Question

As in the previous two questions, your utility function over the goods X and Z takes the following Suppose now that the government wants to discourage consumption of X. It places a tax on X such that the price consumers now face is $4 per unit instead of $3 per unit. Assume that the price of Z remains at $6 per unit and that your income is $900. The consumption bundle that will now maximize your utility subject to your budget constraint is X - 225 and Z 0 (enter only numbers in the blanks, and round to the nearest integer if necessary)

Answer 1

Consumption function is U (X,Z) = X^0.5 + Z^0.5

New Price of X (P_x) = $4/unit , Price of Z (P_z) = $6/unit, Total Income (M) = $900

Budget Constraint: X.P_x + Z.P_z = M

To find the demand of X and Z, first find MRS = partial derivative of U with respect to X / partial derivative of U with respect to Z = MU_x/MU_z

=> MRS = 0.5X^-0.5/ 0.5Z^-0.5 = (Z/X)^0.5

Note that MRS (slope of Indifference curve) = Px/Py (slope of budget line) for optimal bundle

=> (Z/X)^0.5 = 4/6 (divide by 2 throughout) = 2/3 => Z =4X/9 (after squaring both sides) ---(1)

Budget Constraint: X.P_x + Z.P_z = M => 4X + 6Z = 900 --- (2)

Putting (1) in (2) we get,

4X + 6 (4X/9) = 900 => 4X + 8X/3 = 900

(multiplying throughout by 3) 12X + 8X = 2700 => 20X = 2700 => X = 2700/20 = 135

from (1), Z = 4X/9 = 4(135)/9 = 60.

Therefore, Optimal bundle is (X,Z) = (135,60)

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