Suppose that a fast-food junkie derives utility from three goods—soft drinks (x), hamburgers (y),...

Question

Suppose that a fast-food junkie derives utility from three goods—soft drinks (x), hamburgers (y), and ice cream sundaes (z)—according to the Cobb-Douglas utility function

Suppose also that the prices for these goods are given by p_x= 0:25, p_y= 1, and p_z= 2 and that this consumer’s income is given by I = 2.

a. Show that, for z = 0, maximization of utility results in the same optimal choices as Show also that any choice that results in z >0 (even for a fractional z) reduces utility from this optimum.

b. How do you explain the fact that z = 0 is optimal here?

c. How high would this individual’s income have to be in order for any z to be purchased?

David · Accepted Answer

a. Suppose that a fast-food junkie derives utility from three goods: soft drinks (X), 
hamburgers (Y), and ice cream sundaes (Z) according to the Cobb-Douglas Utility function 
U(X, Y, Z) = X
1/2
Y
1/2
 (1+Z)
1/2 
Suppose also that the prices are Px = 0.25, Py =1 and  Pz = 2, and that I = $2.
a. Show that for Z=0, maximization of utility results in the same optimal choices as in 
Example 4.1.  Show also that any choice that results in Z>0 (even for a fractional Z) reduces 
utility from this optimum.
The Lagrangian condition is
L = X
1/2
Y
1/2
 (1+Z)
1/2
 + (2- 0.25X -1Y-2Z) 
 
Setting Z=0 recovers precisely the parameterized problem in 4.1.  More generally,  
 
L/ X  = 1/2(X-1/2Y1/2(1+Z)1/2) - .25 = 0 
 
L/ Y  = 1/2(X1/2Y-1/2(1+Z)1/2) -  = 0 
 
L/ Z  = 1/2(X1/2Y-1/2(1+Z)-1/2) - 2 = 0 
 
L/   = 2-  0.25X  - 1Y - 2(Z)  = 0 
 
Setting Z=0 generates the same first order conditions as in Example 4.1.  More generally, 
however,

Suppose that a fast-food junkie derives utility from three goods—soft drinks (x), hamburgers (y), and ice cream sundaes (z)—according to the Cobb-Douglas utility function Suppose also that the prices...

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