EconS 501 Fall 2016

Felix Munoz

Exercises – Recitation #3

Exercise 1. Find the demanded bundle for a consumer whose utility function is u(x1,x2)= x13/2x2 and her
budget constraint is 3x1+4x2=100.

3
Solution. Making a log transformation of the utility function, ln u ( x1 , x2 )  ln x1  ln x2
2

Write the Lagrangian

3
L( x,  )  ln x1  ln x2   (3x1  4 x2  100)
2
(We can transform u this way, because the Ln function is strictly increasing.) Now, equating the
derivatives with respect to x1, x2, and  to zero, we get three equations in three unknowns
3
 3 ,
2 x1
1
 4 ,
x2
3 x1  4 x2  100.
Solving, we get that the Walrasian demands at price p1  3, p2  4 and income m  100 are
x1 (3, 4,100)  20 , and x2 (3, 4,100)  10 .
Note that if you are going to interpret the Lagrange multiplier as the marginal utility of income, you must
be explicit as to which utility function you are referring to. Thus, the marginal utility of income can be
measured in original ‘utils’ or in ‘ln utils’. Let u*=lnu and, correspondingly, v*=lnv; then
v* ( p, m) u ( p, m)
 and  
m m
Where  denotes the Lagrange multiplier in the Lagrangian
( x,  )  x13/2 x2   (3 x1  4 x2  100).
203/2 1
Check that in this problem we’d get   , , and v(3, 4,100)  203/210 .
4 40

Exercise 2. Use the utility function u(x1,x2)= x11/2x21/3 and the budget constraint m=p1x1+p2x2 to calculate
the Walrasian demand, the indirect utility function, the Hicksian demand, and the expenditure function.

Solution. The Lagrangian for the utility maximization problem is

( x,  )  x11/2 x1/3
2   ( p1 x1  p2 x2  m),
Taking derivatives,

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1 1/2 1/3
x1 x2   p1 ,
2
1 1/2 2/3
x1 x2   p2 ,
3
p1 x1  p2 x2  m.
Solving, we get
3m 2 m
x1 ( p, m)  , x2 ( p, m)  .
5 p1 5 p2
Plugging these demands into the utility function, we get the indirect utility function
1/2 1/3 1/2 1/3
3 m  2 m  m
5/6
 3   2 
v ( p , m)  U  x ( p , m )             .
 5 p1   5 p2  5  p1   p2 
Rewrite the above expression replacing v(p, m) by u and m by e(p, u). Then solve it for e(.) to get
3/5 2/5
p   p2 
e( p , u )  5  1    u 6/5
 3   2 
Finally, since hi  e / pi , the Hicksian demands are
2/5 2/5
p   p2 
h1 ( p, u )   1    u 6/5 ,
 3   2 
3/5 3/5
p   p2 
h2 ( p, u )   1    u 6/5 .
 3   2 

Exercise 3. Consider a two-period model with Dave’s utility given by u  x1, x2  where x1 represents
his consumption during the first period and x2 is his second period’s consumption. Dave is endowed
with  x1, x2  which he could consume in each period, but he could also trade present consumption for
future consumption and vice versa. Thus, his budget constraint is
p1x1  p2 x2  p1x1  p2 x2 ,
where p1 and p2 are the first and second period prices respectively.

a) Derive the Slutsky equation in this model. (Note that now Dave’s income depends on the value
of his endowment which, in turn, depends on prices: m  p1x1  p2 x2 .)

Solution. Differentiate the identity h j ( p, u )  x j  p, e( p, u )  with respect to pi to get

h j ( p, u ) x j ( p, m) x j  p, e( p, u )  e( p, u )
 
pi pi m pi
We must be careful with this last term. Look at the expenditure minimization problem
e( p, u)  min  p( x  x ) : u( x)  u  ph( p, u)  px
By the envelope theorem, we have

