Question

In: Economics

4) Suppose a product can be produced using virgin ore at a marginal cost given by...

4) Suppose a product can be produced using virgin ore at a marginal cost given by MC1 = 0.5q1 and with recycled materials at a marginal cost of MC2 = 5 + 0.1q2.

a. If the inverse demand curve were given by P = 10 ? 0.5(q1 + q2), how many units of the product would be produced with virgin ore and how many with recycled units?

b. How would this change if the inverse demand function were P = 17 ? 0.5(q1 + q2)?

Solutions

Expert Solution

Answer : 4) Given , MC1 = 0.5q1 and MC2 = 5 + 0.1q2

a) Inverse demand function means there is a negative (-) relationship between price and quantity demanded. So, the given inverse demand function becomes, P = 10 - 0.5(q1 + q2)

=> P = 10 - 0.5q1 - 0.5q2

TR1 (Total Revenue) = P*q1 = (10 - 0.5q1 - 0.5q2)*q1 = 10q1 - 0.5q1^2 - 0.5q1q2

MR1 (Marginal Revenue) = TR1 / q1 = 10 - q1 - 0.5q2

At equilibrium condition for virgin ore, MR1 = MC1

=> 10 - q1 - 0.5q2 = 0.5q1

=> 10 - 0.5q2 = 0.5q1 + q1

=> 10 - 0.5q2 = 1.5q1

=> q1 = (10 - 0.5q2) / 1.5

=> q1 = 6.667 - 0.333q2 ...........................(i)

Now, TR2 = P*q2 = (10 - 0.5q1 - 0.5q2)*q2 = 10q2 - 0.5q1q2 - 0.5q2^2

MR2 = TR2 / q2 = 10 - 0.5q1 - q2

At equilibrium condition for recycled materials, MR2 = MC2

=> 10 - 0.5q1 - q2 = 5 + 0.1q2

=> 10 - 5 - 0.5q1 = 0.1q2 + q2

=> 5 - 0.5q1 = 1.1q2

=> q2 = (5 - 0.5q1) / 1.1

=> q2 = 4.545 - 0.455q1 ......................................(ii)

Now, by putting the value of q2 in equation (i), we have,

q1 = 6.667 - 0.333 ( 4.545 - 0.455q1)

=> q1 = 6.667 - 1.513 + 0.152q1

=> q1 - 0.152q1 = 5.154

=> 0.848q1 = 5.154

=> q1 = 5.154 / 0.848

=> q1 = 6.078

From equation (ii) we get,

q2 = 4.545 - 0.455 * 6.078

=> q2 = 1.78

Therefore, virgin ore produces , q1 = 6.078 units and recycled materials produces, q2 = 1.78 units.

b) In case of inverse demand function, P = 17 - 0.5(q1 + q2)

=> P = 17 - 0.5q1 - 0.5q2

TR1 = P*q1 = (17 - 0.5q1 - 0.5q2) *q1 = 17q1 - 0.5q1^2 - 0.5q1q2

MR1 = TR1 / q1 = 17 - q1 - 0.5q2

At equilibrium condition for virgin ore, MR1 = MC1

=> 17 - q1 - 0.5q2 = 0.5q1

=> 17 - 0.5q2 = 0.5q1 + q1

=> 17 - 0.5q2 = 1.5q1

=> q1 = (17 - 0.5q2) / 1.5

=> q1 = 11.333 - 0.333q2 ............................(iii)

Now, TR2 = P*q2 = (17 - 0.5q1 - 0.5q2)*q2 = 17q2 - 0.5q1q2 - 0.5q2^2

MR2 = TR2 / q2 = 17 - 0.5q1 - q2

At equilibrium condition for recycled materials, MR2 = MC2

=> 17 - 0.5q1 - q2 = 5 + 0.1q2

=> 17 - 5 - 0.5q1 = 0.1q2 + q2

=> 12 - 0.5q1 = 1.1q2

=> q2 = (12 - 0.5q1) / 1.1

=> q2 = 10.909 - 0.455q1 ................................(iv)

Now by putting the value of q2 in equation (iii), we have,

q1 = 11.333 - 0.333 (10.909 - 0.455q1)

=> q1 = 11.333 - 3.633 + 0.152q1

=> q1 - 0.152q1 = 7.7

=> 0.848q1 = 7.7

=> q1 = 7.7 / 0.848

=> q1 = 9.08

From equation (iv) we get,

q2 = 10.909 - 0.455 * 9.08

=> q2 = 6.78

Therefore, virgin ore produces, q1 = 9.08 units and recycled materials produces, q2 = 6.78 units.

