Question #267548

1.     Suppose we have the utility function, U = XY + X + Y.

a.      Find the function for the marginal rate of substitution.

  1. If prices are Px = $2 and Py = $4, and if income is M = $18, find the utility maximizing consumption bundle.
1
Expert's answer
2021-11-19T11:11:28-0500

1.). a.). MRS = MUxMUy=PxPy\frac{MU_{x} } {MU_{y}} = \frac{P_{x} } {P_{y}}

MUx = UX\frac{\partial U} {\partial X} = Y + 1


MUy = UY\frac{\partial U} {\partial Y} = X + 1

 

MRS = MUxMUy\frac{MU_{x} } {MU_{y}} = Y + 1/ X + 1

MRS Function = Y + 1/ X + 1

 

1.). Budget constraint: I = PxX + PyY

18 = 2X + 4Y

Set MRS = PxPy\frac{P_{x} } {P_{y}}


Y+1X=24Y + \frac{1} {X} = \frac{2} {4}


Y=X212Y = \frac{X} {2} -\frac{1} {2}

Substitute in the budget constraint to get X:

18 = 2X + 4Y

18 = 2X + 4(X212\frac{X} {2} -\frac{1} {2})

X = 5

Y=X212Y = \frac{X} {2} -\frac{1} {2}

Y = 2

The utility maximizing consumption bundle = (5, 2)


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