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e( p, u )
 hi ( p, u )  xi  xi  p, e( p, u )   xi
pi
Therefore, we have
h j ( p, u ) x j ( p, m) x j  p, e( p, u ) 
   xi ( p, m)  xi 
pi pi m
And reorganizing we get the Slutsky equation
x j ( p, m) h j ( p, u ) x j  p, e( p, u ) 
   xi  xi ( p, m) 
pi pi m

b) Assume that Dave’s optimal choice is such that x1  x . If p1 goes down, will Dave be better off
or worse off? What if p2 goes down?
Solution. The following picture depicts Dave’s optimal allocation h( p, u) for a given price
vector p / p .
1 2

Future’s consumption
x2


p 1

p 2
h( p,u), spending ph( p,u)  e( p,u) which
coincides with expenditure (in$)
along all poonts in BL

x 2

B.L. Today’s consumptiom

x1
Lending
x 1
Borrowing

Intuitively, p  x  p  x measures the extra amount of money that Dave needs to spend after
selling his endowment x , in order to acquire his optimal consumption bundle h( p, u) . Hence,
Dave minimizes the expenditure p  x  p  x at the optimal bundle h( p, u) , i.e., at point A of the
figure.

Therefore, the expenditure function of his EMP is e( p, u)  p  h( p, u)  p  x .

Differentiating with respect to pi , we obtain hi ( p, u )  xi  xi ( p, e( p, u ))  xi

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When p2 goes down, Dave is better off; since there is a region of the new budget line that lies of
the UCS ( x) , i.e., the set of bundle for which Dave is better off than at his original bundle X.

p

x2
BL 2 2

UCS ( x1, x 2)

x 2

x 2

BL 1

x1
x 1 x 1

When p1 goes down, Dave is worse off; since the new budget line, BL2 , unambiguously lies
below the UCS(X)
x2

 p 1

UCS ( x1, x 2)
x 2

x 2

BL 1 BL 2

x1
x 1 x 1

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Exercise 4. The utility function is u  x1, x2   min x2  2 x1, x1  2 x2 .
a) Draw the indifference curve for u  x1, x2   20. Shade the area where u  x1, x2   20.
Solution.

Depicting an indifferent curve. For a given utility level u  20 , consider x2  2 x1  20 and

x1
x1  2 x2  20 . Solving for x2 , we obtain x2  20  2 x1 and x2  10  , respectively. We plot
2
these two lines in the picture below. Line x2  20  2 x1 originates at 20 and has a slope of -2,
x1
whereas x2  10  originates at 10 and has a slope of -1/2.
2

x2  2 x1  20
x2

20

18 A

Indifference curve for U=20 is outside lines

10
9.5
C

B
2 x2  x1  20
1 10 16 20 x1

The indifference curve is the northeast boundary of these two lines (i.e., the upper envelope). In
particularly, for a bundle ( x1 , x2 )  (1,18) , located at point A in the figure, the consumer’s utility
is
min{18  2 1,1  2 18}  min{20,37}  20 .
Similarly, bundle B in the other extreme of the figure, i.e., ( x1 , x2 )  (16, 2) , yields a utility level
of
min{2  2 16,16  2  2}  min{34, 20}  20 .

Note that bundles in the southeast boundary, such as C  (1,9.5) , only provide a utility of
min{9.5  2 1,1  2  9.5}  min{11.5, 20}  11.5  20
So the southwest boundary of the two lines cannot be the indifference curve of u  20 . If we
wanted to depict the indifference curve associated to a utility of u  11.5 , we would need two
lines parallel to the thick lines in the figure but shifted inwards towards to origin so they cross
point C.

Upper contour set. Finally, the upper contour set contains all those bundles to the northeast of the
indifference curve we just depicted

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Felix Munoz

b) For what values of p1 / p2 will the unique optimum be x1  0?

Solution. The slope of a budget line is  p1 / p2 . If the budget line is steeper than 2, in absolute
value, e.g., -5, then we have x1  0 as illustrated in the next figure. Hence, the condition is
p1 / p2  2 .
Corner at x1 =0

45
x2

x1  0
x2  0
2

BL
x1

c) For what values of p1 / p2 will the unique optimum x2  0?

1 1
Solution. Similarly, if the budget line is flatter than in absolute value, e.g.,  , x2 will equal
2 4
0, as illustrated in the next figure. Therefore, the condition is p1 / p2  1/ 2 .


