Chapter 7, Problem 2P

Problem 2. If the consumer preference can be represented by a CES function with δ = 0.5, i.e. u(x, y) = x0.5 + y0.5. Let the prices and income be (px, py, w).  1. Set up the Lagrangian expression.2. Take the first-order conditions.3. Substitute into budget constraint to derive the optimal consumption bundles.

1. A town relies on four different sources for its non-drinking water needs: dam water, reclaimed water, rain water, and desalinated water. The different sources carry different risks and costs. For instance, desalinated water is fully reliable due to abundant sea water, but it is more expensive than other options. Reclaimed water also has relatively lower risk than rain or dam water since a certain amount can be obtained, even during the dry. season, by the treatment of daily generated waste water. Using any of the four options requires an investment in that resource. The return on a particular water source is defined as the amount of water generated by the source per dollar of investment in it. The expected returns and standard deviations of those returns for the four water sources are described in the following table: Water resource Expected return St. Deviation Dam water 2.7481 0.2732 Reclaimed water 1.6005 0.0330 Rain water 0.5477 0.2865 Desalinated water 0.3277 0.0000 Higher…

1. Imagine a society that produces military goods and consumer goods, which we'll call "guns" and "butter." a. Draw a production possibilities frontier for guns and butter. Using the concept of opportunity cost, explain why it most likely has a bowed-out shape. b. Show a point that is impossible for the economy to achieve. Show a point that is feasible but inefficient. c. Imagine that the society has two political parties, called the Hawks (who want a strong military) and the Doves (who want a smaller military). Show a point on your production possibilities frontier that the Hawks might choose and a point the Doves might choose. d. Imagine that an aggressive neighboring country reduces the size of its military. As a result, both the Hawks and the Doves reduce their desired production of guns by the same amount. Which party would get the bigger "peace dividend," measured by the increase in butter production? Explain.