State the first theorem of welfare economics- Microeconomics
Q1. An isolated economy produces just one crop, corn, using a fixed quantity of land and variable quantities of labor and tractors. Labor is also used to produce the tractors.
(a) Which of the following conditions are needed for Pareto optimality:
i. The marginal rate of technical substitution of tractors for labor in the production of corn equals the marginal product of labor in the production of tractors.
ii. The marginal utility of leisure equals the marginal product of labor in corn multiplied by the marginal utility of corn, for each individual.
iii. The marginal rate of substitution of leisure for corn is equal for all consumers.
Explain your answers in detail.
(b) Which of the above conditions would be satisfied in a competitive market economy? How?
Q2. A monopolist faces two types of consumers, with demand curves given by p1 = 3-q1 and p2 = 4-2q2. There are 100 consumers of each type. The (constant) marginal cost of production is 1. Does the profit-maximizing two part tariff involve charging a unit price below marginal cost, once the consumers have paid a fixed fee? Explain why or why not.
Q3. Is it possible to find a two-good utility function such that the two goods are complements? If so, find one; if not, explain why none can be found.
4. (a) State the First Theorem of Welfare Economics for an exchange economy.
(b) Give an example in which the First Theorem does not hold, and explain which assumptions of the theorem are violated in your example.
(c) State the Second Theorem of Welfare Economics for an exchange economy.
(d) Give an example in which the Second Theorem does not hold, and explain which assumptions of the theorem are violated in your example.
Q5. (a) State and prove Hotelling’s Lemma.
(b) Show that a factor of production is inferior if and only if an increase in the price of that factor reduces marginal cost.
Q6. Consider an exchange economy with two goods, fish and cheese, and two people, One and Two. One is endowed with .5 pounds of fish and .5 pounds of cheese, and two has the same endowment. One’s utility function is u(f, c) = fc, and Two’s utility function is u(f, c) = f2+c2.
(a) Can you find a competitive equilibrium for this economy? If so, is it Pareto optimal? If not, explain which assumptions of the First Welfare Theorem are violated.
(b) What is the set of Pareto optimal allocations in this economy? Can all these allocations be supported as competitive equilibria? Explain.
Q7. State whether the following assertions are true, false or ambiguous, and explain why.
(a) Constant relative risk aversion implies that the demand for insurance is a decreasing function of wealth.
(b) If a state seeks to attract new industries by subsidizing production costs, it is more effective to subsidize either labor or capital costs than to allocate the same total to a mixture of labor and capital subsidies.
(c) A firm uses 10 units of labor and 20 units of capital to produce 10 units of output. The marginal product of labor is 0.5. If there are constant returns to scale the marginal product of capital must be 0.25.
