Economics 473/673, Spring 2020

Development Microeconomics

Econ 673 Problem Set #2

(due Friday March 20, 5pm on Canvas)

1. Consider the following example, similar to what we studied in class, except that now the per-unit negative externality is increasing in the level of acres cleared:

Solve for the private and socially optimal levels of output and for the net social loss caused by the externality.  How much of a tax (or, say, price of a permit) should a central government put on land clearing in order to force land clearing to the (lower) socially optimal level?  Use calculus to show the relationship of this optimal tax on land-clearing to the parameters in the model and give intuition behind these relationships.

2. A new crop to a peasant region, Belgian Endive, has a mean expected income of 100 pesos uniformly distributed over [50, 150] but the traditional corn crop has mean expected income of 70 pesos, uniformly distributed over [60, 80].   If subsistence is 62, which will be chosen by a “safety-first” peasant?

3. Suppose a sharecropping peasant, Jorge, has a utility function equal to where Jorge has one unit of time to divide between wage work and own-crop labor, pis output price, and s is the share that Jorge gets to keep from his crop.  K is a fixed portion of capital, L is labor.  Totally differentiating Jorge’s first-order condition, use comparative statics to prove that Jorge spends more time on his crop, the higher his crop price, the lower his outside wage, and the larger his share, i.e. ,,.
4. Consider a household that maximizes utility that is based not only on the yield of its crops, but also taking into account the risk—the variance in the yield—involved in each crop.  Suppose that maize is a stable crop with a yield of and a variance on that yield of zero.  Let’s suppose however, that coffee has a higher yield of , but also a positive variance, The household has one unit of land to divide between the two crops, maize and coffee. The fraction (share) of the land used to cultivate coffee is s. The utility function taking into account the risk-aversion of the household is , where v is the household’s coefficient of risk aversion.

1. a) What is the share of the land that the household should allocated to coffee?
2. b) Show how this share depends on the parameters , , , and
3. c)  If , , = 50, and , what is s*, the fraction of the plot allocated to coffee?

5. Do all exercises 1-4 for Chapter 5 in Games in Economic Development on p.266.

6.  (Mandatory for Masters Students, 10-point bonus for Undergraduates.)

Suppose our peasant Jorge maximizes a production function , where has one unit of time to divide between working as a sharecropper and selling his labor on the wage market, where s is the share that Jorge gets to keep from his own crop, K is capital he allocates to his field (draft animals), L is labor, w is the prevailing wage rate, and r is the rental rate for draft animals.

1. a) Establish first-order conditions for the problem, totally differentiate, and then using Cramer’s rule on the 2×2 matrix, prove Marshallian inefficiency, e.that .

(Remember we always assume that .)

1. b) Do this problem with a working capital constraint equal to W, where working capital is used only to purchase physical capital.  Here you will use Cramer’s rule on a 3×3 matrix.