ((E^(T δ))-λ) ϕ (b δ (δ+ρ)+a (b-p+x) δ (-1+λ) ϕ+(E^(T (δ+ρ))) ((p-x) ρ (δ+ρ)-a (b-p+x) (δ λ+ρ) ϕ)))/((δ-a δ λ+a ϕ-a λ ϕ+a (E^(T δ)) (δ+ϕ)) (-a δ (-1+λ) ϕ+(E^(T (δ+ρ))) (ρ (δ+ρ)+a (δ λ+ρ) ϕ)))),a]]

FullSimplify[{wl->((-b δ λ (δ+ρ)-a (b+x) δ (-1+λ) ϕ+(E^(T (δ+ρ))) ((δ+ρ) (b δ λ+(b+x) ρ)+a (b+x) (δ λ+ρ) ϕ))/(-a δ (-1+λ) ϕ+(E^(T (δ+ρ))) (ρ (δ+ρ)+a (δ λ+ρ) ϕ))),wn->((-b δ (δ+ρ)-a (b+x) δ (-1+λ) ϕ+(E^(T (δ+ρ))) (x ρ (δ+ρ)+a (b+x) (δ λ+ρ) ϕ))/(-a δ (-1+λ) ϕ+(E^(T (δ+ρ))) (ρ (δ+ρ)+a (δ λ+ρ) ϕ)))}/.a->-(((δ+ρ) (-b δ λ+(E^(T (δ+ρ))) (b δ λ+(b-p+x) ρ)))/((δ-δ λ+(E^(T (δ+ρ))) (δ λ+ρ)) ((-p+x) ϕ+b (((E^(T δ))-λ) (δ+ρ)+ϕ))))]

And the wn degenerate solution

FullSimplify[{(((-1+a (-(E^(T δ))+λ)) ϕ (-b δ λ (δ+ρ)-a (b-p+x) δ (-1+λ) ϕ+(E^(T (δ+ρ))) ((δ+ρ) (b δ λ+(b-p+x) ρ)+a (b-p+x) (δ λ+ρ) ϕ)))/((δ-a δ λ+a ϕ-a λ ϕ+a (E^(T δ)) (δ+ϕ)) (-a δ (-1+λ) ϕ+(E^(T (δ+ρ))) (ρ (δ+ρ)+a (δ λ+ρ) ϕ))))<((a ((E^(T δ))-λ) ϕ (b δ (δ+ρ)+a (b-p+x) δ (-1+λ) ϕ+(E^(T (δ+ρ))) ((p-x) ρ (δ+ρ)-a (b-p+x) (δ λ+ρ) ϕ)))/((δ-a δ λ+a ϕ-a λ ϕ+a (E^(T δ)) (δ+ϕ)) (-a δ (-1+λ) ϕ+(E^(T (δ+ρ))) (ρ (δ+ρ)+a (δ λ+ρ) ϕ)))),wl->((-b δ λ (δ+ρ)-a (b+x) δ (-1+λ) ϕ+(E^(T (δ+ρ))) ((δ+ρ) (b δ λ+(b+x) ρ)+a (b+x) (δ λ+ρ) ϕ))/(-a δ (-1+λ) ϕ+(E^(T (δ+ρ))) (ρ (δ+ρ)+a (δ λ+ρ) ϕ))),wn->((-b δ (δ+ρ)-a (b+x) δ (-1+λ) ϕ+(E^(T (δ+ρ))) (x ρ (δ+ρ)+a (b+x) (δ λ+ρ) ϕ))/(-a δ (-1+λ) ϕ+(E^(T (δ+ρ))) (ρ (δ+ρ)+a (δ λ+ρ) ϕ)))}/.a->0]

And the wl degenerate solution

FullSimplify[{(((-1+a (-(E^(T δ))+λ)) ϕ (-b δ λ (δ+ρ)-a (b-p+x) δ (-1+λ) ϕ+(E^(T (δ+ρ))) ((δ+ρ) (b δ λ+(b-p+x) ρ)+a (b-p+x) (δ λ+ρ) ϕ)))/((δ-a δ λ+a ϕ-a λ ϕ+a (E^(T δ)) (δ+ϕ)) (-a δ (-1+λ) ϕ+(E^(T (δ+ρ))) (ρ (δ+ρ)+a (δ λ+ρ) ϕ))))>((a ((E^(T δ))-λ) ϕ (b δ (δ+ρ)+a (b-p+x) δ (-1+λ) ϕ+(E^(T (δ+ρ))) ((p-x) ρ (δ+ρ)-a (b-p+x) (δ λ+ρ) ϕ)))/((δ-a δ λ+a ϕ-a λ ϕ+a (E^(T δ)) (δ+ϕ)) (-a δ (-1+λ) ϕ+(E^(T (δ+ρ))) (ρ (δ+ρ)+a (δ λ+ρ) ϕ)))),wl->((-b δ λ (δ+ρ)-a (b+x) δ (-1+λ) ϕ+(E^(T (δ+ρ))) ((δ+ρ) (b δ λ+(b+x) ρ)+a (b+x) (δ λ+ρ) ϕ))/(-a δ (-1+λ) ϕ+(E^(T (δ+ρ))) (ρ (δ+ρ)+a (δ λ+ρ) ϕ))),wn->((-b δ (δ+ρ)-a (b+x) δ (-1+λ) ϕ+(E^(T (δ+ρ))) (x ρ (δ+ρ)+a (b+x) (δ λ+ρ) ϕ))/(-a δ (-1+λ) ϕ+(E^(T (δ+ρ))) (ρ (δ+ρ)+a (δ λ+ρ) ϕ)))}/.a->1]

Recalibration

The average unemployment duration (2014-2017) for ineligible workers is 18.03 weeks

ϕ->52/18.03

The average unemployment duration for eligible workers is 21.48 weeks

a->(52/21.48)/ϕ/.%

Set T = 1 (one year to get eligibility) and λ = 1 (not retained)

The unemployment rate averages 5.45%

FindRoot[fn+fl==0.0545/.fn->(δ/(δ+ϕ+((δ+a ϕ)/(a (E^(T δ))-a λ))))/.fl->(δ/(δ-a δ λ+a ϕ-a λ ϕ+a (E^(T δ)) (δ+ϕ)))/.ϕ->2.8840820854132003`/.a->0.8393854748603352`/.T->1/.λ->1,{δ,0.15}]

We will normalize p = 1 and set the discount rate to 10%.

The replacement rate is 50%, and the equilibrium condition on a must hold

FullSimplify[(((hl+gl)(Rl)+(hn+gn)Rn)/(hl+gl+hn+gn))/.{Rl->b+x+((b (-1+(E^(T (δ+ρ)))) δ λ (δ+ρ))/(-a δ (-1+λ) ϕ+(E^(T (δ+ρ))) (ρ (ρ+a ϕ)+δ (ρ+a λ ϕ)))),Rn->((b (-(δ^2)-δ ρ+a δ (1+(-1+(E^(T (δ+ρ)))) λ) ϕ+a (E^(T (δ+ρ))) ρ ϕ)+x (-a δ (-1+λ) ϕ+(E^(T (δ+ρ))) (δ ρ+(ρ^2)+a δ λ ϕ+a ρ ϕ)))/(-a δ (-1+λ) ϕ+(E^(T (δ+ρ))) (ρ (ρ+a ϕ)+δ (ρ+a λ ϕ)))),gl->((a (E^(-T δ)) (1+a (-1+(E^(T δ)))) ((E^(T δ))-λ) ϕ)/(δ-a δ λ+a ϕ-a λ ϕ+a (E^(T δ)) (δ+ϕ))),gn->-(((-1+a) a (E^(-T δ)) (-1+(E^(T δ))) ((E^(T δ))-λ) ϕ)/(δ-a δ λ+a ϕ-a λ ϕ+a (E^(T δ)) (δ+ϕ))),hn->-(((-1+a) a (E^(-T δ)) ((E^(T δ))-λ) ϕ)/(δ-a δ λ+a ϕ-a λ ϕ+a (E^(T δ)) (δ+ϕ))),hl->((a (E^(-T δ)) (a ((E^(T δ))-λ)+λ) ϕ)/(δ-a δ λ+a ϕ-a λ ϕ+a (E^(T δ)) (δ+ϕ)))}]

