Harvard Economics 2020a Problem Set 2

API 111 – Econ 2020 – HBS 4010 Fall 2007, Problem Set #2 Due: Wednesday, October 3, 2007. Answer the following 5 questions: 1. Cobb-Douglas Utility: Consider the utility function:

u ( x1 , x2 ) = x1a1 x2a2

Let p1 and p2 be the (strictly positive) prices of x1 and x2, respectively, and assume the consumer’s wealth is w. 1a) 1b) 1c) 1d)

Is this utility function homogeneous in (x1,x2)? If it is, of what degree? If not, why not? Derive the consumer’s Walrasian demand functions Derive the value of the Lagrange multiplier. Derive the consumer’s indirect utility function, v(p,w).

1e) Verify that

dv ( p, w ) =λ. dw

1f) ** optional ** Explain why your answer to part e is an application of the envelope theorem.

2. Another Comparative Statics Exercise A consumer consumes only one good, bananas. Let b denote the number of bananas eaten. The consumer’s willingness to pay (measured in dollars) for b bananas is log(b+1), where “log” denotes the natural logarithm (as it always does in economics). Let p > 0 denote the price of a banana. Thus, the total surplus (i.e., utility) of the consumer that buys and eats b bananas is given by:

u ( b ) = log ( b + 1) − pb .

2a. Given the consumer’s objective function, u(b), derive the first-order conditions for the consumer’s optimal choice of b, which you may denote b*. Note: be sure to address the fact that the optimal choice of b must be non-negative. 2b. Explain whether you can be sure that the b* that satisfies the condition you found in part 3a is really a global maximum. 2c. How will the consumer’s optimal choice of b respond to an increase in the price of bananas? Does your answer depend on p? 2d. Write down an expression for the optimized value of the consumer’s objective function, and explain how the optimized level of the consumer’s utility reacts to a change in p. Does your answer depend on p? Explain and interpret your answer.

3. Perfect Complements: Consider the following utility function: U(x1,x2) = min(ax1,x2) Where a is an unknown positive real number. 3a) Draw a typical indifference curve for this utility function. 3b) Suppose p1 and p2 are the (positive) prices of x1 and x2 respectively. If the consumer has total wealth w>0, state the consumer’s (Walrasian) demand functions. 3c) Consider the utility function: V(x1,x2) = x1α x21-α, where α is a fixed but unknown positive number. You believe that a consumer has either utility function V() or U() as defined above. You observe that when (p1,p2) = (1,1) and income is w = 2, the consumer chooses consumption bundle (x1,x2) = (1,1). Based on this information, can you determine which utility function represents the consumer’s preferences? Why or why not. 3d) Suppose you observe in addition to the information in (3) above that when prices are (p1,p2) = (2,2) and income is 4, the consumer chooses consumption bundle (1,1). Does the additional information allow you to make any new inferences about which utility function represents the consumer’s preferences? Why or why not.

3e) Suppose you observe in addition to the information in (3) and (4) above that when prices are (p1,p2) = (1,2) and income is 3, the consumer chooses consumption bundle (1,1). Does the additional information allow you to make any new inferences about which utility function represents the consumer’s preferences? Why or why not. 3f) In general, how many data points are necessary to distinguish between utilty function U() and V()? What restrictions must be placed on these data points?

4.

Consider the following system of L equations: log x1 = a1 + e11 log p1 + e12 log p2 + … + e1L log pL + e1w log w log x2 = a2 + e21 log p2 + e22 log p2 + … + e2L log pL + e2w log w . . . log xL = aL + eL1 log p1 + eL2 log p2 + … + eLL log pL +eLw log w

You have a data set consisting of observations of the consumption bundle chosen at various levels of prices and income. Through econometric techniques (that you will learn about next semester) you can regress x1, x2, … , xL on prices p1, p2, …, pL in order to achieve estimates of the coefficients a1, …, aL and eij i = 1, …, L, j = 1, …, L. 4a) What is coefficient eij an estimate of? Hint: eij =

∂ log xi ∂x p j = i . ∂ log p j ∂p j xi

4b) What restrictions are imposed on the eij terms by Walras’ Law and homogeneity of degree zero of the Walrasian demand functions? Which restrictions require estimates of all equations in order to test and which do not? 4c) Suppose there are only two goods. If your estimates are e11 = -0.5, e12 = 1, e1w = 0.5, e21 = -0.7, e22 = - 0.2, e2w = -0.7, does this provide evidence in favor or against homogeneity of degree zero holding? Does it provide evidence in favor or against Walras’ law holding? 4d) Suppose that based on the statistical evidence, you believe that homogeneity of degree zero does not hold. Briefly suggest two or three steps you might take to address this problem.

5. There are two commodities, x1 and x2, the prices of which are given by p1 > 0 and p2 > 0, respectively. Consider a utility-maximizing consumer with indirect utility function:

⎛ w ⎞ v ( p, w ) ≡ a ln a + b ln b + ( a + b ) ln ⎜ ⎟ − a ln p1 − b ln p2 ⎝ a+b⎠ where w > 0 is the consumer’s initial wealth and a and b are positive real numbers. 5a) Derive the consumer’s Walrasian demand functions for x1 and x2. 5b) Derive the consumer’s marginal utility of wealth (assuming that the consumer is utility maximizing). 5c) Assume that prices and wealth are currently given by p1 = 5, p2 = 10, and w = 100. Suppose prices change to p1 = 10 and p2 = 5. Under what circumstances is the price change good for the consumer? 5d) Assume that prices and wealth are currently given by p1 = 5 and p2 = 10. Suppose that prices are about to change to p1 = 5 and p2 = y. Derive an expression for the amount of wealth, m*, that the consumer would be willing to give up in order to prevent the price change from occurring. 5e) Under what circumstances will m* be negative? In words, what does it mean for m* to be negative.