3. 2-good Utility Maximization (Show your steps!) In a world of two goods x and y, a utility...

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3. 2-good Utility Maximization (Show your steps!) In a world of two goods x and y, a utility maximizing consumer has utility function u (x; y) = x 1 2 y 1 3 : The price on x and y are respectively px = 2 and py = 1 and she has a budget of m = 10. So the consumer is constrained by pxx + pyy = 10 when maximizing here utility u. (Suppose the consumer always spends her whole budget.) Solve for her optimal choice of x, y and the corresponding utility level. 4. General Cobb-Douglas Utility Maximization. (Show your steps!) A Cobb-Douglas utility function of two goods is of the form u (x; y) = x y after solving a special case in questions 3. Solve for the general problem M ax x;y u (x; y) = x y s:t: pxx + pyy = m Your solution x and y should be functions of the parameters ; ; px; py; m; where all parameters are positive. 5. 3-good Utility Maximization (Show your steps!) Solve the 3 good utility maximization problem M ax x;y;z u (x; y; z) = xy + z s:t: pxx + pyy + pzz = m Your solution x ; y and z should be functions of the parameters px; py; pz; m; where all parameters are positive XXXXXXXXXXinput Cost Minimization (Show your steps!) A Örm is bounded by its production function y = f (x1; x2) = x 1 2 1 x 1 3 2 ; where x1 and x2 are quantities of the 2 inputs used to produce its product. The Örmís cost function is c (x1; x2) = !1x1 + !2x2; where !1 and !2 are input unit prices/costs. Suppose !1 = 2 and !2 = 1 and the Örm targeted at producing y = 10 units of products. Solve for its optimal choice of x1 and x2:
	
Document Preview: 3. 2-good Utility Maximization (Show your steps!)
Inaworldoftwogoodsxandy,autilitymaximizingconsumerhasutilityfunctionu(x;y) =
1 1
2 3
x y : The price on x and y are respectively p = 2 and p = 1 and she has a budget of
x y
m = 10. So the consumer is constrained by p x+p y = 10 when maximizing here utility u.
x y
(Suppose the consumer always spends her whole budget.) Solve for her optimal choice of x,
y and the corresponding utility level.
4. General Cobb-Douglas Utility Maximization. (Show your steps!)
 
A Cobb-Douglas utility function of two goods is of the form u(x;y) = x y after solving a
special case in questions 3. Solve for the general problem
 
Max u(x;y) = x y
x;y
s:t: p x+p y = m
x y
 
Your solution x and y should be functions of the parameters ;;p ;p ;m; where all
x y
parameters are positive.
5. 3-good Utility Maximization (Show your steps!)
Solve the 3 good utility maximization problem
Max u(x;y;z) = xy+z
x;y;z
s:t: p x+p y+p z = m
x y z
  
Your solution x ; y and z should be functions of the parameters p ;p ;p ;m; where all
x y z
parameters are positive.
6. 2-input Cost Minimization (Show your steps!)
1 1
2 3
A ?rm is bounded by its production function y = f (x ;x ) = x x ; where x and x
1 2 1 2
1 2
are quantities of the 2 inputs used to produce its product. The ?rm?s cost function is
c(x ;x ) = ! x +! x ; where ! and ! are input unit prices/costs. Suppose ! = 2 and
 XXXXXXXXXX
! = 1 and the ?rm targeted at producing y = 10 units of products. Solve for its optimal
2
choice of x and x :
1 2
1

Robert · Accepted Answer

Microsoft Word - The consumer.doc
 
 
The consumer’s objective function is given by: 
Max U=x^1/2y^1/3 
Subject to 2x+y=10 
We set the Lagrangian function: 
L=x^1/2y^1/3+l(10-2x-y) 
1
st
 order condition for maximization requires : 
dL/dx=0 
or, 1/2U/x=2l………………1 
dL/dy=0 
or, 1/3 U/y=1………………2 
dividing equation 1 by 2, we get,  
3y/2x=2 
Or, y=4x/3 
Putting this in equation of budget constraint, we get,  
2x+4x/3=10 
Or, 6x+4x=30 
Or, x=3 
Y=12/3=4 
So, optimal value of x=3 and y=4.
 
 
Here, the objective function is iven by: 
Max U=x^ay^b 
Subject to PxX+PyY=M 
We set the lagrangian function as:

3. 2-good Utility Maximization (Show your steps!) In a world of two goods x and y, a utility maximizing consumer has utility function u (x; y) = x 1 2 y 1 3 : The price on x and y are respectively px...

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