FindRoot[{b/(x+(b (a (E^(T (2 δ+ρ))) (δ λ+ρ) (δ+ρ+ϕ)-(E^(T (δ+ρ))) (δ λ+ρ) ((-1+a λ) (δ+ρ)+a (-1+λ) ϕ)-(E^(T δ)) δ (δ+a δ (-1+λ)+ρ+a (-1+λ) (ρ+ϕ))+a δ (-1+λ) (-ϕ+λ (δ+ρ+ϕ))))/((1+(E^(T δ))-λ) (-a δ (-1+λ) ϕ+(E^(T (δ+ρ))) (ρ (δ+ρ)+a (δ λ+ρ) ϕ))))-(1/2),-(((δ+ρ) (-b δ λ+(E^(T (δ+ρ))) (b δ λ+(b-p+x) ρ)))/((δ-δ λ+(E^(T (δ+ρ))) (δ λ+ρ)) ((-p+x) ϕ+b (((E^(T δ))-λ) (δ+ρ)+ϕ))))-a}/.ϕ->2.8840820854132003`/.a->0.8393854748603352`/.T->1/.λ->1/.δ->0.14258496214801308`/.p->1/.ρ->0.1,{x,0.3,0,1},{b,0.02,0,1}]

This implies a wage range of 5% and a differential between eligible and ineligible of 0.8%:

N[{Rl,Rn,b,-1+(Rl/Rn),-1+(Rl/(a(Rl)+(1-a)Rn))}/.{Rl->b+x+((b (-1+(E^(T (δ+ρ)))) δ λ (δ+ρ))/(-a δ (-1+λ) ϕ+(E^(T (δ+ρ))) (ρ (ρ+a ϕ)+δ (ρ+a λ ϕ)))),Rn->((b (-(δ^2)-δ ρ+a δ (1+(-1+(E^(T (δ+ρ)))) λ) ϕ+a (E^(T (δ+ρ))) ρ ϕ)+x (-a δ (-1+λ) ϕ+(E^(T (δ+ρ))) (δ ρ+(ρ^2)+a δ λ ϕ+a ρ ϕ)))/(-a δ (-1+λ) ϕ+(E^(T (δ+ρ))) (ρ (ρ+a ϕ)+δ (ρ+a λ ϕ))))}/.ϕ->2.8840820854132003`/.a->0.8393854748603352`/.T->1/.λ->1/.δ->0.14258496214801308`/.p->1/.ρ->0.1/.x->0.4913890791853736`/.b->0.4964176603772637`]

Voluntary Quits

Steady State Conditions

Worker population is normalized to one.

fn+fl+gn+gl+hn+hl==1

No change to flows through ineligible unemployment.

(gn+gl) δ==ϕ fn

Flows through eligible unemployment now include those who gained eligibility and quit

(hn+hl)δ+γ gn ==fl ϕ a

Flows through ineligible employment are unchanged:

ϕ (1-a)fn==gn(δ+γ)

ϕ a (fn+λ fl)==gl(δ+γ)

Flows from eligible employment at wl are unchanged:

γ gl = δ hl

Flows from eligible employment at wn are now zero:

hn == 0

Solution

FullSimplify[Solve[{(gn+gl) δ==ϕ fn,(hn+hl)δ+γ gn ==fl ϕ a, ϕ (1-a)fn==gn(δ+γ),ϕ a (fn+λ fl)==gl(δ+γ),0==hn,fn+fl+gn+gl+hn+hl==1},{fn,fl,gn,gl,hn,hl}]]

Bellman Equations

For the employed at the low wage, the transition becomes:

FullSimplify[Solve[ρ Vn[w]== w+δ(Un-Vn[w])+γ(Ul-Vn[w])/.Vl[w]->((w+Ul δ)/(δ+ρ)),Vn[w]]]

Use this in the unemployed Bellman equations:

FullSimplify[Solve[{ρ Ul==b+x,ρ Un==x+ϕ(a(Vn[wl]-Un))}/.Vn[wl]->((wl+Ul γ+Un δ)/(γ+δ+ρ)),{Ul,Un}]]

Then substituting for Un and Ul, we obtain expressions for prices:

Simplify[FullSimplify[Solve[{(1-λ) ((wl+Ul δ)/(δ+ρ))+λ(((wl+Ul γ+Un δ)/(γ+δ+ρ)))==Ul,((wn+Ul γ+Un δ)/(γ+δ+ρ))==Un}/.{Ul->((b+x)/ρ),Un->((x ρ (γ+δ+ρ)+a ((b+x) γ+wl ρ) ϕ)/(ρ (ρ (γ+δ+ρ)+a (γ+ρ) ϕ)))},{wl,wn}]]]

Equal Profits

FullSimplify[{(p-wl)((gl+hl)/a),(p-wn)((gn+hn)/(1-a))}/.{wl->b+x+((b δ λ (δ+ρ))/(ρ (γ+δ-γ λ+ρ)+a (γ-γ λ+δ λ+ρ) ϕ)),wn->x+((b γ (-δ+γ (-1+λ)-ρ)+a b (γ-γ λ+δ λ+ρ) ϕ)/(ρ (γ+δ-γ λ+ρ)+a (γ-γ λ+δ λ+ρ) ϕ)),gn->-(((-1+a) a (δ^2) λ ϕ)/(δ (γ+δ) (γ+a δ λ)+a (γ (γ+δ)+δ (a γ+δ) λ) ϕ)),gl->((a δ (γ+a δ) λ ϕ)/(δ (γ+δ) (γ+a δ λ)+a (γ (γ+δ)+δ (a γ+δ) λ) ϕ)),hn->0,hl->((a γ (γ+δ+(-1+a) δ λ) ϕ)/(δ (γ+δ) (γ+a δ λ)+a (γ (γ+δ)+δ (a γ+δ) λ) ϕ))}]

Full Solution

Simplify[Solve[ (((γ+δ) (γ+a δ λ) ϕ (p-x+b (-1-((δ λ (δ+ρ))/(ρ (γ+δ-γ λ+ρ)+a (γ-γ λ+δ λ+ρ) ϕ)))))/(δ (γ+δ) (γ+a δ λ)+a (γ (γ+δ)+δ (a γ+δ) λ) ϕ))==(( a (δ^2) λ ϕ (p-x+((b (-(γ^2) (-1+λ)-a (δ λ+ρ) ϕ+γ (δ+ρ+a (-1+λ) ϕ)))/(ρ (γ+δ-γ λ+ρ)+a (γ-γ λ+δ λ+ρ) ϕ))))/((δ (γ+δ) (γ+a δ λ)+a (γ (γ+δ)+δ (a γ+δ) λ) ϕ))),a]]

For the wl degenerate solution:

FullSimplify[(((γ+δ) (γ+a δ λ) ϕ (p-x+b (-1-((δ λ (δ+ρ))/(ρ (γ+δ-γ λ+ρ)+a (γ-γ λ+δ λ+ρ) ϕ)))))/(δ (γ+δ) (γ+a δ λ)+a (γ (γ+δ)+δ (a γ+δ) λ) ϕ))>(( a (δ^2) λ ϕ (p-x+((b (-(γ^2) (-1+λ)-a (δ λ+ρ) ϕ+γ (δ+ρ+a (-1+λ) ϕ)))/(ρ (γ+δ-γ λ+ρ)+a (γ-γ λ+δ λ+ρ) ϕ))))/( (δ (γ+δ) (γ+a δ λ)+a (γ (γ+δ)+δ (a γ+δ) λ) ϕ)))/.a->1]

For the wn degenerate solution:

FullSimplify[(((γ+δ) (γ+a δ λ) ϕ (p-x+b (-1-((δ λ (δ+ρ))/(ρ (γ+δ-γ λ+ρ)+a (γ-γ λ+δ λ+ρ) ϕ)))))/(δ (γ+δ) (γ+a δ λ)+a (γ (γ+δ)+δ (a γ+δ) λ) ϕ))<(( a (δ^2) λ ϕ (p-x+((b (-(γ^2) (-1+λ)-a (δ λ+ρ) ϕ+γ (δ+ρ+a (-1+λ) ϕ)))/(ρ (γ+δ-γ λ+ρ)+a (γ-γ λ+δ λ+ρ) ϕ))))/( (δ (γ+δ) (γ+a δ λ)+a (γ (γ+δ)+δ (a γ+δ) λ) ϕ)))/.a->0]

Recalibration:

The average unemployment duration (2014-2017) for ineligible workers is 18.03 weeks

ϕ->52/18.03

The average unemployment duration for eligible workers is 21.48 weeks

a->(52/21.48)/ϕ/.%

Set γ = 1 (one year to get eligibility) and λ = 1 (not retained)

The unemployment rate averages 5.45%

Solve[fn+fl==0.0545/.fn->((a (δ^2) (γ+δ) λ)/(δ (γ+δ) (γ+a δ λ)+a (γ (γ+δ)+δ (a γ+δ) λ) ϕ))/.fl->((γ δ (γ+δ))/(δ (γ+δ) (γ+a δ λ)+a (γ (γ+δ)+δ (a γ+δ) λ) ϕ))/.ϕ->2.8840820854132003`/.a->0.8393854748603352`/.γ->1/.λ->1,δ]

We will normalize p = 1 and set the discount rate to 10%.

