Let (G) be a two-player Stackelberg game with leader (L) and follower (F).

[ x\in[0,1],\qquad a\in[0,1],\qquad e\ge 0 ]

where (L) chooses ((x,a)) first and (F) chooses (e) after observing ((x,a)).

Define

[ K(x)=\rho x, \qquad R(x)=1-\rho x, \qquad \rho\in(0,1]. ]

Let the follower’s payoff be

[ u_F(x,a,e)

a e\left[\sigma R(x)+\kappa K(x)\right] -\frac{c_F}{2}e^2, ]

and the leader’s payoff be

[ u_L(x,a,e)

\alpha K(x) + a e\left[\beta R(x)+\gamma K(x)\right] -\frac{c_L}{2}x^2 -\frac{c_A}{2}a^2. ]

All parameters are nonnegative, with

[ c_F,c_L,c_A>0. ]

First derive the follower’s best response

[ e^*(x,a)

\arg\max_{e\ge0}u_F(x,a,e). ]

Then solve

[ (x^,a^)

\arg\max_{(x,a)\in[0,1]^2} u_L\left(x,a,e^*(x,a)\right). ]

Characterize the subgame-perfect equilibrium

[ (x^,a^,e^*). ]

Derive the sign of

[ \frac{\partial e^*}{\partial x} ]

and identify the parameter regions satisfying

[ \frac{\partial e^}{\partial x}<0, \qquad \frac{\partial e^}{\partial x}=0, \qquad \frac{\partial e^*}{\partial x}>0. ]

Find the conditions under which

[ e^*=0, ]

the conditions under which (e^>0) but decreases as (x) increases, and the conditions under which (e^) increases as (x) increases.

Analyze the limiting cases

[ \rho\to0,\qquad \rho\to1,\qquad c_F\to0,\qquad c_F\to\infty, ]

and derive the comparative statics of

[ x^,\quad a^,\quad e^* ]

with respect to

[ \rho,\sigma,\kappa,\beta,\gamma,c_F,c_L,c_A. ]

Interpret the three regimes mathematically as:

[ \kappa<\sigma, \qquad \kappa=\sigma, \qquad \kappa>\sigma. ]

Setup and notation

Define the follower's marginal-return coefficient and the leader's return coefficient:

M(x) = σR(x) + κK(x) = σ + ρx(κ − σ)        (follower's productivity of effort)
N(x) = βR(x) + γK(x) = β + ρx(γ − β)        (leader's return per unit of a·e)

Since σ, κ ≥ 0, ρ ∈ (0,1], x ∈ [0,1], we have M(x) = σ(1−ρx) + κρx ≥ 0 always (a convex combination-like weighting of σ and κ). Same for N(x).

1. Follower's best response

u_F(x,a,e) = a e M(x) − (c_F/2)e² is strictly concave in e. FOC: a M(x) − c_F e = 0, and the constraint e ≥ 0 binds only when a M(x) = 0. Hence

e*(x,a) = a M(x) / c_F  =  a [σ + ρx(κ − σ)] / c_F   ≥ 0.

Sign of ∂e*/∂x

∂e*/∂x = a ρ (κ − σ) / c_F

For a > 0 (otherwise e* ≡ 0 and the derivative is 0):

Region	Sign
κ < σ	∂e*/∂x < 0 (x crowds out effort)
κ = σ (or a = 0)	∂e*/∂x = 0 (neutrality)
κ > σ	∂e*/∂x > 0 (x complements effort)

When is e* = 0?

e* = 0 ⇔ a M(x) = 0 ⇔ a = 0, or M(x) = 0. With nonnegative parameters, M(x) = 0 requires σ = 0 and κρx = 0, i.e. σ = 0 together with (κ = 0 or x = 0). Generically (σ > 0), e* = 0 in equilibrium iff the leader chooses a* = 0 (see §3).

e* > 0 and decreasing in x: a* > 0, σ > 0, and κ < σ.
e* > 0 and increasing in x: a* > 0 and κ > σ (then M(x) ≥ σ ≥ 0, strictly positive if σ > 0 or x > 0).

