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Consider two firms

Inverse functions: $p(q)=max\{a-q,0\} $

$q$ is total production and a is positive constant.

This value of a takes two values ah=2 (high demand) and aL=1 (low demand). Their probabilities are equal (1/2) for each cases.

Firm 1 know the value of a, but firm 2 don’t know.

Firm 1 = incumbent

Firm2 = entrant

Firm 1 move first. Firm 2 observe firm 1’production $q_1$ before deciding $q_2$ and then move.

For all firms, marginal cost and fixed costs are zero.

Considering firm 1’s incentive to signal the state of the world, describe the set of separating equilibria.

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What I have solved:

Case 1:no signaling

Firstly consider the firm 2,

$$\pi 2= max [ (1/2)(ah-q_1^H-q_2)q_2+(1/2)(aL-q_1^L-q_2)q_2]$$

By FOC: $$q_2^*= {3-(q_1^L+q_1^H)\over 4}$$

Now consider the firm 1. There are two states: ah and aL

State 1: $a=ah=2$

$$\pi_1^H = max [ (ah-q_2-q_1^H)q_1^H]$$

where $$q_2^*= {3-(q_1^L+q_1^H)\over 4}$$

I insert this $q_2$ into maximization objective function

By FOC:

$5=6q_1^H- q_1^L$ $\ \ $(Equation 1)

State 2: $a=aL=1$

$$\pi_1^L= max [ (aL-q_2-q_1^L)q_1^L]$$

where $$q_2^*= {3-(q_1^L+q_1^H)\over 4}$$

I insert this $q_2$ into maximization objective function.

FOC:

$6q_1^L+q_1^H=1$ $\ \ $(Equation 2)

When I combine the equations (1) and (2), I obtain that

$$q_1^H+q_1^L=6/5$$ And $q_1^H=31/35$ and $q_1^L=11/35$

$$q_2^*={3-(6/5)\over 4}=9/20$$

When I calculate the profits

$$\pi_2=81/400$$

$$\pi_1^L>0$$

But

$$\pi_1^H<0$$

my first problem is at this point. Why I obtain $\pi_1^H<0$ this profit negative value. This is wrong, but I could not find my mistake.

Case 2: signaling exist

Consider the firm 2

$$\pi_2 = max [ (a-q_2-q_1)q_1]$$

By FOC, $q_2=(1/2)[a-q_1]$

Now consider the firm 1,

$$\pi_1= max [ (1/2)(ah-q_1^H-q_2)q_1^H+(1/2)(aL-q_1^L-q_2)q_1^L]$$

S.t.

$(IC_H):\ $ $(ah-q_2-q_1^H)q_1^H> (ah-q_2-q_1^L)q_1^L$

$(IC_L):\ $ $(aL-q_2-q_1^L)q_1^L> (aL-q_2-q_1^H)q_1^H$

$(PC_H):\ $ $ (ah-q_2-q_1^h)q_2>0$

$(PC_L):\ $ $ (aL-q_2-q_1^L)q_2>0$

Solutions

$IC_G$ must hold because firm 1 cannot deviate.

$PC_L$ must be biding because the firm 2 cannot earns negative profit.

By $IC_G$ and $PC_L$, $PC_G$ is automatically satisfied.

So the optimization problem becomes

$$\pi_1= max [ (1/2)(ah-q_1^H-q_2)q_1^H+(1/2)(aL-q_1^L-q_2)q_1^L]$$

S.t.

$(IC_H):\ $ $(ah-q_2-q_1^H)q_1^H> (ah-q_2-q_1^L)q_1^L$

$(PC_L):\ $ $ (aL-q_2-q_1^L)q_2>0$

Lagrangian

$L= [ (1/2)(ah-q_1^H-q_2)q_1^H+(1/2)(aL-q_1^L-q_2)q_1^L]+ \lambda [(ah-q_2-q_1^H)q_1^H- (ah-q_2-q_1^L)q_1^L]+\mu [ (aL-q_2-q_1^L)q_2]$

FOCs

(I) $dL/dq_1^L=0$

(II) $dL/dq_1^H=0$

(III) $dL/d\lambda\ge 0$ and $(dL/d\lambda)\lambda=0$ For $\lambda \ge 0$

(IV) $dL/d\mu\ge 0$ and $(dL/d\mu)\mu=0$ For $\mu \ge 0$

It must be that $\lambda \ge 0$ and $\mu \ge 0$ such that constraints are binding.

By (III) and (IV), $q_1^H=2$ and $q_1^L=2$

my problem about signaling case, the values of q1 doesn’t not hold with the answers of solution manual book that gives only true value. Where is my mistake in mysolution? I could not find my mistake. Please give me a hint.

I wrote my solution and the points that I have trouble in yellow box. If need more clarification, I immediately add something. Please help me. Thanks.

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