Answer to Question #207951 in Microeconomics for Muka

Question #207951

If an individual has preferences (x1,x2)= x22 +x22, graph the indifference curve for U=20 and U=40


1
Expert's answer
2021-06-18T11:39:39-0400

complete question is:

If an individual’s preferences are described by the utility function U(X1 , X2 ) = X12 + X22, graph the indifference curve for U = 20 and U= 40.Find the optimal consumption quantities if P1 = US$2.50 ;         P2 = US$ 7.50; and M = US$ 60.


Solution

Utility function

"U(x_1,x_2)=x_1^2+x_2^2"

so,


The indifference curve

"U=20\\\\V=40"








Given information

"p_1=US\\$2.50\\\\p_2=US\\$7.50"


"M=US\\$60"


Budget line equation

"p_1x_1+p_2x_2=M..............................................(1)"


substitute these values in the above equation 1

"\\$2.50x_1+\\$7.50x_2=60....................................(2)\\\\From\\space U(x_1,x_2)=x_1^2+x_2^2\\\\MRS(x_1,x_2)=\\frac{MUx_1}{MUx_2}\\\\MUx_1=2x_1,MUx_2=2x_2\\\\MRS(x_1,x_2)=\\frac{2x_1}{2x_2}\\\\=\\frac{x_1}{x_2}"


For optimal bundle

equation slope of budget line and MRS

"\\frac{p_1}{p_2}=MRS_{(x_1,x_2)}\\\\"

"\\frac{2.5}{7.5}=\\frac{x_1}{x_2}"

"x_2(2.5)=(7.5)x_1\\\\x_2=3x_1....................................................(3)"

putting x2 value in eq 2

we get

"\\$2.50x_1+\\$7.50x_2=60\\\\\\$2.50x_1+\\$7.50\\times3x_1=60\\\\\\$2.50x_1+\\$22.5x_1=60\\\\25x_1=60\\\\x_1=\\frac{60}{25}\\\\x_1=2.4"


putting x1 value in eqn 3

"x_2=3x_1\\\\=3\\times2.4\\\\x_2=7.2"


x1=2,4

x2=7.2



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