Economics 711: Microeconomic Theory II - final exams
Q1. A factor of production is called inferior if the conditional demand for that factor falls as output is increased while factor prices are held constant.
(a) Draw an isoquant map showing a technology with an inferior factor of production.
(b) Do you think it is possible that an increase in the price of some factor of production might cause a profit maximizing firm to increase output? Explain.
(c) Analyze the following argument:
"In consumer theory it can happen that an increase in the price of some good leads to an increase in the quantity demanded (if the good is inferior and the income effect of the price increase is stronger than the substitution effect). Similarly it can happen that an increase in the price of a factor of production leads to an increase in the quantity of that factor demanded by a profit-maximizing firm. This can happen if the higher factor price leads to a reduction in output, and if the factor is inferior, so that the reduction in output leads to an increase in the use of this factor."
Q2. (a) Fill in the missing arguments, and explain why the equation is true:
(∂e(p, ?)/∂pi)(∂v(p, ?)/∂Y) + (∂v(p, ?)/∂pi) = 0
where e(·,·) is the expenditure function of a utility-maximizing consumer, and <(·,·) is the indirect utility function.
(b) Suppose a consumer has the following utility function
U(x1, x2 ,...,xk) = αx1 + u2(x2) + u3(x3) + ... + uk(xk)
where α is a positive parameter, and the functions ui(·) are all concave.
i. Is it true that x1 absorbs all income effects: the income elasticity of demand for each of the other goods is zero? Explain.
ii. Is it true that the cross price elasticities of demand are all zero? Explain.
Q3. Consider a one-period pure exchange economy with three agents (1, 2 and 3) and three goods (x, y and z). Agent i's consumption vector is (xi, yi, zi). Agent 1, 2 and 3's endowment vectors are (xe, 0, 0), (0, ye, 0) and (0, 0, ze) (each agent is endowed with only one type of good). The agents' utility functions are as follows:
U1(x1, y1, z1) = x1(y1 + z1)
U2(x2, y2, z2) = x2y2
U3(x3, y3, z3) = x3z3.
(a) Compute a Walrasian equilibrium for this economy under the condition that good z cannot be traded.
(b) Assume now that there exists a complete set of markets in all three goods. Compute a Walrasian equilibrium for this case.
(c) Can you Pareto-rank the equilibria which you found? If so, in which order? Explain your findings.
4. State whether the following assertion is true, false or ambiguous, and explain why.
"There are many firms which produce wooden chairs, and many firms which produce wooden tables. If these firms are all separate, and if they all maximize profits, taking prices as given, then the equilibrium cannot be efficient.
This is because when the chair firms increase output they bid up the price of wood, which reduces the profits of the table-producing firms. But the chair firms ignore the effect of their output decisions on the profits of the table firms.
An efficient equilibrium would be achieved if all of the firms produced both tables and chairs."
Q5. Consider the function
Y(p1, p2, p3, y) = (y-p1-3p2-2p3/(p1).2(p2).3(p3).5)
Is this a valid indirect utility function for a consumer who buys three commodities at prices p1, p2, p3, with an income of y? If so, find the indirect utility function. If not, explain why.
Q6. A profit maximizing monopoly firm sells to many identical buyers. The marginal cost of production is a constant, c, with c < .5. Each buyer's demand function is p = 1 - q. The firm decides to charge a two-part tariff, so that a consumer who buys a quantity q must pay the amount y = a + bq. How would the firm choose a and b?
(a) Now suppose a second firm enters the industry. This firm has the same production cost as the first firm. Can you find a Nash equilibrium in which each firm chooses a two-part tariff which is optimal, given the other firm's two-part tariff?