45
x2

Corner at x2 =0

BL
x1
x1  0
x2  0

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d) If neither x1 nor x2 is equal to zero, and the optimum is unique, what must be the value of
x1 / x2 ?
Solution. If the optimum is unique, it must occur where at the kink x2  2 x1  x1  2 x2 . Since
line x2  2 x1 crosses x1  2 x2 at the 45 -line, the interior optimum occurs at x1  x2 , so that
x1 / x2  1 .


45
x2

BL
x x
1 2

x1

Plugging this result, x1  x2 , into the budget line, we obtain 𝑝1 𝑥2 + 𝑝2 𝑥2 = 𝑤. Solving for 𝑥2 ,
yields a Walrasian demand of
𝑤
𝑥2 = .
𝑝1 + 𝑝2
which coincides with the Walrasian demand of good 1 since 𝑥1 = 𝑥2 at the kink.

Exercise 5. Under current tax law some individuals can save up to $2,000 a year in an Individual
Retirement Account (I.R.A.), a savings vehicle that has an especially favorable tax treatment. Consider
an individual at a specific point in time who has income Y, which he or she wants to spend on
consumption, C, I.R.S. savings, S1 , or ordinary savings S 2 . Suppose that the “reduced form” utility
function is taken to be:

U  C , S1, S2   S1 S2 C  .

(This is a reduced form since the parameters are not truly exogenous taste parameters, but also include the
tax treatment of the assets, etc.) The budget constraint of the consumer is given by:

C  S1  S2  Y ,

and the limit that he or she can contribute to the I.R.A. is denoted by L.

a) Derive the demand functions for S1 and S 2 for a consumer for whom the limit L is not binding.
Solution. Building the Lagrangian, we obtain:

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Felix Munoz
L   ln s1   ln s2   ln C   (Y  C  s1  s2 ) .
Take all derivatives with respect to  , C , s1 , s2 to find s1 , s2 . This is an ordinary Cobb-Douglas
demand:
 
S1  Y and S2  Y.
       

b) Derive the demand function S1 and S 2 for a consumer for whom the limit L is binding.
Solution. Since s2 has reached the maximum allowed, L, we plug S2=L in the utility function
U (C , S1 , L)  S1 L C  . Note that the L term is just a constant, so applying the standard Cobb-

Douglas formula S1  Y.
 

Exercise 3.E.7. Show that if a preference relation is quasilinear with respect to good 1, the Hicksian
demand functions for the remaining goods 2, 3, …, L do not depend on u. What is the form of the
expenditure function in this case?

Solution. Exercise 3.C.5(b) in MWG shows that every quasilinear preference with respect to good
1 can be represented by a utility function of the form u  x   x1  u  x2 , , xL  . Let
e1  1, 0, , 0  L
. We shall prove that for every p  0 with p1  1, u  ,   ,
and x   ,    L 1
 , if x  h  p, u  , then x   e1  h  p, u    . Note first that
u  x   e1   u   , that is, x   e1 satisfies the constraint of the EMP for  p, u    . Let
y L
 and u  y   u   . Then u  y   e1   u. Hence, p   y   e1   p  x. Thus
p  y  p   x   e1  . Hence x   e1  h  p, u    .

Therefore, for every good 2 , L , u  , and u  , h  p, u   h  p, u . That is, the hicksian
demand functions for goods 2, L are independent of the utility level that the individual must reach in
his EMP. Thus, if we define the hicksian demand of reaching a zero utility level as h  p   h  p, 0  ,
then the hicksian demand of reaching a positive utility level u  0 , h( p, u) , is h  p, u   h  p   ue1 ,
where the positive utility originates from units of good 1.
We can extend the above argument by saying that the hicksian of reaching an even farther utility level
u   , h( p, u   ) , is h  p, u     h  p, u    e1, that is, the hicksian from reaching utility level u
plus additional units of good 1. Thus, we have that the expenditure function of such hicksian demand,
h( p, u   ) is e  p, u     e  p, u    , which indicates that, in order to increase the utility level from
u to u   , the consumer must increase his minimal expenditure from e( p, u ) to e( p, u )   . Thus, if
we define the expenditure of reaching a zero utility level as e  p   e  p, 0  , then the minimal
expenditure of reaching a positive utility level u  0 is e  p, u   e  p   u .