By comparing the values of q1 and q2 for given question's a and b's demand functions, it is clear that in case of b's demand function both the virgin ore and recycled materials produces more units than a's demand function's production levels.


Related Solutions

1. Suppose that a competitive firm’s marginal cost function is given by MC(q) = 4 +...
1. Suppose that a competitive firm’s marginal cost function is given by MC(q) = 4 + aq, where a > 0 and the market price is 16. a) What is the firm’s profit-maximizing level of output when a = 4? b) Suppose the firm’s fixed costs increase and the value of a decreases. How will this affect the firm’s profit-maximizing level of output? c) Suppose the firm’s producer surplus at its profit-maximizing level of output is 36. What is the...
Suppose a company’s product might either be produced unsustainably with marginal costs of ?? = 1...
Suppose a company’s product might either be produced unsustainably with marginal costs of ?? = 1 (with probability 50%) or sustainably with ?? = 4, and that if the company is transparent about its production process, it has to install a block chain technology which costs ?? ∗ ? where ? equals 6 if the company is unsustainable and 1/2 if it not. The market for the product is perfectly competitive such that ? = ?[??] where ? is the...
A certain monopoly has a marginal cost that depends on the quantity produced. The marginal cost...
A certain monopoly has a marginal cost that depends on the quantity produced. The marginal cost is MC = 4Q The marginal revenue curve is: MR = 40 – 4Q The demand curve is: D = 40 - 2Q Fixed cost of production $10, variable cost is $5 per unit produced. a) Graph the MR, MC and demand curves! b) Which quantity the monopoly will produce at which price? c) Calculate the profit!
When marginal product is increasing: Marginal cost is increasing          c. marginal cost is constant Marginal cost is...
When marginal product is increasing: Marginal cost is increasing          c. marginal cost is constant Marginal cost is decreasing         d average product is decreasing
Marginal Cost and Marginal Benefit Suppose you are given a benefit equation as B(t) = -3t2...
Marginal Cost and Marginal Benefit Suppose you are given a benefit equation as B(t) = -3t2 + 15t + 160 and cost equation as C(t) = t2 -5t + 60, where t = time in hours. Find a NB equation. b. Solve for MB, MC, and find a MNB expression. c. Find the optimal amount of time you should spend on the activity and the value of NB associated with that optimal time. d. If you were given the following...
Suppose the inverse demand for a product produced by a single firm is given by P...
Suppose the inverse demand for a product produced by a single firm is given by P = 200 − 5Q and this firm has a marginal cost of production of MC = 20 + 2Q. a. If the firm cannot price-discriminate, what is the profit-maximizing price and level of output for this monopolist? What are the levels of producer and consumer surplus in the market? What is the deadweight loss? b. If the monopolist can practice perfect price discrimination, what...
Suppose the inverse demand for a product produced by a single firm is given by: P...
Suppose the inverse demand for a product produced by a single firm is given by: P = 76 – 4(Q) and this firm has a marginal cost of production of: MC = 10 1.  If the firm cannot price-discriminate , what is the profit-maximizing a)price     b)and level of output?     2. If the firm cannot price-discriminate , what is : a)the consumer surplus     , b)the producer surplus     c)the dead-weight loss     3. If the firm can practice perfect price discrmination,...
Suppose the inverse demand for a product produced by a single firm is given by: P...
Suppose the inverse demand for a product produced by a single firm is given by: P = 76 – 4(Q) and this firm has a marginal cost of production of: MC = 10 1.  If the firm cannot price-discriminate , what is the profit-maximizing a)price?    b)and level of output?     2. If the firm cannot price-discriminate , what is : a)the consumer surplus?    b)the producer surplus?    c)the dead-weight loss?    3. If the firm can practice perfect price...
Suppose that the market for bike locks is served by a monopolist with marginal cost given...
Suppose that the market for bike locks is served by a monopolist with marginal cost given by MC = 20. It is also the case that inverse demand for bike locks is given by P = 100 – 0.25Q. (20 points) What are the equilibrium price and quantity? (15 points) What is the monopoly profit? (15 points) What is the deadweight loss associated with monopoly power?
Suppose a firm's hourly marginal product of labour is given by MPN = A(200 – N)....
Suppose a firm's hourly marginal product of labour is given by MPN = A(200 – N). Here, A measures labour productivity and N is the number of workers. If A = .2 and the real wage is $10 per hour, how much labour (number of workers N) will the firm want to hire? Suppose the real wage rate rises to $20 per hour. How much labour (number of workers N) will the firm want to hire? With the real wage...
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT