# Calculating optimal output and market price

Propylene is used to make plastic. The propylene industry is perfectly competitive and each producer has a long run total cost function given by

LTC= 1/3 Q^3 -6Q^(2 )+40Q

Where Q denotes the output of the individual firm.

The market demand for propylene is

X =2200 -100P

Where X and P denote the market output and price respectively.

1. Calculate the optimal output produced by each firm at the long run competitive equilibrium (LRCE).

2. Calculate the market price and market output at the LRCE.

3. Calculate the number of firms at the LRCE.

Suppose the demand curve shifts to

X =A -100P

Where A is a positive number.

Calculate how large A would have to be so that in the new LRCE, the number of firms is twice what it was in the initial equilibrium.

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#### Solution Preview

1. Calculate the optimal output produced by each firm at the long run competitive equilibrium (LRCE).

At LRCE, long run marginal Cost is equal to long run average cost.

Given, LTC= 1/3 Q^3 -6Q^(2 )+40Q

Long run marginal cost (LRMC) is given by

LRMC=d(LTC)/dQ=Q^2-12Q+40

Long run average cost (LRAC) is given by

LRAC=LTC/Q=1/3 ...

#### Solution Summary

Solution describes the steps to calculate optimal output and market price. It also calculates number of firms present in the market in the given scenario.

Bundling & Intra-firm Pricing Question

The Xerxes Company is composed of a marketing division and a production division. The marketing division packages and distributes a plastic item made by the production division. The demand curve for the finished product sold by the marketing division is

P0=200-3Q0

Where P0 is the price sold (in dollars per pound) of the finished product and Q0 is the quantity sold (in thousands of pounds). Excluding the production cost of the basic plastic item, the marketing division's total cost function is

TC0=100+15Q0

Where TC0 is the marketing division's total cost (in thousands of dollars). The productions division's total cost function is

TC1 =5+3Q1+0.4Q1^2

Where TC1 is total production cost (in thousands of dollars) and Q1 is the total quantity produced of the basic plastic item (in thousands of pounds). There is a perfectly competitive market for the basic plastic item, the price being $20 per pound.

What is the optimal output for the production division?

What is the optimal output for the marketing division?

What is the optimal transfer price for the basic plastic item?

At what price should the marketing division sell its product?