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Exericse 3.E.8. For the Cobb-Douglas utility function, verify that the following relationships in (3.E.1)
and (3.E.3) respectively hold.
e(p,v(p,w))=w and v(p,e(p,u))=u, and
h(p,u)=x(p,e(p,u)) and x(p,w)=h(p,v(p,w))

Note that the expenditure function can be derived by simply inverting the indirect utility function, and
vice versa.

1 
Solution. We use the utility function u  x   x1 x2 . To prove (3.E.1),
e  p, v  p, w      1   
 1  1
p1 p2  
1   1 
p1 p2 1w  w,

v  p, e  p, u      1     u   u.
1
p1 p2 1 
1    1 p1 p12

To prove (3.E.3),


x  p, e  p, u      1   
 1

p1 p12 u  p1 , 1    p2 

 1  
 
 p   1    p 
 u   h  p, u  ,
2 1
 u, 
  1    p1    p  
 2

 1  
  p2   1    p1 
h  p, v  p, w    
1

1    p1 p2 1w    ,   
   1    p1    p2  
 
 w  p1 , 1    p2   x  p, w  .

Exercise 3.E.9. Use the relations in 3.E.1:

e(p,v(p,w))=w and v(p,e(p,u))=u

to show that the properties of the indirect utility function e(p,u) identified in Proposition 3.E.2:

1. Homogeneous of degree one in prices.

2. Strictly increasing in u and nondecreasing in pk for any good k.

3. Concave in prices.

4. Continuous in p and w.

imply the properties of the expenditure function v(p,w) identified in Proposition 3.D.3:

1. Homogeneity of degree zero.

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2. Strictly increasing in w and nonincreasing in pk for any good k.

3. Quasiconvex; that is, the set {(p,w): v(p,w)≤v} is convex for any v.

4. Continuous in p and w.

Likewise, use the relations

e(p,v(p,w))=w and v(p,e(p,u))=u

to prove that the properties of v(p,w) identified in Proposition 3.D.3 imply the properties of e(p,u)
identified in Proposition 3.E.2.

Solution. First, we shall prove that Proposition 3.D.3 implies Proposition 3.E.2 via (3.E.1). Let
p  0, p  0, u  , u  , and   0.

(i) Homogeneity of
degree one in p: Let   0. Define
w  e  p, u  , then u  v  p, w by the second relation of (3.E.1). Hence

e  p, u   e  p, v  p, w    e  p, v  p,  w     w   e  p, u  ,
where the second equality follows from the homogeneity of v  ,  and the third from the first
relation of (3.E.1).

(ii) Monotonicity: Let u  u. Define w  e  p, u  and w  e  p, u  , then u  v  p, w

and u  v  p, w  . By the monotonicity of v  ,  in w, we must have w  w, that is,
e  p, u   e  p, u  .

Next let p  p. Define w  e  p, u  and w  e  p, u  , then, by the second relation of

(3.E.1), u  v  p, w  v  p, w  . By the monotonicity of v  ,  , we must have w  w, that
is, e  p, u   e  p, u  .

(iii) Concavity: Let   0,1. Define w  e  p, u  and w  e  p, u  , then

u  v  p, w  v  p, w . Define p   p  1    p and w   w  1    w. Then, by
the quasiconvexity of v  ,  , v  p, w   u  v( p, e( p, u)) . Hence, by the monotonicity of
v  ,  in w and the second relation of (3.E.1), w  e  p, u  . That is,

e  p  1    p, u    e  p, u   1    e  p, u  .

(iv) Continuity: It is sufficient to prove the following statement: For any sequence

   p , u    p, u   

pn , u n with n n
and any w, if e p n , u n  w for every n, then
n1

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   
e  p, u   w ; if e p n , u n  w for every n, then e  p, u   w . Suppose e p n , u n  w for
every n. Then, by the monotonicity of v  ,  in w, and the second relation of (3.E.1), we have

 
u n  v p n , w for every n. By the continuity of v ,  , u  v  p, w . By the second relation
of (3.E.1) and the monotonicity of v  ,  in w, we must have e  p, u   w. The same

 
argument can be applied for the case with e p n , u n  w for every n.

