Arwen-i has income=96 and faces prices Px=4 for Good X and apt=10 for good Y. Fi

Question

Arwen-i has income=96 and faces prices Px=4 for Good X and apt=10 for good Y. Find Arwen's Marginal Rate of Substituion and her utility maximizing bundle for each of the following utility functions.
a) Arwen-1: U(X,Y)= X + 5Y b) Arwen-2: U(X,Y)= X + 2Y c) Arwen-3: U(X,Y) = min[2X,Y] d) Arwen-4: U(X,Y) = X^(1/2) + Y e) Arwen-5: U(X,Y) = XY Arwen-i has income=96 and faces prices Px=4 for Good X and apt=10 for good Y. Find Arwen's Marginal Rate of Substituion and her utility maximizing bundle for each of the following utility functions.
a) Arwen-1: U(X,Y)= X + 5Y b) Arwen-2: U(X,Y)= X + 2Y c) Arwen-3: U(X,Y) = min[2X,Y] d) Arwen-4: U(X,Y) = X^(1/2) + Y e) Arwen-5: U(X,Y) = XY
a) Arwen-1: U(X,Y)= X + 5Y b) Arwen-2: U(X,Y)= X + 2Y c) Arwen-3: U(X,Y) = min[2X,Y] d) Arwen-4: U(X,Y) = X^(1/2) + Y e) Arwen-5: U(X,Y) = XY

Explanation / Answer

(a) U = X + 5Y

Budget line: 96 = 4X + 10Y

MRS = MUX / MUY

MUX = dU / dX = 1

MUY = dU / dY = 5

MRS = 1/5 = 0.20

Here, X & Y are perfect substitututes. So, MRS = PX / PY rule is invalid.

Instead, utility is maximized at corner points.

When X = 0, Y = 9.6 & U = 5 x 9.6 = 48

When Y = 0, X = 96/4 = 24 & U = 24

So, utility is maximized with bundle (X, Y) = (0, 48)

(b) U = X + 2Y

MUX = dU / dX = 1

MUY = dU / dY = 2

MRS = 1/2 = 0.5

96 = X + 2Y

As before, utility is maximized by corner solution (X & Y being substitutes).

When X = 0, Y = 96 / 2 = 48

So, U = 2Y = 96

When Y = 0, X = 96, So U = 96

Utility is maximized at either of the two corners of indifference curve.

(c) U = Min [2X, Y]

For perfect complements, the indifference curve is L-shaped & MRS cannot be calculated because the function is not differentiable.

The optimal bundle is where 2X = Y

So,

96 = 4X + 10 x 2X = 24X

X = 96 / 24 = 4

Y = 2X = 8

Maximum utility = X + 2Y = 4 + 16 = 20

(d) U = X1/2 + Y

MUX = dU / dX = (1/2)X- 1/2

MUY = dU / dY = 1

MRS = (1/2)X- 1/2

Now, utility maximization rule states: MRS = PX / PY

Or, (1/2)X- 1/2 = 4 / 10 = 0.4

X- 1/2 = 0.8

(1 / X)1/2 = 0.8

X = (1 / 0.8)2 = 1.5625

When X = 1.5625,

96 = 4X + 10Y = 6.25 + 10Y

10Y = 89.75

Y = 8.98

U = X1/2 + Y = 1.25 + 8.98 = 10.23

(e) U = XY

MUX = Y

MUY = X

MRS = Y / X

We know that MRS = PX / PY

Y / X = 4 / 10 = 2 / 5

Y = 2X / 5

So.

96 = 4X + 10Y

= 4X + 10 x (2X / 5) = 8X

X = 96 / 8 = 12

Y = 2X / 5 = 24 / 5 = 4.8

U = XY = 12 x 4.8 = 57.6

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