Answer to Question #267882 in Microeconomics for sentayew

Question #267882

1.     Given utility function U(x,y) = X1/4Y3/4, Where price of X=8Br and Price of Y = 2 Br and income of the consumer is 480 Br;

A.    Compute the utility maximizing level of X and Y

B.    Calculate the marginal rate of substitution of X to Y at equilibrium and interpret the result

C.    Compute the total utility at equilibrium



1
Expert's answer
2021-11-18T10:21:41-0500

A. The utility maximizing level of X and Y;

"MU_x=0.25x^{-0.75}y^{0.75}"




"MU_y=0.75x^{0.25}y^{-0.25}"




"{\\frac{MU_x}{p_x}}=\\frac {MU_y}{p_y}""x \\times p_x+ y \\times p_y=M"




"y=4x"




"8 \\times x+2 \\times y=480"




"x=30, y=120."



B.  The marginal rate of substitution of X to Y at equilibrium;

"\\frac {\\partial U}{\\partial x \\partial y} =\\frac {0.1875\\times y^{0.5}} {(x)^{0.5}}"




"MRS x.y = \\frac {\\partial U} {\\partial x \\partial y} = \\frac {0.1875\\times 120^{0.5}}{( 30)^{0.5}}=0.375"



C.    The total utility at equilibrium;


="30^{0.25}\\times120^{0.75}"


=84.85




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