The replacement rate is 50%, and the equilibrium condition on a has to hold

FullSimplify[(((hl+gl)(Rl)+(hn+gn)Rn)/(hl+gl+hn+gn))/.{Rl->b+x+((b δ λ (δ+ρ))/(ρ (γ+δ-γ λ+ρ)+a (γ-γ λ+δ λ+ρ) ϕ)),Rn->x+((b γ (-δ+γ (-1+λ)-ρ)+a b (γ-γ λ+δ λ+ρ) ϕ)/(ρ (γ+δ-γ λ+ρ)+a (γ-γ λ+δ λ+ρ) ϕ)),gn->-(((-1+a) a (δ^2) λ ϕ)/(δ (γ+δ) (γ+a δ λ)+a (γ (γ+δ)+δ (a γ+δ) λ) ϕ)),gl->((a δ (γ+a δ) λ ϕ)/(δ (γ+δ) (γ+a δ λ)+a (γ (γ+δ)+δ (a γ+δ) λ) ϕ)),hn->0,hl->((a γ (γ+δ+(-1+a) δ λ) ϕ)/(δ (γ+δ) (γ+a δ λ)+a (γ (γ+δ)+δ (a γ+δ) λ) ϕ))}]

FindRoot[{b/(b+x+((b δ λ (-γ δ (-1+λ) ρ+δ ρ (δ+ρ)-(γ^2) (δ λ+ρ)-a δ (-(γ^2) (-1+λ)+(δ+ρ) (δ λ+ρ)+γ (δ+δ λ+2 ρ))))/((γ (γ+δ)+δ (a γ+δ) λ) (-ρ (δ+ρ)-a (δ λ+ρ) ϕ+γ (-1+λ) (ρ+a ϕ)))))-(1/2),(b (γ^2) (δ^2) λ+b γ (δ^3) λ-b (γ^2) (δ^2) (λ^2)+b γ (δ^3) (λ^2)+b (δ^4) (λ^2)+b (γ^2) δ λ ρ-p (γ^2) δ λ ρ+x (γ^2) δ λ ρ+3 b γ (δ^2) λ ρ-p γ (δ^2) λ ρ+x γ (δ^2) λ ρ+b (δ^3) λ ρ-b (γ^2) δ (λ^2) ρ+p (γ^2) δ (λ^2) ρ-x (γ^2) δ (λ^2) ρ+b (δ^3) (λ^2) ρ+b γ δ λ (ρ^2)-p γ δ λ (ρ^2)+x γ δ λ (ρ^2)+b (δ^2) λ (ρ^2)+b (γ^3) ϕ-p (γ^3) ϕ+x (γ^3) ϕ+b (γ^2) δ ϕ-p (γ^2) δ ϕ+x (γ^2) δ ϕ-b (γ^3) λ ϕ+p (γ^3) λ ϕ-x (γ^3) λ ϕ+b γ (δ^2) λ ϕ-p γ (δ^2) λ ϕ+x γ (δ^2) λ ϕ+b (γ^2) ρ ϕ-p (γ^2) ρ ϕ+x (γ^2) ρ ϕ+b γ δ ρ ϕ-p γ δ ρ ϕ+x γ δ ρ ϕ+√(-4 (b-p+x) (γ^2) δ (γ+δ) λ (γ (-1+λ)-δ λ-ρ) ((p-x) ρ (γ+δ-γ λ+ρ)-b ((δ^2) λ+δ (1+λ) ρ+ρ (γ-γ λ+ρ))) ϕ+((((p-x) γ ((γ^2) (-1+λ) ϕ+γ (δ (-λ ρ+(λ^2) ρ-ϕ)-ρ ϕ)-δ (δ λ (ρ+ϕ)+ρ (λ ρ+ϕ)))-b (-(δ^2) λ (δ+ρ) (δ λ+ρ)+(γ^2) (δ+ρ) (δ (-1+λ) λ-ϕ)+(γ^3) (-1+λ) ϕ-γ δ ((δ^2) λ (1+λ)+δ λ (3 ρ+ϕ)+ρ (λ ρ+ϕ)))))^2)))/(2 (b-p+x) γ δ λ (γ (-1+λ)-δ λ-ρ) ϕ)-a}/.ϕ->2.8840820854132003`/.a->0.8393854748603352`/.γ->1/.λ->1/.δ->0.13988602686244667`/.p->1/.ρ->0.1,{x,0.3,0,1},{b,0.4,0,1}]

This will yield a fraction of eligible workers as 89%, which is a little high, but fine:

N[(γ/(γ+a δ λ))==L/.ϕ->2.8840820854132003`/.a->0.8393854748603352`/.γ->1/.λ->1/.δ->0.13988602686244667`]

This implies a wage range of 5% and a differential between eligible and ineligible of 0.8%:

N[{-1+(Rl/Rn),-1+(Rl/(a(Rl)+(1-a)Rn))}/.{Rl->b+x+((b δ λ (δ+ρ))/(ρ (γ+δ-γ λ+ρ)+a (γ-γ λ+δ λ+ρ) ϕ)),Rn->x+((b γ (-δ+γ (-1+λ)-ρ)+a b (γ-γ λ+δ λ+ρ) ϕ)/(ρ (γ+δ-γ λ+ρ)+a (γ-γ λ+δ λ+ρ) ϕ))}/.ϕ->2.8840820854132003`/.a->0.8393854748603352`/.γ->1/.λ->1/.δ->0.13988602686244667`/.p->1/.ρ->0.1/.x->0.47106105921460306`/.b->0.49809823613279625`]

UI Funding

Steady state conditions are unchanged.

Balanced Budget Constraint

FullSimplify[Solve[fl b== τ(gl+gn+hl+hn)/.{fn->((a (δ^2) λ)/(a δ λ (δ+ϕ)+γ (δ+a ϕ))),fl->((γ δ)/(a δ λ (δ+ϕ)+γ (δ+a ϕ))),gn->-(((-1+a) a (δ^2) λ ϕ)/((γ+δ) (a δ λ (δ+ϕ)+γ (δ+a ϕ)))),gl->((a δ (γ+a δ) λ ϕ)/((γ+δ) (a δ λ (δ+ϕ)+γ (δ+a ϕ)))),hn->-(((-1+a) a γ δ λ ϕ)/((γ+δ) (a δ λ (δ+ϕ)+γ (δ+a ϕ)))),hl->((a γ (γ+δ+(-1+a) δ λ) ϕ)/((γ+δ) (a δ λ (δ+ϕ)+γ (δ+a ϕ))))},b]]

Bellman Equations

The only change is that workers are charged τ while employed.

