Question

3. Sunnyvale Orchards is one of many small, perfectly competitive firms growing plums for the U.S. market. The forecasted price of plums in 2018 is $24 a crate. The management of Sunnyvale Orchards estimates its short run cost function to be: TC - 3000 + 249 -0,04572 +0.000057', where q is output per week Find the profit maximizing output. How much profits will Sunnyvale make? Output = Profits = $ Market conditions deteriorate. Sunnyvale should shut down if the price of plums falls below $. Then they make losses of $_

Answer 1

In a perfectly competitive market, profit maximization is achieved at the output where price is equal to marginal cost.

The profit maximization rule states that profit is maximized when marginal revenue equals marginal cost.

In perfectly competitive market, for profit maximization to take place, Price = Marginal cost = Marginal revenue

Total cost = 3000 + 24Q - 0.045Q² + 0.00005Q³

Marginal cost = 24 - 0.09Q + 0.00015Q²

Equating Price and Marginal cost

$ 24 = 24 - 0.09Q + 0.00015Q²

0.09 Q - 0.00015Q² = 0

Q ( 0.09 - 0.00015Q ) = 0

0.00015Q = 0.09

Q = 0.09 $\div$ 0.00015

Q = 600 units

Profit = Revenue - Total cost

Profit = 600 units x $ 24 - [ 3000 + 24 x 600 - 0.045 x (600)² + 0.00005 x (600)³ ]

Profit = $ 2400

Output = 600 units ; Profit = $ 2400

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The shutdown price for a perfectly competitive firm is the minimum point on the average variable cost curve.

The total cost = 3000 + 24Q - 0.045Q² + 0.00005Q³

The variable cost component of the total cost equation = 24Q - 0.045Q² + 0.00005Q³

Variable cost = 24Q - 0.045Q² + 0.00005Q³

Dividing the above variable cost equation by Q , we get the average variable cost equation

AVC = 24 - 0.045Q + 0.00005Q²

Taking the first derivative of above equation

dAVC/dq = -0.045 + 0.0001Q

Equating the above equation to zero

-0.045 + 0.0001Q = 0

Q = 0.045 $\div$ 0.0001

Q = 450 units

At Q = 450 units, the average variable cost is minimum

We know that , AVC = 24 - 0.045Q + 0.00005Q²

Substituting the value of Q = 450 in the above, we find the average variable cost at Q = 450

The average variable cost at 450 units = 24 - 0.045 x 450  + 0.00005 x 450²

The average variable cost at 450 units = $ 13.875

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Sunnyvale should shutdown if the price of plums falls below $ 13.875 .

Loss = Revenue - total cost

Loss = $ 13.875 x 450 units - [ 3000 + 24 x 450  - 0.045 x (450)² + 0.00005 x (450)³ ]

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Shutdown price = $ 13.875

Loss = - $ 3000