An apple farmer must decide how many apples to harvest for the world apple market.He knows that there is a one-third probability that the world price will be $1, a one-third probability that it will be $1.5, and a one-third probability that it will be $2.His cost function is C(Q) = 0.01Q².The expected profit maximizing quantity is:

Question

Quizplus · Accepted Answer

To find the expected profit-maximizing quantity, we need to calculate the expected profit for each possible quantity, and then choose the quantity that gives us the highest expected profit.

Let Q be the quantity of apples harvested for the world market. Then the revenue function is:

R(Q) = p(Q) * Q

where p(Q) is the world price of apples, which is given by:

p(Q) = 1 with probability 1/3
p(Q) = 1.5 with probability 1/3
p(Q) = 2 with probability 1/3

So the expected revenue function is:

E[R(Q)] = (1/3) * 1 * Q + (1/3) * 1.5 * Q + (1/3) * 2 * Q
         = 1.5Q

The cost function is given by:

C(Q) = 0.01Q(S1U1P1^2)S1(S1P0)

To find the profit function, we need to subtract the cost function from the expected revenue function:

π(Q) = E[R(Q)] - C(Q)
     = 1.5Q - 0.01Q(S1U1P1^2)S1(S1P0)

To find the expected profit, we need to take the expected value of the profit function:

E[π(Q)] = (1/3) * (1.5Q - 0.01Q(S1U1P1^2)S1(S1P0)) 
         + (1/3) * (1.5Q - 0.01Q(S1U1P1^2)S1(S1P0))
         + (1/3) * (1.5Q - 0.01Q(S1U1P1^2)S1(S1P0))
         = 0.5(1.5Q - 0.01Q(S1U1P1^2)S1(S1P0))

Now we need to find the quantity that maximizes the expected profit. To do this, we take the derivative of the expected profit function with respect to Q:

dE[π(Q)]/dQ = 0.5(1.5 - 0.01(S1U1P1^2)S1(S1P0)) = 0

Solving for Q, we get:

Q = 75

Therefore, the expected profit-maximizing quantity is 75.

An Apple Farmer Must Decide How Many Apples to Harvest