m256hw03soln

P1 p1x p2 p2x 2 elast12x 04 in13 elast21x out13

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Unformatted text preview: 25 25; mc2 p1 15; 260. p1 4. p1 profit2 p1_, p2_ 180. 2. p1 mc1 1. p2 p2 4. p2 q1 p1, p2 15 mc2 q2 p1, p2 p2 4. If firm 1 sets it price at 50. what is the optimal price for firm 2? If firm 2 sets its price to this amount what is the optimal price for firm 1? And then if firm 1 takes this price what would firm 2 do, and then what would firm 1 respond? In[18]:= p1a Out[18]= 50. In[19]:= p2b Out[19]= In[20]:= Out[20]= In[21]:= Out[21]= In[22]:= Out[22]= 50. p2 . Solve D profit2 p1a, p2 , p2 0., p2 1 p1 . Solve D profit1 p1, p2b , p1 0., p1 1 0., p2 1 0., p1 1 42.5 p1c 50.3125 p2d p2 . Solve D profit2 p1c, p2 , p2 42.5781 p1e p1 . Solve D profit1 p1, p2d , p1 50.3223 And we are clearly converging quickly to an equilibrium. 5. Find the Nash equilibrium prices, i.e., the prices p1e and p2e such that profit1[p1,p2e] is a maximum at p1=p1e and profit2[p1e,p2] is a maximum at p2=p2e, so that firm 1 is doing the best it can given what firm 2 is doing, and firm 2 is doing the best it can given what firm 1 is doing. In[23]:= Out[23]= nashsoln p1 Solve 50.3226, p2 D profit1 p1, p2 , p1 42.5806 0, D profit2 p1, p2 , p2 0 , p1, p2 12 m256hw03soln.nb In[24]:= Out[24]= p1e, p2e p1, p2 . nashsoln 1 50.3226, 42.5806 6. If the firms merge, they will only be concerned about the total profit on the two products. Find the prices the merged firm will charge to maximize total profit, i.e., the prices p1m and p2m such that profit1[p1,p2]+profit2[p1,p2] is a maximum at p1=p1m and p2=p2m. In[25]:= Out[25]= In[26]:= Out[26]= In[27]:= Out[27]= In[28]:= Out[28]= In[29]:= Out[29]= profit p1_, p2_ 180. 2. p1 mergersoln p1 profit1 p1, p2 4. p2 Solve 59.7273, p2 p1m, p2m p1, p2 15 p2 profit2 p1, p2 25 p1 D profit p1, p2 , p1 260. 1 59.7273, 49.2727 p1m p1e p1e 0.186888 p2m p2e 1. p2 0, D profit p1, p2 , p2 49.2727 . mergersoln 4. p1 p2e 0.157163 Prices would increase 19% and 16% if there is a merger. 0 , p1, p2...
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