ρ Vl[w]== w-τ+δ(Ul-Vl[w])

ρ Vn[w]== w-τ+δ(Un-Vn[w])+γ(Vl[w]-Vn[w])

Solution:

The Bellman Equations simplify in several ways.  First, we can combine like terms:

Solve[ρ Vl[w]== w-τ+δ( Ul-Vl[w]),Vl[w]]

For the employed who are ineligible:

FullSimplify[Solve[ρ Vn[w]== w-τ+δ(Un-Vn[w])+γ(Vl[w]-Vn[w])/.Vl[w]->((w+Ul δ-τ)/(δ+ρ)),Vn[w]]]

The unemployed Bellman equations become:

FullSimplify[Solve[{ρ Ul==b+x,ρ Un==x+ϕ(a(Vn[wl]-Un))}/.Vn[wl]->((Ul γ δ+Un δ (δ+ρ)+wl (γ+δ+ρ)-(γ+δ+ρ) τ)/((δ+ρ) (γ+δ+ρ))),{Ul,Un}]]

Then substituting for Ul and Un into the indifference conditions, we obtain expressions for prices:

Simplify[FullSimplify[Solve[{(1-λ) ((wl+Ul δ-τ)/(δ+ρ))+λ(((Ul γ δ+Un δ (δ+ρ)+wl (γ+δ+ρ)-(γ+δ+ρ) τ)/((δ+ρ) (γ+δ+ρ))))==Ul,((Ul γ δ+Un δ (δ+ρ)+wn (γ+δ+ρ)-(γ+δ+ρ) τ)/((δ+ρ) (γ+δ+ρ)))==Un}/.{Ul->((b+x)/ρ),Un->((x ρ (δ+ρ) (γ+δ+ρ)+a (b γ δ+x γ δ+ρ (γ+δ+ρ) (wl-τ)) ϕ)/(ρ (δ+ρ) (ρ (γ+δ+ρ)+a (γ+ρ) ϕ)))},{wl,wn}]]]

FullSimplify[((-b γ δ+a b (γ+δ λ+ρ) ϕ+(x+τ) (ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ))/(ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ))==b+x+τ-((b (γ+ρ) (δ+ρ))/(ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ))]

Equal Profit Condition

FullSimplify[{(p-wl)((gl+hl)/a),(p-wn)((gn+hn)/(1-a))}/.{wl->b+x+τ+((b δ λ (δ+ρ))/(ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ)),wn->((-b γ δ+a b (γ+δ λ+ρ) ϕ+(x+τ) (ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ))/(ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ)),fn->((a (δ^2) λ)/(a δ λ (δ+ϕ)+γ (δ+a ϕ))),fl->((γ δ)/(a δ λ (δ+ϕ)+γ (δ+a ϕ))),gn->-(((-1+a) a (δ^2) λ ϕ)/((γ+δ) (a δ λ (δ+ϕ)+γ (δ+a ϕ)))),gl->((a δ (γ+a δ) λ ϕ)/((γ+δ) (a δ λ (δ+ϕ)+γ (δ+a ϕ)))),hn->-(((-1+a) a γ δ λ ϕ)/((γ+δ) (a δ λ (δ+ϕ)+γ (δ+a ϕ)))),hl->((a γ (γ+δ+(-1+a) δ λ) ϕ)/((γ+δ) (a δ λ (δ+ϕ)+γ (δ+a ϕ))))}/.b->((a (γ+δ λ) τ ϕ)/(γ δ))]

Full Solution

Dispersed Solution is:

FullSimplify[(((γ+a δ λ) ϕ (p-x-τ-((a (γ+δ λ) τ ϕ)/(γ δ))-((a λ (γ+δ λ) (δ+ρ) τ ϕ)/(γ ρ (γ+δ+ρ)+a γ (γ+δ λ+ρ) ϕ))))/(a δ λ (δ+ϕ)+γ (δ+a ϕ)))==-(((-1+a) a δ λ ϕ (p-((ρ (γ+δ+ρ) (x+τ)+a (x (γ+δ λ+ρ)+ρ τ) ϕ+(((a^2) (γ+δ λ) (γ+δ λ+ρ) τ (ϕ^2))/(γ δ)))/(ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ))))/((1-a) (a δ λ (δ+ϕ)+γ (δ+a ϕ))))]

We rearrange this for simplification in the text:

Collect[(-p (γ^2)+x (γ^2))(δ ρ (γ+δ+ρ)+a δ (γ+δ λ+ρ) ϕ)+τ ((a^2) (δ^2) (λ^2) (δ+ρ) (δ λ+ρ) ϕ+(γ^3) (δ+a ϕ) (ρ+a ϕ)+a γ δ λ ϕ ((δ+ρ) (δ λ+2 a δ λ+ρ+a ρ)+a (δ λ+ρ) ϕ)+(γ^2) (δ ρ (δ+ρ)+a ((2+a) (δ^2) λ+δ (2+(2+a) λ) ρ+(ρ^2)) ϕ+(a^2) (2 δ λ+ρ) (ϕ^2)))==0,a]

FullSimplify[(-p (γ^2)+x (γ^2)) δ ρ (γ+δ+ρ)+(γ^3) δ ρ τ+(γ^2) δ ρ (δ+ρ) τ]+FullSimplify[a ((-p (γ^2)+x (γ^2)) δ (γ+δ λ+ρ) ϕ+(γ^3) δ τ ϕ+2 (γ^2) (δ^2) λ τ ϕ+(γ^3) ρ τ ϕ+2 (γ^2) δ ρ τ ϕ+2 (γ^2) δ λ ρ τ ϕ+(γ^2) (ρ^2) τ ϕ+γ (δ^2) (λ^2) (δ+ρ) τ ϕ+γ δ λ ρ (δ+ρ) τ ϕ)]+FullSimplify[(a^2) ((γ^2) (δ^2) λ τ ϕ+(γ^2) δ λ ρ τ ϕ+2 γ (δ^2) (λ^2) (δ+ρ) τ ϕ+γ δ λ ρ (δ+ρ) τ ϕ+(δ^2) (λ^2) (δ+ρ) (δ λ+ρ) τ ϕ+(γ^3) τ (ϕ^2)+γ δ λ (δ λ+ρ) τ (ϕ^2)+(γ^2) (2 δ λ+ρ) τ (ϕ^2))]

FullSimplify[((x-p+τ) γ δ (γ+δ λ+ρ)+(γ+δ λ) (γ ρ+(δ+ρ) (δ λ+ρ)) τ)==(-p γ δ (γ+δ λ+ρ)+x γ δ (γ+δ λ+ρ)+((γ^2) (δ+ρ)+δ λ (δ+ρ) (δ λ+ρ)+γ (2 (δ^2) λ+2 δ (1+λ) ρ+(ρ^2))) τ)]

We can also explicitly solve the quadratic equation.  Only the second solution is non-negative.

Simplify[Solve[-(γ^2) δ ρ (γ+δ+ρ) (p-x-τ)+a γ (-p γ δ (γ+δ λ+ρ)+x γ δ (γ+δ λ+ρ)+((γ^2) (δ+ρ)+δ λ (δ+ρ) (δ λ+ρ)+γ (2 (δ^2) λ+2 δ (1+λ) ρ+(ρ^2))) τ) ϕ+(a^2) (γ+δ λ) (γ+δ λ+ρ) τ ϕ (δ λ (δ+ρ)+γ ϕ)==0,a]]

The wl degenerate equilibrium is:

FullSimplify[{(((γ+a δ λ) ϕ (p-x-τ-((a (γ+δ λ) τ ϕ)/(γ δ))-((a λ (γ+δ λ) (δ+ρ) τ ϕ)/(γ ρ (γ+δ+ρ)+a γ (γ+δ λ+ρ) ϕ))))/(a δ λ (δ+ϕ)+γ (δ+a ϕ)))>(( a δ λ ϕ (p-((ρ (γ+δ+ρ) (x+τ)+a (x (γ+δ λ+ρ)+ρ τ) ϕ+(((a^2) (γ+δ λ) (γ+δ λ+ρ) τ (ϕ^2))/(γ δ)))/(ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ))))/((a δ λ (δ+ϕ)+γ (δ+a ϕ)))),wl->b+x+τ+((b δ λ (δ+ρ))/(ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ)),wn->((-b γ δ+x γ ρ+x δ ρ+x (ρ^2)+a (b+x) (γ+δ λ+ρ) ϕ)/(ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ))+τ}/.b-> ((a (γ+δ λ) τ ϕ)/(γ δ))/.a->1]

