In this week's problem set there are several mathematical problems. Please illustrate your answers with diagrams and interpret your results.

- Suppose the taxi industry in a city consists of many individual drivers who rent
their vehicles, pay a fee to a dispatching service, purchase insurance, and pay for
their petrol and maintenance costs. They are free to set their own prices.
To encourage the use of cabs rather than private vehicles, the local government
implements a policy of rebating 50p per litre of fuel tax to cab drivers.

- What is the effect on the market in the short run? Use a diagram or diagrams in your answer.
- What is the effect on the market in the
*long*run? Again use a diagram. - How do your answers change if the city requires drivers to have a special tax licence, with the number of licenses fixed, and small enough to have an impact on equilibrium prices?

- In a competitive industry, all firms have cost functions
*C*(*q*) = 32 + 2*q*^{2}. The market demand function is*Q*= 3000 - 75*P*.

- Write down the profit function for an individual firm when the market price is
*P*, and hence show that the firm's supply curve is*q*=*P*/4. - Find the firm's average cost function, and determine the price in long-run industry equilibrium.
- In the short-run there are 200 firms. Find the industry supply function, and hence the equilibrium price, and the output and profits of each firm.
- How many firms will there be in long-run equilibrium?
- Suppose that a trade association could limit the number of firms entering the industry.
- Find the equilibrium price and quantity, in terms of the number of firms
*n*. - What upper limit would the trade association set if it wanted to maximise industry revenue?

- Write down the profit function for an individual firm when the market price is
- A publisher pays the author of a book a royalty of 15% of the total revenue. Demand for the
book is
*q*= 200 - 5*p*and the production cost is*c*(*q*) = 10 + 2*q*+*q*^{2}. Think of the market for this book as a monopoly.

- Find the average and marginal cost of production.
- Write down the expression for profits and find the optimal quantity of books sold from the publisher's perspective, given that the publisher is a monopolist who maximises profits.
- How many books would the author like to sell?

- EU consumers have a demand function for umbrellas given by
*D*(*p*) = 200 - 50*p*, where*p*is the price per unit. Umbrellas are supplied by EU firms and US firms though there is no demand for umbrellas in the US. There are 50 umbrella firms in the EU and 50 in the US. Assume that all firms are identical and have a cost function given by*C*(*q*) =*q*^{2}.

- What is the aggregate supply function for umbrellas in the EU?
- What is the equilibrium price and quantity sold in the EU?
- What is the elasticity of demand at the equilibrium price?
- Now suppose the EU imposes a 1 euro tariff per unit umbrella on US imports. What is the new EU price for umbrellas paid by the consumers?
- How many umbrellas are supplied by US firms and how many are supplied by EU firms?
- Estimate the effect of the tariff on the welfare of EU umbrella producers and consumers.

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