Economics 601 Harvey Lapan Microeconomic Analysis I Fall 2016 Problem Set No. 5 Due by: Thursday, September 29, 2016 Finish problems 7 and 8 from problem set 4. 1. Stone-Geary utility function Assume that, for the 2-good case, the utility function has the so-called Stone-Geary form: ( ) ( ) ( )1 2 1 1 2 2,u x x x xα βγ γ= − −

a) Explain why we should have 0α > and 0β > , and, without loss of generality, why it is possible to put 1α β+ = . (For the remainder of the problem, maintain these conditions).

b) Standard theory does not restrict the sign of parameters 1γ and 2γ (why?). Here, however, assume that 0iγ > , 1,2i = . Given that, define an appropriate consumption set for these preferences and provide a graphical illustration.

For the remainder of this problem, assume that an interior solution applies. c) Set up and solve the UMP (utility maximizing problem), and derive the indirect utility function.

Verify that the indirect utility function satisfies the appropriate homogeneity property.

d) Set up and solve the EMP (expenditure minimimizing problem), and derive the expenditure function. Verify that the expenditure function satisfies the appropriate curvature property.

e) Verify that the indirect utility function can be alternatively obtained by inverting the expenditure function, and, correspondingly, that the expenditure function can be obtained by inverting the indirect utility function. Briefly explain why this procedure is legitimate.

2. Consider the choice problem of a consumer with a linear budget constraint and strictly convex

preferences defined on 3+ , and the standard notation whereby ip denote prices, w is income, and u

is a utility level. Suppose that the Hicksian demand functions for goods 1 and 3 are:

1 4 1 2 1 42 22 3 2 1

1 1 2 3 33 4 1 21 3

( , , , ) ;h hu p p u p px p p p u x

p p

βα= =

a) Using the properties of Hicksian demands, determine the values of the parameters ,α β

b) Using the properties of Hicksian demands, find 2 1

hx p∂ ∂ and 2 3hx p∂ ∂ .

c) Find the Hicksian demand for good 2 (using integration).

3. In a 3 good world, suppose you have the following Marshallian demand for good 1 and Hicksian

demand for good 2:

1

( )1 21 2 1 2 1 3 1 32 1 3

1 22 31 2

;2 2

H MU p p p wx x

p p+

= =

(a) Using the Hicksian demand for good 1, find the expenditure function by integration; don’t forget

about the constant of integration, ( )2 3, , .p p uφ

(b) Using the expenditure function, what restriction on the function φ must hold to insure the Marshallian demand for good 2 is derived from these preferences.

(c) Suppose the Hicksian demand for good 3 is: ( )1 21 2 1 2 1 3 1 3 1 2 1 2

2 1 3 13 2 3 1 2

3 32H

U p p p p Uxp p

+= + . Verify

that this is consistent with the expenditure function you found in part (a), and find the Marshallian demands for goods 1 and 3.

4. Utility function and revealed preferences (MWG, 3.D.7)

In a two-good world, a consumer makes choices in two periods with budget sets of, respectively,

0 0,p wB and 1 1,p wB , where 0 (2,1)p = , 0 12w = , 1 (2,4)p =

and 1 28w = . The observed choice vector

with 0 0,p wB is 0 (4,4)x = . When the budget set is 1 1,p wB all we know is that the consumer picks a

vector 1x such that 1 1 1p x w⋅ = . [Hint: the answer to the following questions can be obtained with the aid of properly drawn diagrams.] (a) Determine the region of permissible choices for 1x so that 0 1( , )x x satisfy the WARP. For the remainder of this problem, assume that preferences are represented by a differentiable utility function ( )u x (b) Determine the region of permissible choices for 1x so that 0 1( , )x x are consistent with the

maximization of preferences that are quasilinear with respect to the first good. (c) Determine the region of permissible choices for 1x so that 0 1( , )x x are consistent with the

maximization of preferences that are quasilinear with respect to the second good. (d) Determine the region of permissible choices for 1x so that 0 1( , )x x are consistent with the

maximization of preferences for which both goods are normal. (e) Determine the region of permissible choices for 1x so that 0 1( , )x x are consistent with the

maximization of preferences that are homothetic. 5. WARP vs. SARP

In a 3 good world suppose the consumer makes the following choices:

2

At ( )0 2,1,1p =

she buys: ( )0 6,7,5 ;x =

At ( )1 1,2,1p = she buys: ( )1 5,6,7 ;x =

At ( )2 1,1,2p = she buys: ( )2 13 ,5, ;x α α= −

(a) For what values of α do these data satisfy the Weak Axiom of Revealed Preference? (answer by comparing bundles on a pairwise basis – that is, compare 0x and 1x and conclude whether one is revealed preferred to the other; then do the same for 0x and 2x and finally for 1x and 2x ). Consider all positive values of α .

(b) Are there any values of α for which this data satisfies WARP but violates transitivity? If so, what are they? Which assumption is stronger – WARP or SARP?

6. In a 3 good world, suppose you have the following Marshallian demand curves for goods 1 and 2:

( ) ( )3 21 3 1 3 1 1 3 1 31 2 3 1 2 3

1 2 1 2 31 2 3 2 3

4; ; 0, 0

p p p p p pwx xp p p p p

β

β

αα β

+

+

+ += − = > >

Assume w is such that an interior solution holds.

(d) Find the Marshallian demand for good 3.

(e) What must the values of α and β be to satisfy the integrability conditions?

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