Question

[Pricing Strategy] A local fruit orchard sells peaches to both a local farmer’s market and to a grocery store chain. The demand function for the farmer’s market is q_F=800-200p_F, and demand for the grocery store chain is q_G=600-100p_G. The total cost of supplying peaches to market is TC(Q)=2Q+400 where Q=q_F+q_G. Note that quantity is measured in pounds of peaches. What is the profit-maximizing price the orchard should set in each market? How many pounds of peaches will be sold to each market? What is the orchard’s profit?

Answer 1

Solution:

Profit function for the orchard = Total revenue - total cost

We have qf = 800 - 200*Pf, so Pf = 4 - 0.005*qf

Similarly, with qg = 600 - 100*Pg, so Pg = 6 - 0.01*qg

Total revenue, TR = Pf*qf + Pg*qg

TR = (4 - 0.005*qf)*qf + (6 - 0.01*qg)*qg

So, profit function is:

Profit, Z = 4*qf - 0.005qf² + 6*qg - 0.01*qg² - 2*(qf + qg) - 400

Now, profit maximizing quantities can be found using the first order conditions: = 0, = 0

So, = 4 - 2*0.005*qf - 2

So, FOC gives us 2 - 0.01*qf = 0

qf = 2/0.01 = 200 pounds of peaches

Similarly, = 6 - 2*0.01*qg - 2

Then, FOC gives us 4 - 0.02*qg = 0

qg = 4/0.02 = 200 pounds of peaches

Then, prices charged in each market can be derived as:

Pf = 4 - 0.005*200 = $3

Pg = 6 - 0.01*200 = $4

Then, total cost = 2*(200 + 200) + 400 = $1,200

And total revenue = 3*200 + 4*200 = $1,400

Orchard's profit, Z = 1400 - 1200 = $200