FullSimplify[{(((γ+a δ λ) ϕ (p-x-τ-((a (γ+δ λ) τ ϕ)/(γ δ))-((a λ (γ+δ λ) (δ+ρ) τ ϕ)/(γ ρ (γ+δ+ρ)+a γ (γ+δ λ+ρ) ϕ))))/(a δ λ (δ+ϕ)+γ (δ+a ϕ)))>(( a δ λ ϕ (p-((ρ (γ+δ+ρ) (x+τ)+a (x (γ+δ λ+ρ)+ρ τ) ϕ+(((a^2) (γ+δ λ) (γ+δ λ+ρ) τ (ϕ^2))/(γ δ)))/(ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ))))/((a δ λ (δ+ϕ)+γ (δ+a ϕ)))),wl->b+x+τ+((b δ λ (δ+ρ))/(ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ)),wn->((-b γ δ+x γ ρ+x δ ρ+x (ρ^2)+a (b+x) (γ+δ λ+ρ) ϕ)/(ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ))+τ}/.b-> ((a (γ+δ λ) τ ϕ)/(γ δ))/.a->0]

Recalibration:

The average unemployment duration (2014-2017) for ineligible workers is 18.03 weeks

ϕ->52/18.03

The average unemployment duration for eligible workers is 21.48 weeks

a->(52/21.48)/ϕ/.%

Set γ = 1 (one year to get eligibility) and λ = 1 (not retained)

The unemployment rate averages 5.45%

Solve[fn+fl==0.0545/.fn->((a (δ^2) λ)/(a δ λ (δ+ϕ)+γ (δ+a ϕ)))/.fl->((γ δ)/(a δ λ (δ+ϕ)+γ (δ+a ϕ)))/.ϕ->2.8840820854132003`/.a->0.8393854748603352`/.γ->1/.λ->1,δ]

We will normalize p = 1 and set the discount rate to 10%.

The replacement rate is 50%, and the equilibrium condition on a has to hold

FullSimplify[(((hl+gl)(Rl)+(hn+gn)Rn)/(hl+gl+hn+gn))/.{Rl->b+x+τ+((b δ λ (δ+ρ))/(ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ)),Rn->((-b γ δ+x γ ρ+x δ ρ+x (ρ^2)+a (b+x) (γ+δ λ+ρ) ϕ)/(ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ))+τ,gn->-(((-1+a) a (δ^2) λ ϕ)/((γ+δ) (a δ λ (δ+ϕ)+γ (δ+a ϕ)))),gl->((a δ (γ+a δ) λ ϕ)/((γ+δ) (a δ λ (δ+ϕ)+γ (δ+a ϕ)))),hn->-(((-1+a) a γ δ λ ϕ)/((γ+δ) (a δ λ (δ+ϕ)+γ (δ+a ϕ)))),hl->((a γ (γ+δ+(-1+a) δ λ) ϕ)/((γ+δ) (a δ λ (δ+ϕ)+γ (δ+a ϕ))))}]

FindRoot[{b/(((ρ (b γ (γ+δ+δ λ+ρ)+(γ+δ λ) (γ+δ+ρ) (x+τ))+a (γ+δ λ+ρ) (b δ λ (δ+ρ)+(γ+δ λ) (b+x+τ) ϕ))/((γ+δ λ) (ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ))))-(1/2),-((-p (γ^3) δ ϕ+x (γ^3) δ ϕ-p (γ^2) (δ^2) λ ϕ+x (γ^2) (δ^2) λ ϕ-p (γ^2) δ ρ ϕ+x (γ^2) δ ρ ϕ+(γ^3) δ τ ϕ+2 (γ^2) (δ^2) λ τ ϕ+γ (δ^3) (λ^2) τ ϕ+(γ^3) ρ τ ϕ+2 (γ^2) δ ρ τ ϕ+2 (γ^2) δ λ ρ τ ϕ+γ (δ^2) λ ρ τ ϕ+γ (δ^2) (λ^2) ρ τ ϕ+(γ^2) (ρ^2) τ ϕ+γ δ λ (ρ^2) τ ϕ-√((γ^2) ϕ ((((-p γ δ (γ+δ λ+ρ)+x γ δ (γ+δ λ+ρ)+((γ^2) (δ+ρ)+δ λ (δ+ρ) (δ λ+ρ)+γ (2 (δ^2) λ+2 δ (1+λ) ρ+(ρ^2))) τ))^2) ϕ+4 δ (γ+δ λ) ρ (γ+δ+ρ) (γ+δ λ+ρ) (p-x-τ) τ ((δ^2) λ+δ λ ρ+γ ϕ))))/(2 (γ+δ λ) (γ+δ λ+ρ) τ ϕ ((δ^2) λ+δ λ ρ+γ ϕ)))-a}/.b->((a (γ+δ λ) τ ϕ)/(γ δ))/.ϕ->2.8840820854132003`/.a->0.8393854748603352`/.γ->1/.λ->1/.δ->0.1423923515533254`/.p->1/.ρ->0.1,{x,0.3,0,1},{τ,0.02,0,1}]

This yields a benefit of:

b->((a (γ+δ λ) τ ϕ)/(γ δ))/.τ->0.025572220969249996`/.ϕ->2.8840820854132003`/.a->0.8393854748603352`/.γ->1/.λ->1/.δ->0.1423923515533254`/.p->1/.ρ->0.1

This will yield a fraction of eligible workers as 89%, which is a little high, but fine:

N[(γ/(γ+a δ λ))==L/.ϕ->2.8840820854132003`/.a->0.8393854748603352`/.γ->1/.λ->1/.δ->0.1423923515533254`]

This implies a wage range of 5% and a differential between eligible and ineligible of 0.8%:

N[{Rl,Rn,b,-1+(Rl/Rn),-1+(Rl/(a(Rl)+(1-a)Rn))}/.{Rl->b+x+τ+((b δ λ (δ+ρ))/(ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ)),Rn->((-b γ δ+x γ ρ+x δ ρ+x (ρ^2)+a (b+x) (γ+δ λ+ρ) ϕ)/(ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ))+τ}/.b->((a (γ+δ λ) τ ϕ)/(γ δ))/.ϕ->2.8840820854132003`/.a->0.8393854748603352`/.γ->1/.λ->1/.δ->0.1423923515533254`/.p->1/.ρ->0.1/.x->0.4665783148899482`/.τ->0.025572220969249996`]

For comparison to the base model, we look at wages  in each:

Manipulate[Grid[{{"wl","wn","a","b"},{"Base model",((b γ ρ+x ρ (γ+δ+ρ)+b (δ+ρ) (δ λ+ρ)-p γ ϕ)/(ρ (γ+δ+ρ)-γ ϕ)),((δ λ (-b γ δ+x ρ (γ+δ+ρ))-γ (-p (γ+ρ)+b (γ+δ λ+ρ)+x (γ+δ λ+ρ)) ϕ)/(δ λ (ρ (γ+δ+ρ)-γ ϕ))),((γ ((p-x) ρ (γ+δ+ρ)-b (γ ρ+(δ+ρ) (δ λ+ρ))))/((γ+δ λ+ρ) (b δ λ (δ+ρ)+(b-p+x) γ ϕ))),b}/.b->0.4980982361327964`,{"Funded model",b+x+τ+((b δ λ (δ+ρ))/(ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ)),((-b γ δ+x γ ρ+x δ ρ+x (ρ^2)+a (b+x) (γ+δ λ+ρ) ϕ)/(ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ))+τ,a,b}/.b-> ((a (γ+δ λ) τ ϕ)/(γ δ))/.a->-((-p (γ^3) δ ϕ+x (γ^3) δ ϕ-p (γ^2) (δ^2) λ ϕ+x (γ^2) (δ^2) λ ϕ-p (γ^2) δ ρ ϕ+x (γ^2) δ ρ ϕ+(γ^3) δ τ ϕ+2 (γ^2) (δ^2) λ τ ϕ+γ (δ^3) (λ^2) τ ϕ+(γ^3) ρ τ ϕ+2 (γ^2) δ ρ τ ϕ+2 (γ^2) δ λ ρ τ ϕ+γ (δ^2) λ ρ τ ϕ+γ (δ^2) (λ^2) ρ τ ϕ+(γ^2) (ρ^2) τ ϕ+γ δ λ (ρ^2) τ ϕ-√((γ^2) ϕ ((((-p γ δ (γ+δ λ+ρ)+x γ δ (γ+δ λ+ρ)+((γ^2) (δ+ρ)+δ λ (δ+ρ) (δ λ+ρ)+γ (2 (δ^2) λ+2 δ (1+λ) ρ+(ρ^2))) τ))^2) ϕ+4 δ (γ+δ λ) ρ (γ+δ+ρ) (γ+δ λ+ρ) (p-x-τ) τ ((δ^2) λ+δ λ ρ+γ ϕ))))/(2 (γ+δ λ) (γ+δ λ+ρ) τ ϕ ((δ^2) λ+δ λ ρ+γ ϕ)))}],{{p,1},0,1},{{τ,0.025572220969249996`},0,.5},{{x,0.4665783148899482`},0,1},{{γ,1},0,2},{{δ,0.1423923515533254`},0.001,0.4},{{ϕ,2.8840820854132003`},0,4},{{ρ,0.1},0.001,0.2},{{λ,1},0,1}]

Graphs

Figure 2 graphs are generated here with the recalibrated parameters.

