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.


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


$5=6q_1^H- q_1^L$ $\ \ $(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.


$6q_1^L+q_1^H=1$ $\ \ $(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





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,

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


$(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$


$IC_G$ 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

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


$(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$


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


(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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