Question

Q:

Retailers and manufacturers have bargaining over the price of coffee. Assume that retailers’ willingness to sell is 50 and manufacturers’ willingness to buy is 100. What price maximizes the Nash product if they have the equal bargaining power?

Show the solution steps please

Answer #1

Manufacturers have a maximum willingness to buy at 100 while
Retailers must receive at least 40. The bargained price must
prevail in between these two values. Assume that the optimum price
is *p* so that Manufacturer’s net surplus is 100 -
*p.*

If retailers can sell at this price their net surplus is
*p* – 40. Hence, the Nash product becomes:

NP = (100 - *p*)( *p* – 40)

Nash bargaining solution has a price that maximizes the Nash product which implies:

dNP /dp = 0

(100*p* – 4000 – *p*^2 + 40*p*)/d*p*
= 0

d(*p^*2 – 140*p* + 4000)/d*p* = 0

2*p* – 140 = 0

*p* = 70

Hence, the price that maximizes the Nash product is *p* =
70

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