Manipulate[Grid[{{Plot[{If[p<(x+τ),0],If[(γ^2) δ ρ (γ+δ+ρ) (p-x-τ)+(p-x) (γ^2) δ (γ+δ λ+ρ) ϕ-(δ (((γ+δ λ))^3)+((γ^3)+(δ^3) (λ^2) (1+λ)+(γ^2) δ (2+3 λ)+γ (δ^2) λ (2+3 λ)) ρ+(((γ+δ λ))^2) (ρ^2)) τ ϕ-γ (γ+δ λ) (γ+δ λ+ρ) τ (ϕ^2)>0,1],If[0≤-((-p (γ^3) δ ϕ+x (γ^3) δ ϕ-p (γ^2) (δ^2) λ ϕ+x (γ^2) (δ^2) λ ϕ-p (γ^2) δ ρ ϕ+x (γ^2) δ ρ ϕ+(γ^3) δ τ ϕ+2 (γ^2) (δ^2) λ τ ϕ+γ (δ^3) (λ^2) τ ϕ+(γ^3) ρ τ ϕ+2 (γ^2) δ ρ τ ϕ+2 (γ^2) δ λ ρ τ ϕ+γ (δ^2) λ ρ τ ϕ+γ (δ^2) (λ^2) ρ τ ϕ+(γ^2) (ρ^2) τ ϕ+γ δ λ (ρ^2) τ ϕ-√((γ^2) ϕ ((((-p γ δ (γ+δ λ+ρ)+x γ δ (γ+δ λ+ρ)+((γ^2) (δ+ρ)+δ λ (δ+ρ) (δ λ+ρ)+γ (2 (δ^2) λ+2 δ (1+λ) ρ+(ρ^2))) τ))^2) ϕ+4 δ (γ+δ λ) ρ (γ+δ+ρ) (γ+δ λ+ρ) (p-x-τ) τ ((δ^2) λ+δ λ ρ+γ ϕ))))/(2 (γ+δ λ) (γ+δ λ+ρ) τ ϕ ((δ^2) λ+δ λ ρ+γ ϕ)))≤1,-((-p (γ^3) δ ϕ+x (γ^3) δ ϕ-p (γ^2) (δ^2) λ ϕ+x (γ^2) (δ^2) λ ϕ-p (γ^2) δ ρ ϕ+x (γ^2) δ ρ ϕ+(γ^3) δ τ ϕ+2 (γ^2) (δ^2) λ τ ϕ+γ (δ^3) (λ^2) τ ϕ+(γ^3) ρ τ ϕ+2 (γ^2) δ ρ τ ϕ+2 (γ^2) δ λ ρ τ ϕ+γ (δ^2) λ ρ τ ϕ+γ (δ^2) (λ^2) ρ τ ϕ+(γ^2) (ρ^2) τ ϕ+γ δ λ (ρ^2) τ ϕ-√((γ^2) ϕ ((((-p γ δ (γ+δ λ+ρ)+x γ δ (γ+δ λ+ρ)+((γ^2) (δ+ρ)+δ λ (δ+ρ) (δ λ+ρ)+γ (2 (δ^2) λ+2 δ (1+λ) ρ+(ρ^2))) τ))^2) ϕ+4 δ (γ+δ λ) ρ (γ+δ+ρ) (γ+δ λ+ρ) (p-x-τ) τ ((δ^2) λ+δ λ ρ+γ ϕ))))/(2 (γ+δ λ) (γ+δ λ+ρ) τ ϕ ((δ^2) λ+δ λ ρ+γ ϕ)))]},{τ,0,0.65},PlotRange->Full,AxesOrigin->{0,-0.02},AxesLabel->{"τ","(SuperStar[a])"},PlotStyle->{Directive[Darker[Red],Thick,Dashing[{.05,.03}]],Directive[Darker[Blue],Thick,Dotted],Directive[Darker[Green],Thick]}],Plot[{If[ p<(x+τ),{x+τ,x+τ}],If[(γ^2) δ ρ (γ+δ+ρ) (p-x-τ)+(p-x) (γ^2) δ (γ+δ λ+ρ) ϕ-(δ (((γ+δ λ))^3)+((γ^3)+(δ^3) (λ^2) (1+λ)+(γ^2) δ (2+3 λ)+γ (δ^2) λ (2+3 λ)) ρ+(((γ+δ λ))^2) (ρ^2)) τ ϕ-γ (γ+δ λ) (γ+δ λ+ρ) τ (ϕ^2)>0,{x+τ+((1/δ)+(λ/γ)) τ ϕ+((λ (γ+δ λ) (δ+ρ) τ ϕ)/(γ ρ (γ+δ+ρ)+γ (γ+δ λ+ρ) ϕ)),x+((τ (γ δ ρ (γ+δ+ρ)+γ δ ρ ϕ+(γ+δ λ) (γ+δ λ+ρ) (ϕ^2)))/(γ δ (ρ (γ+δ+ρ)+(γ+δ λ+ρ) ϕ)))}],If[0≤-((-p (γ^3) δ ϕ+x (γ^3) δ ϕ-p (γ^2) (δ^2) λ ϕ+x (γ^2) (δ^2) λ ϕ-p (γ^2) δ ρ ϕ+x (γ^2) δ ρ ϕ+(γ^3) δ τ ϕ+2 (γ^2) (δ^2) λ τ ϕ+γ (δ^3) (λ^2) τ ϕ+(γ^3) ρ τ ϕ+2 (γ^2) δ ρ τ ϕ+2 (γ^2) δ λ ρ τ ϕ+γ (δ^2) λ ρ τ ϕ+γ (δ^2) (λ^2) ρ τ ϕ+(γ^2) (ρ^2) τ ϕ+γ δ λ (ρ^2) τ ϕ-√((γ^2) ϕ ((((-p γ δ (γ+δ λ+ρ)+x γ δ (γ+δ λ+ρ)+((γ^2) (δ+ρ)+δ λ (δ+ρ) (δ λ+ρ)+γ (2 (δ^2) λ+2 δ (1+λ) ρ+(ρ^2))) τ))^2) ϕ+4 δ (γ+δ λ) ρ (γ+δ+ρ) (γ+δ λ+ρ) (p-x-τ) τ ((δ^2) λ+δ λ ρ+γ ϕ))))/(2 (γ+δ λ) (γ+δ λ+ρ) τ ϕ ((δ^2) λ+δ λ ρ+γ ϕ)))≤1,{b+x+τ+((b δ λ (δ+ρ))/(ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ)),((-b γ δ+x γ ρ+x δ ρ+x (ρ^2)+a (b+x) (γ+δ λ+ρ) ϕ)/(ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ))+τ}/.b-> ((a (γ+δ λ) τ ϕ)/(γ δ))/.a->-((-p (γ^3) δ ϕ+x (γ^3) δ ϕ-p (γ^2) (δ^2) λ ϕ+x (γ^2) (δ^2) λ ϕ-p (γ^2) δ ρ ϕ+x (γ^2) δ ρ ϕ+(γ^3) δ τ ϕ+2 (γ^2) (δ^2) λ τ ϕ+γ (δ^3) (λ^2) τ ϕ+(γ^3) ρ τ ϕ+2 (γ^2) δ ρ τ ϕ+2 (γ^2) δ λ ρ τ ϕ+γ (δ^2) λ ρ τ ϕ+γ (δ^2) (λ^2) ρ τ ϕ+(γ^2) (ρ^2) τ ϕ+γ δ λ (ρ^2) τ ϕ-√((γ^2) ϕ ((((-p γ δ (γ+δ λ+ρ)+x γ δ (γ+δ λ+ρ)+((γ^2) (δ+ρ)+δ λ (δ+ρ) (δ λ+ρ)+γ (2 (δ^2) λ+2 δ (1+λ) ρ+(ρ^2))) τ))^2) ϕ+4 δ (γ+δ λ) ρ (γ+δ+ρ) (γ+δ λ+ρ) (p-x-τ) τ ((δ^2) λ+δ λ ρ+γ ϕ))))/(2 (γ+δ λ) (γ+δ λ+ρ) τ ϕ ((δ^2) λ+δ λ ρ+γ ϕ)))]},{τ,0,0.65},PlotRange->Full,AxesLabel->{"τ","wage"},PlotStyle->{Directive[Darker[Red],Thick,Dashing[{.05,.03}]],Directive[Darker[Blue],Thick,Dotted],Directive[Darker[Green],Thick]}]}}],{{p,1},0,1},{{x,0.4665783148899482`},0,1},{{γ,1},0,1},{{δ,0.1423923515533254`},0.001,0.4},{{ϕ,2.8840820854132003`},0,4},{{ρ,0.1},0.001,0.2},{{λ,1},0,1}]

Welfare

Think of optimality as being defined in a utilitarian form, maximizing the sum of worker utilities.  Firm profits are zero anyhow in the endogenous arrival world.

