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LONDON SCHOOL OF ECONOMICS Leonardo Felli Department of Economics 32L.4.02; x
Assignment 5 EC487 Advanced Microeconomics Part I
1. Consider a technology that produces a single output y using two inputs x 1 and x 2. Assume that the techology has the following characteristics. Each unit of output y can be produced either by means of 3 units of input x 1 and a quantity of input x 2 such that 0 ≤ x 2 ≤ 1 , or by means of 1 unit of input x 2 and a quantity of input x 1 such that 0 ≤ x 1 ≤ 3.
(i) Draw the map of isoquants associated with this technology. (ii) Is an input requirement set for this technology convex? Explain your answer.
(iii) Identify the cost function (the solution to the cost minimization prob- lem) for this technology distinguishing the case in which (3 w 1 ) ≥ w 2 and the case (3 w 1 ) ≤ w 2.
(iv) Draw the cost function you computed in (iii) above as a function of y. Is this cost function an homogenous function of a given degree in y?
(v) From your answer to (iv) what type of returns to scale does this tech- nology exhibit? Explain.
2. Consider a perfectly-competitive pure-exchange economy with one com- modity and two consumers. Assume that the two consumers’ endow- ments of the commodity are strictly positive. What is the Walrasian equi- librium allocation of this economy? What is the equilibrium relative price? Explain.
3. Consider an exchange economy with two consumers, A and B, and two goods, X and Y. The economy is endowed with 5 units of X and 7 units of Y. Consumer A’s utility function is
UA(xA, yA) = xA yA,
where xA is her consumption of X and yA is her consumption of Y. Con- sumer B’s utility function is
UB (xB , yB ) = 2 xB + 3 yB ,
where xB is his consumption of X and yB is his consumption of Y. Is the allocation (xA, yA, xB , yB ) = (2, 4 , 3 , 3) Pareto efficient? Explain your answer carefully.
4. Consider a different exchange economy with two consumers, A and B, and two goods, X and Y such that the economy is endowed with 2 units of X and 3 units of Y. Consumer A’s utility function is
UA(xA, yA) = 3xA + yA,
and consumer B’s utility function is
UB (xB , yB ) = xB + 3 yB.
Find the Pareto efficient allocations.
5. Once upon a time students in economics received no grades and worked only for the pleasure of learning. In those days of yore, each student received cardinal utility G(x) from working at the level of effort of x hours per day. (i) At what level of effort did each student choose to work? Then one dark day, grades were introduced. Now each student cares not only about learning, but also about his relative standing in the class. If y