2. Leader's reduced objective

Substituting e*(x,a):

u_L(x, a, e*(x,a)) = αρx + (a²/c_F) Φ(x) − (c_L/2)x² − (c_A/2)a²,

where

Φ(x) ≡ M(x)N(x) = [σ + ρx(κ−σ)][β + ρx(γ−β)]
     = σβ + ρA x + ρ²B x²,
A ≡ σ(γ−β) + β(κ−σ) = σγ + βκ − 2σβ,
B ≡ (κ−σ)(γ−β).

Choice of a: bang-bang

The a-terms are a² [Φ(x)/c_F − c_A/2] — purely quadratic, no linear term (because e* is itself proportional to a). Hence for fixed x:

a*(x) = 1   if 2Φ(x) > c_A c_F
a*(x) = 0   if 2Φ(x) < c_A c_F      (indifferent at equality)

The equilibrium is found by comparing the two candidate values:

V₀ = max_x [ αρx − (c_L/2)x² ]                       (a = 0 branch)
V₁ = max_x [ αρx + Φ(x)/c_F − (c_L/2)x² ] − c_A/2    (a = 1 branch)

a* = 1 iff V₁ ≥ V₀; intuitively iff Φ (leader's return × follower's productivity) is large relative to c_A c_F.

Choice of x

Branch a = 0: x* = min(αρ/c_L, 1), the pure direct-benefit optimum. Interior value V₀ = α²ρ²/(2c_L).

Branch a = 1: g(x) = αρx + Φ(x)/c_F − (c_L/2)x², with

g'(x) = αρ + ρ(A + 2ρB x)/c_F − c_L x,
g''(x) = 2ρ²B/c_F − c_L.

SOC (concavity): D ≡ c_L c_F − 2ρ²B > 0. Automatic whenever B ≤ 0 (e.g. κ<σ, γ>β, or κ>σ, γ<β, or κ=σ).
Interior solution (when D > 0 and the result lands in [0,1]):

x* = ρ (α c_F + A) / D  =  ρ [α c_F + σγ + βκ − 2σβ] / [c_L c_F − 2ρ²(κ−σ)(γ−β)]

clipped to [0,1]; x* = 0 if αc_F + A ≤ 0, x* = 1 if g'(1) ≥ 0.

If D < 0 (requires κ>σ and γ>β with 2ρ²(κ−σ)(γ−β) > c_L c_F): g is convex → corner solution x* ∈ {0,1}, whichever of g(0) = σβ/c_F, g(1) = αρ + κγ... (i.e. Φ(1)/c_F + αρ − c_L/2) is larger. This is the "double complementarity" bang-bang regime.

SPE summary

(x*, a*)  as above (compare V₀, V₁);    e* = a* M(x*) / c_F.

3. Limiting cases

ρ → 0: K → 0, R → 1. M → σ, N → β, Φ → σβ, and x has no upside (αρx → 0), so x* → 0; a* = 1 iff 2σβ > c_A c_F; e* → a*σ/c_F. The game decouples from x entirely.

ρ → 1: K(x) = x, R(x) = 1−x — full reallocation. M(x) = σ + x(κ−σ) interpolates exactly between σ (x=0) and κ (x=1); the crowding-out/complementarity tension is maximal, and the convexity condition D < 0 is easiest to satisfy (B enters as 2B vs 2ρ²B).

c_F → 0: e* = aM(x)/c_F → ∞ whenever aM(x) > 0. The term Φ(x)/c_F dominates u_L, so a* = 1 and x* → argmax_{[0,1]} Φ(x); the leader's payoff diverges to +∞ (effort is free to elicit).

c_F → ∞: e* → 0; the interaction term vanishes, V₁ → V₀ − c_A/2 < V₀, so a* = 0, x* → min(αρ/c_L, 1), e* = 0. The game collapses to the leader's stand-alone problem.

