As the manager of Smith Construction, you need to make a decision on the number

Question

As the manager of Smith Construction, you need to make a decision on the number of homes to build in a new residential area where you are the only builder. Unfortunately, you must build the homes before you learn how strong demand is for homes in this large neighborhood. There is a 60 percent chance of low demand and a 40 percent chance of high demand. The corresponding (inverse) demand functions for these two scenarios are P = 300,000 400Q and P = 500,000 275Q, respectively. Your cost function is C(Q) = 140,000 + 240,000Q.

How many new homes should you build, and what profits can you expect?

As the manager of Smith Construction, you need to make a decision on the number of homes to build in a new residential area where you are the only builder. Unfortunately, you must build the homes before you learn how strong demand is for homes in this large neighborhood. There is a 60 percent chance of low demand and a 40 percent chance of high demand. The corresponding (inverse) demand functions for these two scenarios are P = 300,000 400Q and P = 500,000 275Q, respectively. Your cost function is C(Q) = 140,000 + 240,000Q.

Explanation / Answer

As there are 2 scenarios for demand function,

expected demand function = 60%*(300000-400Q)+40%*(500000-275Q)

=380,000-350Q

so P =380000-350Q

so how many new homes build depend on maximising profit

as profit =revenue -cost

=> profit = (380000-350Q)*Q - (140000+240000Q)

=> for maximising d(profit)/dQ =0

=> d(profit)/dQ = 380000-350*2*Q-240000

so 380000-350*2*Q-240000 =0

=> Q=200

so number of houses that should be build =200

and profit = (380000-350Q)*Q - (140000+240000Q)

=((380000-350*200)*200) - (140000+(240000*200))

=$13,860,000

so answer : new houses build =200

and profit =$13,860,000

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