Let’s next prove that Proposition 3.E.2 implies Proposition 3.D.3 via (3.E.1). Let
p  0, p  0, w  , w  , and   0.

i. Homogeneity: Let   0. Define u  v  p, w . Then, by the first relation of

(3.E.1), e  p, u   w. Hence

v  p,  w   v  p,  e  p, w    v  p, e  p, u    u  v  p, w  ,

where the second equality follows from the homogeneity of e  ,  and the third from
the second relation of (3.E.1).

ii. Monotonicity: Let w  w. Define u  v  p, w and u  v  p, w  , then

e  p, u   w and e  p, u   w. By the monotonicity of e  ,  and w  w , we must
have u   u , that is, v  p, w   v  p, w .

Next, assume that p  p . Define u  v  p, w and u  v  p, w , then

e  p, u   e  p, u  w. By the monotonicity of e  ,  and p  p , we must have
u   u , that is, v  p, w  v  p, w
.

iii. Quasiconvexity: Quasiconvexity means that the lower contour set ( LCS) is
convex. Let   0,1 . Define u  v  p, w and u  v  p, w  . Then e  p, u   w
and e  p, u   w. Without loss of generality, assume that u   u . Define
p   p  1    p and. w   w  (1   )w Then

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Felix Munoz
e  p, u  

  e  p, u    1    e  p, u  

  e  p, u   1    e  p, u  

  w  1    w  w,

where the first inequality follows from the concavity of e  , u  the second from the
monotonicity of e  ,  in u and u  u . We must thus have v  p, w   u  v( p, w).

iv. Continuity: It is sufficient to prove the following statement. For any sequence

 p , w   p , w    p, w and any u, if v  p , w   u for every n,


n n n n n n
with
n1

then v  p, w  u ; if v  p , w   u for every n, then v  p, w  u . Suppose

n n

 
v p n , wn  u for every n. Then, by the monotonicity of e  ,  in u and the first

relation of (3.E.1), we have wn  e p n , u   for every n. By the continuity of

e ,  , w  e  p, u  . We must thus have v  p, w  u. The same argument can be


applied for the case with v p n , wn  u for every n. 
Alternative: An alternative, simpler way to show the equivalence on the
concavity/quasiconvexity and the continuity uses what is sometimes called the epigraph.

For the concavity/quasiconvexity, the concavity of e  , u  is equivalent to the convexity

 p, w : e  p, u   w and the quasi-convexity of v  is the equivalent to the

of the set
convexity of the set  p, w  : v  p, w   u for every u. But (3.E.1) and the
monotonicity imply that v  p, w  u if and only if e  p, u   w. Hence the two sets
coincide and the quasiconvexity of v  is equivalent to the concavity of e  , u  .

As for the continuity, the function e  is continuous if and only if both

 p, w, u  : e  p, u   w and  p, w, u  : e  p, u   w are closed sets. The function

v   is continuous if and only if both  p, w, u  : v  p, w  u and

 p, w, u  : v  p, w  u are closed sets. But, again by (3.E.1) and the monotonicity,

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Felix Munoz

 p, w, u  : e  p, u   w   p, w, u  : v  p, w   u;

 p, w, u  : e  p, u   w   p, w, u  : v  p, w   u
Hence the continuity of e  is equivalent to that of v  .

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Felix Munoz     Fall 2008 
EconS 501 

Microeconomic Theory – Recitation #3 – Exercises. 
 
1.  Jan’s utility function for goods X and Y is  U = 7200 X .75Y .25 .   She must pay $90 for a 
unit of good X and $30 for a unit of good Y.  Jan’s income is $1200. 
 
a. Determine the amounts of goods X and Y Jan purchases to maximize her utility 
given her budget constraint. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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Felix Munoz     Fall 2008 
EconS 501 

b. Determine the maximum amount of utility Jan receives. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
c. Determine the value of  λ *  associated with this problem. 
 