4. Comparative statics (interior a* = 1 case: x* = ρ(αc_F + A)/D, P ≡ ρ(αc_F+A) > 0, D > 0)

Recall A = σγ + βκ − 2σβ, B = (κ−σ)(γ−β), e* = [σ + ρx*(κ−σ)]/c_F.

θ	∂x*/∂θ	∂a*/∂θ (regime margin)	∂e*/∂θ
ρ	sign of (αc_F+A)(c_Lc_F + 2ρ²B); > 0 if B ≥ 0	↑ if Φ↑	sign of (κ−σ)·∂(ρx*)/∂ρ
σ	[ρ(γ−2β)D − 2ρ²(γ−β)P]/D² — ambiguous	weakly ↑ (Φ↑)	direct + (1−ρx*)/c_F > 0; total ambiguous if κ≠σ
κ	[ρβD + 2ρ²(γ−β)P]/D²; > 0 if γ ≥ β	weakly ↑	ρx/c_F + ρ(κ−σ)x_κ /c_F; > 0 if κ ≥ σ, γ ≥ β
β	[ρ(κ−2σ)D − 2ρ²(κ−σ)P]/D² — ambiguous	weakly ↑	via x* only: sign = sign[(κ−σ)·x*_β]
γ	[ρσD + 2ρ²(κ−σ)P]/D²; > 0 if κ ≥ σ	weakly ↑	via x* only: sign = sign[(κ−σ)·x_γ]; > 0 if κ > σ*
c_F	−ρ(2ρ²αB + c_LA)/D²; < 0 if A,B ≥ 0	weakly ↓	< 0 dominantly (e* ∝ 1/c_F; exactly −σ/c_F² when κ=σ)
c_L	−c_F P/D² < 0	≈ 0 / weakly ↑ via V₀↓	sign = −sign(κ−σ)·(x* effect)
c_A	0 (conditional on a*=1)	↓ (switches a* to 0 when c_A > 2V₁'/…, i.e. large)	0 until regime switch, then drops to 0

Also: ∂x*/∂α = ρc_F/D > 0 in both branches.

Key monotone facts:

Φ(x) is pointwise nondecreasing in each of σ, κ, β, γ ⇒ V₁, and hence the region where a* = 1 (and e* > 0), expands in each of these and shrinks in c_A, c_F.
Conditional on a* = 1 and κ = σ: e* = σ/c_F exactly — increasing in σ, decreasing in c_F, independent of everything else.

5. The three regimes

κ < σ (substitution / crowding-out). M′(x) = ρ(κ−σ) < 0: every unit of x converts high-powered base R into low-powered K from the follower's perspective, so e* falls in x. The leader's marginal calculus is αρ + ρ(A + 2ρBx)/c_F = c_L x; the indirect term can be negative (A < 0 when βκ + σγ < 2σβ), pushing x* below the stand-alone level αρ/c_L: the leader restrains x to preserve follower incentives. g is concave whenever γ ≥ β (B ≤ 0), so the solution is interior and unique.

κ = σ (neutrality). M(x) ≡ σ: the follower's incentive is invariant to x, ∂e*/∂x = 0, B = 0, and the problem separates: e* = a*σ/c_F and x* = ρ(αc_F + σ(γ−β))/(c_L c_F) — x is still tilted by the leader's own return asymmetry (γ vs β), but has no incentive spillover.

κ > σ (complementarity). M′(x) > 0: x raises follower effort, so x* exceeds the stand-alone level whenever N > 0 (A > 0 when βκ + σγ > 2σβ). If additionally γ > β, then B > 0 and Φ is convex in x; once 2ρ²(κ−σ)(γ−β) > c_L c_F the leader's problem is convex and the equilibrium is bang-bang in x (x* ∈ {0,1}), typically x* = 1: leader and follower returns reinforce each other, so the leader goes all-in on x, sets a* = 1 (if 2Φ(1) > c_A c_F net of costs), and effort e* = κ/c_F (at ρ = 1) is maximal.