Question

In: Economics

Consider the constant-elasticity demand function Q = p−ε, where ε > 0. Solve for the inverse...

Consider the constant-elasticity demand function Q = p−ε, where ε > 0.

Solve for the inverse demand function p(Q).
Calculate the demand price elasticity.
Show that p(Q)/MR(Q) is independent of the output level Q. (Hint: Use the relationship between marginal revenue and the elasticity of demand.)

Expert Solution

The demand function is given as .

The inverse demand function would be as or or or or .

The demand price elasticity would be or or or or or or .

The TR would be as or or , and hence MR would be or or or . Thus, or or , which is not a function of Q, and hence is independent of Q.

Rahul Sunny answered 2 years ago

Consider the following inverse demand function, p(Q) = a-bQ, Q = q1 +q2, where a and...

Consider the following inverse demand function, p(Q) = a-bQ, Q = q1 +q2, where a and b are positive parameters and qi denotes firm i's output, i = 1, 2. Assume that the total cost of firm i is cqi2/2, with c > 0. Firms choose quantities simultaneously and non cooperatively (Cournot competition). The Cournot game described above is infinitely repeated. Firms use grim trigger strategies (infinite Nash reversion). Firms discount future profits at a rate r > 0. a)...

Consider a market where the inverse demand function is p =100 – Q, Q = q1+q2....

Consider a market where the inverse demand function is p =100 – Q, Q = q1+q2. Both firms in the market have a constant marginal cost of $10 and no fixed costs. Suppose these two firms are engaged in Cournot competition. Now answer the following questions: a) Define best response function. Find the best response function for each firm. b) Find Cournot-Nash equilibrium quantities and price. c) Compare Cournot solution with monopoly and perfect competitive solutions.

Consider a market where the inverse demand function is P = 100 - Q. All firms...

Consider a market where the inverse demand function is P = 100 - Q. All firms in the market have a constant marginal cost of $10, and no fixed costs. Compare the deadweight loss in a monopoly, a Cournot duopoly with identical firms, and a Bertrand duopoly with homogeneous products.

Consider a market where inverse demand is given by P = 40 − Q, where Q...

Consider a market where inverse demand is given by P = 40 − Q, where Q is the total quantity produced. This market is served by two firms, F1 and F2, who each produce a homogeneous good at constant marginal cost c = $4. You are asked to analyze how market outcomes vary with industry conduct: that is, the way in which firms in the industry compete (or don’t). First assume that F1 and F2 engage in Bertrand competition. 1....

If the demand curve is Q(p)=pa (where a<0), what is the elasticity of demand? If the...

If the demand curve is Q(p)=pa (where a<0), what is the elasticity of demand? If the marginal cost is $1, and a=-3, what is the profit-maximizing price?

For a Monopolist, the inverse demand function is P=80 – Q (Or, the Demand function is...

For a Monopolist, the inverse demand function is P=80 – Q (Or, the Demand function is Q = 80 – P, if you prefer). The firm’s total cost function is 20 + 5Q + .5Q2. Identify profit maximizing quantity and price Q = _____, P = _____ Graph the relevant curves in the area provided (Hint: If you have difficulty drawing the graph, try plotting a couple of points along the curve) Identify CS and PS in the graph Calculate...

Consider a Cournot duopoly with the following inverse demand function:p(Q) =a−Q where p is the price...

Consider a Cournot duopoly with the following inverse demand function:p(Q) =a−Q where p is the price of the product and Q is the total amount of goods exchanged in the market. The total costs areC(q1) = 300q1, C(q2) = 300q2 for firm 1 and firm 2, respectively. But the demand is uncertain (i.e., a new product may be introduced soon which will decrease the demand drastically). Firm 1 learns whether demand will be high (a =1800) or small (a=900) before...

The inverse demand function in a market is given by p=32-Q where Q is the aggregate quantity produced.

The inverse demand function in a market is given by p=32-Q where Q is the aggregate quantity produced. The market has 3 identical firms with marginal and average costs of 8. These firms engage in Cournot competition. a) How much output does each firm produce? b) What is the equilibrium price in the market? c) How much profit does each firm make? d) Consider a merger between two firms. Assuming that due to efficiency gains from the merger,...

Two firms compete in a market with inverse demand P(Q) = a − Q, where the...

Two firms compete in a market with inverse demand P(Q) = a − Q, where the aggregate quantity is Q = q1 + q2. The profit of firm i ∈ {1, 2} is πi(q1, q2) = P(Q)qi − cqi , where c is the constant marginal cost, with a > c > 0. The timing of the game is: (1) firm 1 chooses its quantity q1 ≥ 0; (2) firm 2 observes q1 and then chooses its quantity q2 ≥...

The inverse demand function for a medication is p = 10000 - 0.5Q, where p is...

The inverse demand function for a medication is p = 10000 - 0.5Q, where p is the market price and Q is quantity demanded MC = 100 + 0.05Q. What is the competitive market price and quantity in this market, and producer and consumer surplus and social welfare? Suppose production creates an externality. You can assume marginal external costs are simply MEC = 400. Based on this information, what is your estimate for a regulated market outcome where marginal social...