 
 
 
 
 
 
 
 
 
 
 
d. Interpret the value of  λ *  you computed in part c. as it specifically applies to Jan. 
 
 
 
 
 
 
 

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Felix Munoz     Fall 2008 
EconS 501 

2.  a.  Formulate the dual constrained expenditure minimization problem associated 

with 4.3 and determine the optimal amounts of goods X and Y Jan should 
purchase. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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Felix Munoz     Fall 2008 
EconS 501 

b.  Determine the minimum amount of expenditure made by Jan. 
 
 
 
 
 
 
 
 
 
c.  Determine the optimal value of  λ D  and provide a written interpretation of this 
value as it specifically applies to Jan in this problem. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
d.  Compare the optimal values of X, Y and  λ  you computed in exercise 4.3 with 
those you computed in parts a. and c. of this exercise. 
 
 
 
 
 
 
 
 
 

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Felix Munoz     Fall 2008 
EconS 501 

 
3.  Raymond derives utility from consuming goods X and Y, where his utility function is 
U = 80 X .25Y .25 .   He spends all of his income, I, on his purchases of goods X and Y, 
and he must pay prices of  Px  and  Py  for each unit of these goods, respectively.  
Assume that his income is $3200, the unit price of good X is $100, and the unit price 
of good Y is $100. 
 
a. Determine the amounts of goods X and Y that Raymond should purchase to 
maximize his utility given his budget constraint. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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Felix Munoz     Fall 2008 
EconS 501 

 
b. Determine the maximum amount of utility Raymond can receive. 
 
 
 
 
 
 
 
4.  Refer to your response to exercise 5.1. 
 
a. Derive Raymond’s own‐price demand curve for good X. 
 
 
 
 
 
 
 
 
 
 
 
b. Derive Raymond’s own‐price demand curve for good Y. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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Felix Munoz     Fall 2008 
EconS 501 

 
5.  Refer to your responses to exercise 5.1. 
 
a. Derive Raymond’s Engel curve for good X. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
b. Is good X a normal good or an inferior good?  Justify your response 
mathematically. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

7 
 
Felix Munoz     Fall 2008 
EconS 501 

 
c. Derive Raymond’s Engel curve for good Y. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
d. Is good Y a normal good or an inferior good?  Justify your response 
mathematically. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

8 
 
Felix Munoz     Fall 2008 
EconS 501 

6.  Assume an individual’s own‐price demand function for good X is 
( )
X = X Px , Py , I = 200 − 4 Px − 1.5 PY + 0.008 I  where of  Px  and  Py  denote the unit 
prices of goods X and Y, respectively, and I denotes the consumer’s money income. 
 
a. Compute the individual’s cross‐price demand curve for good X when the unit 
price of good X is $2 and the consumer’s income is $40,000. 
 
 
 
 
 
 
 
 
 
 
 
b. Are goods X and Y gross substitutes or gross complements?  Justify your 
response mathematically. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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Felix Munoz     Fall 2008 
EconS 501 

7.  Recall from exercise 5.1 Raymond’s utility function, when he consumes goods X and 
Y, is  U = 80 X .25Y .25 .   Once again, assume the unit price of good X,  Px , is $100, and 
the unit price of good Y,  Py , is $100.  Determine the quantities of goods X and Y 
Raymond should purchase that will minimize his expenditures on these goods and 
yield 320 units of utility to him. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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Felix Munoz     Fall 2008 
EconS 501 

 
8.  Refer to your response to exercise 5.5. 
 
a. Determine Raymond’s compensated demand curve for good X. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
b. Determine Raymond’s compensated demand curve for good Y. 
 
 
 
 
 

11 
 
Felix Munoz     Fall 2008 
EconS 501 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
9.  Is it possible for an individual’s demand curve for a good to be positively sloped?  
Support your response with an appropriate graphical analysis. 
 
 
 
 
 
 
 
 
 
 
 
 

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Felix Munoz     Fall 2008 
EconS 501 

 
 
 
 

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