Microeconomics Problem Set

Task 1: Utility Function and Elasticity of Demand

1. The utility function for goods y and x is the following: u(x, y) = eyb where a + b = 1. Noting that price elasticities are derived from the uncompensated demand function x(p., Pp 0, 

a) calculate the price elasticity of demand for good x;

b) calculate the income elasticity of demand for good x; and

c) calculate the cross-price elasticity of demand for good x. 

2. Consider a person with £12 at his disposal for purchasing two goods, x and y. His utility function is given by U(x, y) = 2x + y, where x and y denote the quantity of the two goods, respectively.

a) Given the following prices, find graphically the optimal bundle the person will choose: p. = E1/kg and p, = E1/kg.

b) If prices were p. = £2/kg and py = £1/kg.. what would the optimal bundle be?

c) Decompose the effect of the price increase on the person's optimal choice in terms of the substitution and income   effects. 

3. Consider a firm with its production function of the Cobb-Douglas form as the following: q = 1000•45K015 where q is output, and L and K are labour and non-labour inputs, respectively.

a) Find the equation of the isoquant for q = 100 and sketch the graph of the isoquant.

b) Give an economic interpretation to the slope and curvature of the isoquant.

c) Can you tell if this production function exhibits increasing, constant, or decreasing returns to scale? Explain. 

4. Suppose that the market demand for a product is given by Qi, = D(P) = A - BP. Suppose also that the typical firm's cost function is given by C(q) = k + aq + bq2. 

a) Compute the long-run equilibrium output and price for the typical firm in this market.

b) Calculate the equilibrium number of firms in this market as a function of all the parameters in this problem.

c) Describe how changes in the demand parameters A and B affect the equilibrium number of firms in this market. 

5. A monopolist can produce at constant average and marginal costs of AC = MC = 5. The firm faces a market demand curve given by 0 = 53 - P: 

(a) Calculate the profit-maximising price-quantity combination for the monopolist. Also calculate the monopolist's profits.

(b) What output level would be produced by this industry under perfect competition (where price = marginal cost)?

(c) Calculate the consumer surplus obtained by consumers in case (b). Show that this exceeds the sum of the monopolist's profits and the consumer surplus received in case (a).

(d) What is the value of the "deadweight loss' from monopolization? 

6. The figure below shows a production possibility frontier (PPF) for a society producing two goods. Food and Clothing. 

(a) Explain why points on the PPF are technically efficient, and why producers have the incentive to be technically efficient.

(b) How does competition ensure that an efficient product mix is attained? Explain, with graphic illustration, with respect to the behaviour of both the consumers and the producers.

(c) Assume that the society is composed of only two individuals, A and B. who are equal in terms of power status. Now, if point 'a" is the efficient product mix, draw a possible Edgeworth box and indifference curves for A and 8 on the figure.