First compute the dispersed equilibrium welfare:

FullSimplify[fn Un+fl Ul+gn(((Ul γ δ+Un δ (δ+ρ)+Rn (γ+δ+ρ)-(γ+δ+ρ) τ)/((δ+ρ) (γ+δ+ρ))))+gl(((Ul γ δ+Un δ (δ+ρ)+Rl (γ+δ+ρ)-(γ+δ+ρ) τ)/((δ+ρ) (γ+δ+ρ))))+hn((Rn+Ul δ-τ)/(δ+ρ))+hl((Rl+Ul δ-τ)/(δ+ρ))/.{Ul->((b+x)/ρ),Un->((x ρ (δ+ρ) (γ+δ+ρ)+a (b γ δ+x γ δ+ρ (γ+δ+ρ) (Rl-τ)) ϕ)/(ρ (δ+ρ) (ρ (γ+δ+ρ)+a (γ+ρ) ϕ)))}/.{Rl->b+x+τ+((b δ λ (δ+ρ))/(ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ)),Rn->((-b γ δ+a b (γ+δ λ+ρ) ϕ+(x+τ) (ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ))/(ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ))}/.{fn->((a (δ^2) λ)/(a δ λ (δ+ϕ)+γ (δ+a ϕ))),fl->((γ δ)/(a δ λ (δ+ϕ)+γ (δ+a ϕ))),gn->-(((-1+a) a (δ^2) λ ϕ)/((γ+δ) (a δ λ (δ+ϕ)+γ (δ+a ϕ)))),gl->((a δ (γ+a δ) λ ϕ)/((γ+δ) (a δ λ (δ+ϕ)+γ (δ+a ϕ)))),hn->-(((-1+a) a γ δ λ ϕ)/((γ+δ) (a δ λ (δ+ϕ)+γ (δ+a ϕ)))),hl->((a γ (γ+δ+(-1+a) δ λ) ϕ)/((γ+δ) (a δ λ (δ+ϕ)+γ (δ+a ϕ))))}]

Next compute the wn degenerate welfare:

FullSimplify[((x+((b (γ δ ρ (γ+δ+ρ)+a ((γ^2) (δ+ρ)+a δ λ (δ+ρ) (δ λ+ρ)+γ ((1+a) (δ^2) λ+δ (2+λ+a λ) ρ+(ρ^2))) ϕ+(a^2) (γ+δ λ) (γ+δ λ+ρ) (ϕ^2)))/((ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ) (a δ λ (δ+ϕ)+γ (δ+a ϕ)))))/ρ)-(((-1+a) a δ λ ϕ (p-x-τ-((a (γ+δ λ) τ ϕ)/(γ δ))+((a (γ+δ λ) (γ+ρ) (δ+ρ) τ ϕ)/(γ δ (ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ)))))/((1-a) (a δ λ (δ+ϕ)+γ (δ+a ϕ))))/.b-> ((a (γ+δ λ) τ ϕ)/(γ δ))/.a->0]

And the compute the wl degenerate welfare:

FullSimplify[((x+((b (γ δ ρ (γ+δ+ρ)+a ((γ^2) (δ+ρ)+a δ λ (δ+ρ) (δ λ+ρ)+γ ((1+a) (δ^2) λ+δ (2+λ+a λ) ρ+(ρ^2))) ϕ+(a^2) (γ+δ λ) (γ+δ λ+ρ) (ϕ^2)))/((ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ) (a δ λ (δ+ϕ)+γ (δ+a ϕ)))))/ρ)/.b-> ((a (γ+δ λ) τ ϕ)/(γ δ))/.a->1]

We can plot these to readily see that they are maximized at the tax rate where the dispersed equilibrium barely emerges.

