Economics D10-1 Answer to Problem Set #1 Fall 1993
Economics D10-1 Answers to Problem Set #5 Fall 1993
1. Assume that a consumer's preferences are monotonic and strictly convex. Prove that his market demand behavior obeys WARP. (HINT: Suppose not, ...)
Let [pic] and [pic] be the bundles that maximize consumer preferences at [pic] and [pic]. By WARP we know that if [pic] and [pic], then [pic]
[pic].
Proof: Suppose not, i.e., [pic]. Let's define [pic]. By strict convexity we have [pic] and [pic], and besides [pic] is affordable at [pic], i.e.,
[pic].
But this contradicts the assumption that [pic] maximizes preferences at [pic].
2. Given that demand functions are homogeneous of degree zero and that all income is spent on commodities, but without assuming preference maximization, prove that the Slutsky matrix is symmetric in a two commodity world. What is the significance of this result?
We 're in a two-commodity world ; call the two demands [pic].
By the two assumption made in the question, we have the following results:
• [pic][pic] (5.1)
Budget constraint is satisfied with equality. All income is spent.
• Demand functions are homogeneous of degree 0. (5.2)
Differentiating (5.1) with respect to [pic] and m gives us:
[pic] (5.3)
[pic]. (5.4)
Zeros homogeneity and Euler's Theorem require:
[pic] (5.5)
[pic] (5.6)
From (5.3), we get [pic]. Substituting into (5.6) yields:
[pic]
[pic] (5.7)
Multiplying (5.4) through by [pic], substituting (m-[pic]) for [pic] and rearranging gives:
[pic] (5.8)
From (5.7) and (5.8), we can establish Slutsky symmetry for the two-good system:
[pic]
Therefore the Slutsky matrix is symmetric for two goods. we know that WARP implies that the Slutsky matrix is negative semidefinite but not necessarily symmetric. For the two commodity case we have proved that it is symmetric, i.e., our demand system is integrable. Then we can conclude that for the case of two commodities, a demand system that satisfies WARP is also consistent with preference maximization, and can be used to recover underlying utility function.
3. Preferences are said to be additively separable if they can be represented by a utility function of the form [pic].
a. Show that at most one subutility function [pic]can exhibit increasing marginal utility if preferences are convex.
[pic]
The bordered Hessian of the utility function is negative definite, and therefore determinants of principal minors must alternate in sign. Let's take any principal minor of order 2. For additively separable functions we have:
[pic].
Hence, there cannot be two [pic]'s that are both positive because otherwise the above condition would be violated.
b. Analyze the demand behavior of a consumer with additively separable preferences.
• The demand functions are not additively separable. First order conditions for interior solutions are:
[pic],
[pic].
The solution for the n+1 system of equations are [pic] and [pic].
• If all goods exhibit diminishing marginal utility, and the utility function is additively separable, then all goods are normal.
Proof: The FOCs of the MAX problem evaluated at the optimum (for interior solutions) are:
[pic]
Claim: [pic] for concave utility functions. This is the counterpart of increasing marginal cost for convex cost functions. This additively separable utility function with decreasing marginal utilities is concave (its Hessian is diagonal and all diagonal elements are negative.) This result can be proved by differentiating FOCs to obtain the standard comparative statics result using the fact that principal minors of this bordered Hessian have a alternating signs for a local maximum.
Differentiate FOC's w.r.t. m and we get:
[pic].
So that [pic].
• [pic].
Let's differentiate FOC for good i w.r.t. [pic]:
[pic].
Substitute [pic] according to the result of the above section to get
[pic].
The same type of relationship holds for good j, and the result follows straightforward.
• [pic].
Note that [pic]
[pic]
and [pic]
But [pic] by Young's Theorem.
Therefore, [pic].
Now, using the expression the previous part and write [pic] as
[pic]
[pic]
[pic]
by substituting the expression obtained before for [pic].
Let r=p/m be the vector of normalized prices, and define the indirect utility function V(r) = v(p/m,1). Then V is additively separable if it can be written [pic].
c. Prove that if both u and V are additively separable, preferences must be homothetic.
Observe that if v(p,m) is additively separable in p, we know that [pic] Then,
[pic].
So that [pic].
Now, computing [pic], it is straightforward to show that [pic].
Therefore, we can put together this result and that obtained in the third subpart of part b to show :
[pic],
which holds [pic].
Observe that the above equality implies
[pic],
[pic], [pic],
i.e., income elasticities are equal across commodities. Therefore, the preferences must be homothetic.
4. Which of the following sets of observations of price-quantity data are consistent with utility maximization? Which are consistent with WARP?
In general compute [pic]. Then apply WARP for [pic]:
if [pic], then x R x'.
a. [pic]
[pic]
[pic]
[pic].
These observations are consistent with WARP and utility maximization. Given strict inequalities and by monotonicity [pic].
b. [pic]
Here [pic] and [pic] but [pic]. Given that the inequalities are strict, we can not have both [pic] and [pic] by monotonicity of preferences, so that this data is inconsistent with utility maximization. For i,j=2,3, if [pic] and [pic], then it is not the case that [pic] or [pic] and WARP does not hold.
c. [pic]
Here [pic], [pic], and [pic], so that these data are consistent with both WARP and utility maximization.
5. Sarah and Rob are married but have no children. Sarah makes all the decisions about how to spend their family income m and divide the purchases. She does this in such a way that she maximizes her utility function subject to the constraint that Rob's utility level is at least as great as [pic]. (Sarah's best friend has promised Rob that he would attain that level of utility if he left Sarah for her.)
Let [pic] denote Sarah's consumption vector and [pic] Rob's consumption vector. Define [pic]=[pic]+[pic].
a. Formulate and analyze Sarah's decision problem. Write down the LaGrangian expression and the Kuhn-Tucker conditions.
[pic]
[pic]
FONCs:
[pic];
[pic];
[pic];
[pic].
b. Show that a version of Roy's identity holds for Sarah's indirect utility function [pic].
By the Envelope Theorem:
[pic] and [pic].
Therefore,
[pic].
c. Demonstrate that the family expenditure function [pic]satisfies all the usual properties, where [pic]represents some specified level of Sarah's utility.
[pic]=[pic]
Denote [pic] and [pic] the solutions to this problem.
c.1)[pic] is nondecreasing in p. Let [pic] and let [pic], [pic] be the expenditure minimizing bundles associated with p and p', respectively. Then, [pic] [pic] [pic][pic] by the definition of a minimum and because [pic].
c.2) [pic] is homogeneous of degree 1 in p. [pic]=[pic]
=[pic]
= [pic].
c.3) [pic] is concave in p.
Let's denote ([pic]) the expenditure minimizing bundles at [pic] and t=0,1,(; [pic]. Then
[pic]= [pic][pic]
= [pic]
[pic]
= (1-() [pic]+( [pic].
c.4) [pic] is continuous in p for [pic].
This is true by the Theorem of the Maximum, since both the objective function and constraints are continuous everywhere in their domains if [pic].
c.5) [pic].
This result follows from the Envelope Theorem.
d. Does the vector of family consumption levels [pic]satisfy the Slutsky condition? Provide either a proof or a counter example.
Let's assume that the following solutions exist:
([pic],[pic])[pic][pic]
([pic], [pic] ) [pic][pic]
Assume: [pic], [pic] are continuous functions and that preferences are locally non-satiated(LNS). Then [pic]. Proof:
[pic]Suppose x solves MAX. Then x = h. Suppose not and let h = x'[pic]x. Then px'[pic] and px' [pic]m, [pic][pic][pic] since ph>0 and the preferences are continuous. Thus, [pic] t, 0 ................
................
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