Math  /  Algebra

Question2. Richard Petty has a utility function u(x,y,z)=xy(1+z)u(x, y, z)=x y(1+z) where xx is food, yy is clothing, and zz is automobiles. The price of a unit of food is $1\$ 1, a unit of clothes is $2\$ 2, and a 1 unit of cars is $2000\$ 2000. Cars must be bought in discrete units. Richard has an income of $9000\$ 9000. Calculate the bundle of goods that maximizes Richard's utility.

Studdy Solution

STEP 1

1. Richard Petty's utility function is given by u(x,y,z)=xy(1+z) u(x, y, z) = xy(1+z) , where x x is the quantity of food, y y is the quantity of clothing, and z z is the number of automobiles.
2. The prices are: px=1 p_x = 1 dollar per unit of food, py=2 p_y = 2 dollars per unit of clothing, and pz=2000 p_z = 2000 dollars per automobile.
3. Richard's total income is I=9000 I = 9000 dollars.
4. The problem requires finding the optimal bundle (x,y,z) (x, y, z) that maximizes the utility function subject to the budget constraint.

STEP 2

1. Formulate the budget constraint.
2. Analyze the utility function to understand how utility changes with different values of z z .
3. Determine the optimal z z by considering the discrete nature of automobile purchases.
4. Solve for the optimal x x and y y given the chosen z z .
5. Verify the solution satisfies the budget constraint and provides maximum utility.

STEP 3

Formulate the budget constraint.
Given the prices and income:
x+2y+2000z9000 x + 2y + 2000z \leq 9000

STEP 4

Analyze the utility function u(x,y,z)=xy(1+z) u(x, y, z) = xy(1+z) to understand how utility changes with different values of z z .

STEP 5

Consider the discrete nature of z z (automobiles) and evaluate potential values of z z (0, 1, 2, 3, and 4) to determine which might maximize utility.

STEP 6

For z=0 z = 0 :
x+2y9000 x + 2y \leq 9000

STEP 7

For z=1 z = 1 :
x+2y+20009000 x + 2y + 2000 \leq 9000
x+2y7000 x + 2y \leq 7000

STEP 8

For z=2 z = 2 :
x+2y+40009000 x + 2y + 4000 \leq 9000
x+2y5000 x + 2y \leq 5000

STEP 9

For z=3 z = 3 :
x+2y+60009000 x + 2y + 6000 \leq 9000
x+2y3000 x + 2y \leq 3000

STEP 10

For z=4 z = 4 :
x+2y+80009000 x + 2y + 8000 \leq 9000
x+2y1000 x + 2y \leq 1000

STEP 11

Evaluate the utility u(x,y,z)=xy(1+z) u(x, y, z) = xy(1+z) for each possible z z by solving for x x and y y within the budget constraint.

STEP 12

For z=0 z = 0 :
Maximize u(x,y,0)=xy u(x, y, 0) = x y subject to x+2y9000 x + 2y \leq 9000 .
If y=0 y = 0 , x=9000 x = 9000 , utility =0 = 0 .
If x=0 x = 0 , y=4500 y = 4500 , utility =0 = 0 .
Balanced approach: set x=2y x = 2y , then 2y+2y=90004y=9000y=2250x=4500 2y + 2y = 9000 \rightarrow 4y = 9000 \rightarrow y = 2250 \rightarrow x = 4500 .
Utility =4500×2250=10125000 = 4500 \times 2250 = 10125000 .

STEP 13

For z=1 z = 1 :
Maximize u(x,y,1)=2xy u(x, y, 1) = 2xy subject to x+2y7000 x + 2y \leq 7000 .
Balanced approach: set x=2y x = 2y , then 2y+2y=70004y=7000y=1750x=3500 2y + 2y = 7000 \rightarrow 4y = 7000 \rightarrow y = 1750 \rightarrow x = 3500 .
Utility =2×3500×1750=12250000 = 2 \times 3500 \times 1750 = 12250000 .

STEP 14

For z=2 z = 2 :
Maximize u(x,y,2)=3xy u(x, y, 2) = 3xy subject to x+2y5000 x + 2y \leq 5000 .
Balanced approach: set x=2y x = 2y , then 2y+2y=50004y=5000y=1250x=2500 2y + 2y = 5000 \rightarrow 4y = 5000 \rightarrow y = 1250 \rightarrow x = 2500 .
Utility =3×2500×1250=9375000 = 3 \times 2500 \times 1250 = 9375000 .

STEP 15

For z=3 z = 3 :
Maximize u(x,y,3)=4xy u(x, y, 3) = 4xy subject to x+2y3000 x + 2y \leq 3000 .
Balanced approach: set x=2y x = 2y , then 2y+2y=30004y=3000y=750x=1500 2y + 2y = 3000 \rightarrow 4y = 3000 \rightarrow y = 750 \rightarrow x = 1500 .
Utility =4×1500×750=4500000 = 4 \times 1500 \times 750 = 4500000 .

STEP 16

For z=4 z = 4 :
Maximize u(x,y,4)=5xy u(x, y, 4) = 5xy subject to x+2y1000 x + 2y \leq 1000 .
Balanced approach: set x=2y x = 2y , then 2y+2y=10004y=1000y=250x=500 2y + 2y = 1000 \rightarrow 4y = 1000 \rightarrow y = 250 \rightarrow x = 500 .
Utility =5×500×250=625000 = 5 \times 500 \times 250 = 625000 .

STEP 17

Compare the calculated utilities for each z z .
Highest utility is for z=1 z = 1 , with x=3500 x = 3500 , y=1750 y = 1750 , and z=1 z = 1 .

STEP 18

Verify the solution satisfies the budget constraint:
3500+2×1750+2000×1=3500+3500+2000=9000 3500 + 2 \times 1750 + 2000 \times 1 = 3500 + 3500 + 2000 = 9000
The solution satisfies the budget constraint.
Solution: The bundle of goods that maximizes Richard's utility is (x,y,z)=(3500,1750,1) (x, y, z) = (3500, 1750, 1) .

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