Manipulate[Grid[{{Plot[{If[p<(x+τ),0],If[(γ^2) δ ρ (γ+δ+ρ) (p-x-τ)+(p-x) (γ^2) δ (γ+δ λ+ρ) ϕ-(δ (((γ+δ λ))^3)+((γ^3)+(δ^3) (λ^2) (1+λ)+(γ^2) δ (2+3 λ)+γ (δ^2) λ (2+3 λ)) ρ+(((γ+δ λ))^2) (ρ^2)) τ ϕ-γ (γ+δ λ) (γ+δ λ+ρ) τ (ϕ^2)>0,(((γ+δ λ) τ ϕ)/(γ δ))],If[0≤-((-p (γ^3) δ ϕ+x (γ^3) δ ϕ-p (γ^2) (δ^2) λ ϕ+x (γ^2) (δ^2) λ ϕ-p (γ^2) δ ρ ϕ+x (γ^2) δ ρ ϕ+(γ^3) δ τ ϕ+2 (γ^2) (δ^2) λ τ ϕ+γ (δ^3) (λ^2) τ ϕ+(γ^3) ρ τ ϕ+2 (γ^2) δ ρ τ ϕ+2 (γ^2) δ λ ρ τ ϕ+γ (δ^2) λ ρ τ ϕ+γ (δ^2) (λ^2) ρ τ ϕ+(γ^2) (ρ^2) τ ϕ+γ δ λ (ρ^2) τ ϕ-√((γ^2) ϕ ((((-p γ δ (γ+δ λ+ρ)+x γ δ (γ+δ λ+ρ)+((γ^2) (δ+ρ)+δ λ (δ+ρ) (δ λ+ρ)+γ (2 (δ^2) λ+2 δ (1+λ) ρ+(ρ^2))) τ))^2) ϕ+4 δ (γ+δ λ) ρ (γ+δ+ρ) (γ+δ λ+ρ) (p-x-τ) τ ((δ^2) λ+δ λ ρ+γ ϕ))))/(2 (γ+δ λ) (γ+δ λ+ρ) τ ϕ ((δ^2) λ+δ λ ρ+γ ϕ)))≤1,((a (γ+δ λ) τ ϕ)/(γ δ))/.a-> -((-p (γ^3) δ ϕ+x (γ^3) δ ϕ-p (γ^2) (δ^2) λ ϕ+x (γ^2) (δ^2) λ ϕ-p (γ^2) δ ρ ϕ+x (γ^2) δ ρ ϕ+(γ^3) δ τ ϕ+2 (γ^2) (δ^2) λ τ ϕ+γ (δ^3) (λ^2) τ ϕ+(γ^3) ρ τ ϕ+2 (γ^2) δ ρ τ ϕ+2 (γ^2) δ λ ρ τ ϕ+γ (δ^2) λ ρ τ ϕ+γ (δ^2) (λ^2) ρ τ ϕ+(γ^2) (ρ^2) τ ϕ+γ δ λ (ρ^2) τ ϕ-√((γ^2) ϕ ((((-p γ δ (γ+δ λ+ρ)+x γ δ (γ+δ λ+ρ)+((γ^2) (δ+ρ)+δ λ (δ+ρ) (δ λ+ρ)+γ (2 (δ^2) λ+2 δ (1+λ) ρ+(ρ^2))) τ))^2) ϕ+4 δ (γ+δ λ) ρ (γ+δ+ρ) (γ+δ λ+ρ) (p-x-τ) τ ((δ^2) λ+δ λ ρ+γ ϕ))))/(2 (γ+δ λ) (γ+δ λ+ρ) τ ϕ ((δ^2) λ+δ λ ρ+γ ϕ)))]},{τ,0,0.8},PlotRange->Full,AxesLabel->{"τ","Benefit"},PlotStyle->{Directive[Darker[Red],Thick,Dashing[{.05,.03}]],Directive[Darker[Blue],Thick,Dotted],Directive[Darker[Green],Thick]}],Plot[{If[ p<x+τ ,(x/ρ)],If[p-x>(τ ((γ^2) δ ρ (γ+δ+ρ)+(δ (((γ+δ λ))^3)+((γ^3)+(δ^3) (λ^2) (1+λ)+(γ^2) δ (2+3 λ)+γ (δ^2) λ (2+3 λ)) ρ+(((γ+δ λ))^2) (ρ^2)) ϕ+γ (γ+δ λ) (γ+δ λ+ρ) (ϕ^2)))/((γ^2) δ (ρ (γ+δ+ρ)+(γ+δ λ+ρ) ϕ)),((x+((τ ϕ ((γ^2) (δ+ϕ) (ρ+ϕ)+δ λ (δ λ+ρ) ϕ (δ+ρ+ϕ)+γ (δ+ρ+ϕ) (δ ρ+2 δ λ ϕ+ρ ϕ)))/(γ δ (δ+ϕ) (ρ (γ+δ+ρ)+(γ+δ λ+ρ) ϕ))))/ρ)],If[0≤-((γ δ λ (δ+ρ) (δ λ+ρ) τ ϕ+(γ^3) (-p δ+x δ+(δ+ρ) τ) ϕ+(γ^2) (-(p-x) δ (δ λ+ρ)+(2 (δ^2) λ+2 δ (1+λ) ρ+(ρ^2)) τ) ϕ-√((γ^2) ϕ ((((-p γ δ (γ+δ λ+ρ)+x γ δ (γ+δ λ+ρ)+((γ^2) (δ+ρ)+δ λ (δ+ρ) (δ λ+ρ)+γ (2 (δ^2) λ+2 δ (1+λ) ρ+(ρ^2))) τ))^2) ϕ+4 δ (γ+δ λ) ρ (γ+δ+ρ) (γ+δ λ+ρ) (p-x-τ) τ (δ λ (δ+ρ)+γ ϕ))))/(2 (γ+δ λ) (γ+δ λ+ρ) τ ϕ (δ λ (δ+ρ)+γ ϕ)))≤1,((x+((b (γ δ ρ (γ+δ+ρ)+a ((γ^2) (δ+ρ)+a δ λ (δ+ρ) (δ λ+ρ)+γ ((1+a) (δ^2) λ+δ (2+λ+a λ) ρ+(ρ^2))) ϕ+(a^2) (γ+δ λ) (γ+δ λ+ρ) (ϕ^2)))/((ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ) (a δ λ (δ+ϕ)+γ (δ+a ϕ)))))/ρ)/.b-> ((a (γ+δ λ) τ ϕ)/(γ δ))/.a->-((γ δ λ (δ+ρ) (δ λ+ρ) τ ϕ+(γ^3) (-p δ+x δ+(δ+ρ) τ) ϕ+(γ^2) (-(p-x) δ (δ λ+ρ)+(2 (δ^2) λ+2 δ (1+λ) ρ+(ρ^2)) τ) ϕ-√((γ^2) ϕ ((((-p γ δ (γ+δ λ+ρ)+x γ δ (γ+δ λ+ρ)+((γ^2) (δ+ρ)+δ λ (δ+ρ) (δ λ+ρ)+γ (2 (δ^2) λ+2 δ (1+λ) ρ+(ρ^2))) τ))^2) ϕ+4 δ (γ+δ λ) ρ (γ+δ+ρ) (γ+δ λ+ρ) (p-x-τ) τ (δ λ (δ+ρ)+γ ϕ))))/(2 (γ+δ λ) (γ+δ λ+ρ) τ ϕ (δ λ (δ+ρ)+γ ϕ)))]},{τ,0,0.6},PlotRange->Full,AxesLabel->{"τ","Welfare"},PlotStyle->{Directive[Darker[Red],Thick,Dashing[{.05,.03}]],Directive[Darker[Blue],Thick,Dotted],Directive[Darker[Green],Thick]}]}}],{{p,1},0,1},{{x,0.4665783148899482`},0,1},{{γ,1},0,1},{{δ,0.1423923515533254`},0.001,0.4},{{ϕ,2.8840820854132003`},0,4},{{ρ,0.1},0.001,0.2},{{λ,1},0,1}]

To report the welfare at that peak, we find:

Manipulate[Grid[{{FindRoot[1==-((γ δ λ (δ+ρ) (δ λ+ρ) τ ϕ+(γ^3) (-p δ+x δ+(δ+ρ) τ) ϕ+(γ^2) (-(p-x) δ (δ λ+ρ)+(2 (δ^2) λ+2 δ (1+λ) ρ+(ρ^2)) τ) ϕ-√((γ^2) ϕ ((((-p γ δ (γ+δ λ+ρ)+x γ δ (γ+δ λ+ρ)+((γ^2) (δ+ρ)+δ λ (δ+ρ) (δ λ+ρ)+γ (2 (δ^2) λ+2 δ (1+λ) ρ+(ρ^2))) τ))^2) ϕ+4 δ (γ+δ λ) ρ (γ+δ+ρ) (γ+δ λ+ρ) (p-x-τ) τ (δ λ (δ+ρ)+γ ϕ))))/(2 (γ+δ λ) (γ+δ λ+ρ) τ ϕ (δ λ (δ+ρ)+γ ϕ))),{τ,0.04}],N[((x+((b (γ δ ρ (γ+δ+ρ)+a ((γ^2) (δ+ρ)+a δ λ (δ+ρ) (δ λ+ρ)+γ ((1+a) (δ^2) λ+δ (2+λ+a λ) ρ+(ρ^2))) ϕ+(a^2) (γ+δ λ) (γ+δ λ+ρ) (ϕ^2)))/((ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ) (a δ λ (δ+ϕ)+γ (δ+a ϕ)))))/ρ)/.b-> ((a (γ+δ λ) τ ϕ)/(γ δ))/.a->1/.FindRoot[1==-((γ δ λ (δ+ρ) (δ λ+ρ) τ ϕ+(γ^3) (-p δ+x δ+(δ+ρ) τ) ϕ+(γ^2) (-(p-x) δ (δ λ+ρ)+(2 (δ^2) λ+2 δ (1+λ) ρ+(ρ^2)) τ) ϕ-√((γ^2) ϕ ((((-p γ δ (γ+δ λ+ρ)+x γ δ (γ+δ λ+ρ)+((γ^2) (δ+ρ)+δ λ (δ+ρ) (δ λ+ρ)+γ (2 (δ^2) λ+2 δ (1+λ) ρ+(ρ^2))) τ))^2) ϕ+4 δ (γ+δ λ) ρ (γ+δ+ρ) (γ+δ λ+ρ) (p-x-τ) τ (δ λ (δ+ρ)+γ ϕ))))/(2 (γ+δ λ) (γ+δ λ+ρ) τ ϕ (δ λ (δ+ρ)+γ ϕ))),{τ,0.04}]],N[((x+((b (γ δ ρ (γ+δ+ρ)+a ((γ^2) (δ+ρ)+a δ λ (δ+ρ) (δ λ+ρ)+γ ((1+a) (δ^2) λ+δ (2+λ+a λ) ρ+(ρ^2))) ϕ+(a^2) (γ+δ λ) (γ+δ λ+ρ) (ϕ^2)))/((ρ (γ+δ+ρ)+a (γ+δ λ+ρ) ϕ) (a δ λ (δ+ϕ)+γ (δ+a ϕ)))))/ρ)/.b-> ((a (γ+δ λ) τ ϕ)/(γ δ))/.a->0.8393854748603352`/.τ->0.025572220969249996`]}}],{{p,1},0,1},{{x,0.4665783148899482`},0,1},{{γ,1},0,20},{{δ,0.1423923515533254`},0.001,0.4},{{ϕ,2.8840820854132003`},0,4},{{ρ,0.1},0.001,0.2},{{λ,1},0,1}]