Compensating Variation (CV) vs. Equivalent Variation (EV)
1) Compensating Variation (CV)
Definition. Compensating Variation is the amount of money that must be given to (or taken from) a consumer after a price change so that the consumer can reach their original utility level.
Formal expression (expenditure function).
\( CV = E(p_{1}, u_{0}) - E(p_{1}, u_{1}) \)
- \( E(p, u) \): expenditure function (minimum expenditure to achieve utility \( u \) at prices \( p \))
- \( p_{1} \): new price vector
- \( u_{1} \): new utility level
- \( u_{0} \): initial utility level
Economic interpretation.
- Prices change first, and compensation is provided afterwards.
- \(CV \) answers: “How much compensation is required to keep the consumer just as well off as before?”
Example.
- Rebates. In many cases, prices are increased or fixed exogenously—due to taxes, regulations, or market conditions—and rebates are subsequently offered to offset the resulting welfare loss.
2) Equivalent Variation (EV)
Definition. Equivalent Variation is the amount of money that must be given to (or taken from) a consumer before a price change so that the consumer reaches the new utility level under the old prices.
Formal expression (expenditure function).
\( EV = E(p_{0}, u_{1}) − E(p_{0}, u_{0}) \)
- \( p_{0}\): initial price vector
Economic interpretation.
- \(EV \) answers: “How much money would make the consumer as well off as they would be after the price change?”
- the maximum lump-sum tax instead of a commodity tax
3) Relevant formulas
- Size of Deadweight Loss = | Tax revenues - Rebates |
- Current Budget = Initial Budget + Rebates
- \(EV \) = \( \int_{p_1^1}^{p_1^0} x^c (p_1, p_{-1}, u_1 ) dp_1 \)
- \(CV \) = \( \int_{p_1^1}^{p_1^0} x^c (p_1, p_{-1}, u_0 ) dp_1 \), since \(F(p_0, u_0) = F(p_1, u_1) \)
- \(CS \) = \( \int_{p_1^1}^{p_1^0} x^d (p_1, p_{-1}, M ) dp_1 \), where \(CS \) denotes consumer surplus
4) Intuition
- When the price decreases, the consumer is given an amount of money equal to \(CV_1 \) or \(EV_1 \). When the price increases, the consumer is required to give up an amount of money equal to \(CV_2 \) or \(EV_2 \).
- In absolute value, | \(CV_1 \) | = | \(EV_2 \) |, and | \(EV_1 \) | = | \(CV_2 \) |.
- Therefore, in many applications, the sign of \(CV \) and \(EV \) is less important than their magnitudes.
5) Deriving EV
Case 1 : The direct utility function is given
Step 1. Derive Marshallian demand \(x_d \), \(y_d \)
\(max \) \(U s.t. p_x x + p_y y = M \)
or else,
find \(x_d \), \(y_d \) that satisfies both \(MRS_{xy} = \frac{p_x}{p_y} \)and \(p_x x + p_y y = M \)
At this step, check whether a corner solution exists.
Step 2. Derive Hicksian (compensated) demand \(x_h \), \(y_h \)
Solve the expenditure minimization problem. There are two main approaches, as described below.
1) \(min \) \(E = p_x x + p_y y s.t. u(x, y) = \bar{u} \)
or else, find \(x_h \), \(y_h \) that satisfies both \(MRS_{xy} = \frac{p_x}{p_y} \)and \( u(x, y) = \bar{u} \)
2) Use Shephard's Lemma. \( x^h = \frac{ \partial E}{ \partial p_x}, y^h = \frac{ \partial E}{ \partial p_y} \)
Step 3. Compute \(u_1 \) after prices change.
Step 4. Compute \(x^h, y^h \) at the initial prices that achieve the post-change utility level.
Step 5. Compute the expenditure required to obtain the bundle needed from Step 4.
Step 6. EV = (Initial budget) - (Expenditure from Step 5)
Case 2 : The indirect utility function is given
Step 1. Compute the utility level after prices change by plugging \(p_1 \) into the indirect utility function.
Step 2. The lump-sum tax \(T \) imposed before the price change that equalizes utility defines the Equivalent Variation.
Example. \(V = \frac{I}{\sqrt{p_1 p_2}} \), \(p_1 \) changed from 100 to 400. \(p_2 \) remains 100, \(I \) remains 400. In this case, we can compute \(EV \) easily by calculating \(T \) that satisfies \( \frac{400}{\sqrt{400 \times 100}} = \frac{400-T}{\sqrt{100 \times 100}} \).
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