Question

Suppose that Billy's preferences over baskets containing milk (good x), and coffee (good
y ), are described by the utility function U(x; y ) = xy +2x. Billy's corresponding marginal
utilities are, MUx = y + 2 and MUy = x:
Use Px to represent the price of milk, Py to represent the price of coffee, and I to represent
Billy's income.

Suppose that Px = $1 and I = $40. Find the equivalent variation for an
increase in the price of coffee from Py1 = $4 to Py2 = $5

Answer 1

Question#65485

Suppose that Billy's preferences over baskets containing milk (good x), and coffee (good y), are described by the utility function $U(x; y) = xy + 2x$. Billy's corresponding marginal utilities are, $MUx = y + 2$ and $MUy = x$.

Use Px to represent the price of milk, Py to represent the price of coffee, and I to represent Billy's income.

Suppose that $Px = \$1$ and $I = \$40$. Find the equivalent variation for an increase in the price of coffee from $Py1 = \$4$ to $Py2 = \$5$.

**Solution:** The budget constraint is: $p_x x + p_y y = I$. Or: $x + 4y = 40$. In the point of the local consumer market equilibrium the following equation must be implemented:

$$MUx / MUy = Px / Py.$$

So, before an increase in the price of coffee we have the next equation:

$$(y + 2) / x = 1 / 4. \text{ So, } x = 4(y + 2).$$

After substitution of the last expression to the budget constraint we obtain the following: $4(y + 2) + 4y = 40$, $8y + 8 = 40$, $y = (40 - 8) / 8 = 4$.

$$x = 4(y + 2) = 4(4 + 2) = 24.$$

So, the utility maximizing bundle is $x = 24$, $y = 4$.

After increase in the price of coffee the new budget constraint is: $x^* + 5y^* = 40$, $(y^* + 2) / x^* = 1 / 5$, $x^* = 5(y^* + 2)$. So, $5(y^* + 2) + 5y = 40$, $10y^* + 10 = 40$, $y^* = 3$, $x^* = 5(3 + 2) = 25$.

The new utility maximizing bundle is (25;3).

Such the consumption bundle cost before an increase in the price of coffee:

$$I^* = x^* + 4y^* = 25 + 4*3 = 37.$$

So, the equivalent variation for an increase in the price of coffee from $Py1 = \$4$ to $Py2 = \$5$ is:

$$EV = I - I^* = 40 - 37 = 3.$$

**Answer:** The equivalent variation for an increase in the price of coffee is 3 units of income.

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