Suppose that consumer has utility function U(X1, X2)=X½,X½. He originally face price (1,1) and has income 100. Then the price of good1 increase to 2.
Follow steps to find compensating and equivalent variation?
To find the compensating and equivalent variation, we need to determine the consumer's new optimal consumption bundle after the price change. Here are the steps to follow:
Step 1: Find the initial optimal consumption bundle.
Given the utility function U(X1, X2) = X1^(1/2) * X2^(1/2), and the initial prices p1 = 1 and p2 = 1, we need to find the consumer's initial optimal consumption bundle.
To find the initial optimal consumption bundle, we can use the utility maximization condition, which states that the consumer will allocate their income in a way that maximizes their utility, given the prices of goods.
Mathematically, this can be expressed as follows:
Maximize U(X1, X2) = X1^(1/2) * X2^(1/2)
subject to the budget constraint p1X1 + p2X2 = I, where I = 100.
Substituting the values of p1, p2, and I into the budget constraint, we have:
1X1 + 1X2 = 100
X1 + X2 = 100
From here, we can solve for X1 and X2 using either substitution or graphical methods. Let's solve it graphically:
Graphically, the budget constraint X1 + X2 = 100 is a straight line that passes through the points (100, 0) and (0, 100) on an XY coordinate plane.
To find the consumer's optimal consumption bundle, we need to find the point on the budget constraint that maximizes their utility.
For a utility function of U(X1, X2) = X1^(1/2) * X2^(1/2), the consumer's optimal consumption bundle will be where the marginal rate of substitution is equal to the relative price of goods.
The marginal rate of substitution (MRS) is the rate at which the consumer is willing to exchange one good for another while keeping the utility constant.
In this case, the MRS can be calculated as:
MRS = MU(X1) / MU(X2) = (dU/dX1) / (dU/dX2) = (1/2) * (X2/X1)^(1/2)
The relative price of goods is the ratio of their prices:
Relative Price = p1 / p2 = 1 / 1 = 1
Setting the MRS equal to the relative price, we have:
(1/2) * (X2/X1)^(1/2) = 1
Simplifying this equation, we get:
(X2/X1)^(1/2) = 2
Squaring both sides, we obtain:
X2/X1 = 4
From the budget constraint equation X1 + X2 = 100, we can substitute the value of X2/X1 to get:
X1 + 4X1 = 100
5X1 = 100
X1 = 20
Substituting X1 = 20 into the budget constraint equation, we get:
20 + X2 = 100
X2 = 80
Therefore, the initial optimal consumption bundle is X1 = 20 and X2 = 80.
Step 2: Find the new optimal consumption bundle.
Now, let's consider the price change, where the price of good 1 (p1) increases from 1 to 2. The price of good 2 (p2) remains the same at 1.
To find the new optimal consumption bundle, we repeat the same process as in step 1, but with the new prices.
The new budget constraint is:
2X1 + X2 = 100
The new MRS equation is:
(1/2) * (X2/X1)^(1/2) = 2
To solve these equations, we need to find X1 and X2.
From the new budget constraint:
2X1 + X2 = 100
From the MRS equation:
Simplifying (X2/X1)^(1/2) = 4, we get X2/X1 = 16
Substituting X1 = 100 - X2 into the X2/X1 equation, we have:
X2 / (100 - X2) = 16
Solving this equation for X2, we get:
X2 = 80
Substituting X2 = 80 into the budget constraint equation, we have:
2X1 + 80 = 100
2X1 = 100 - 80
2X1 = 20
X1 = 10
Therefore, the new optimal consumption bundle is X1 = 10 and X2 = 80.
Step 3: Calculate Compensating Variation (CV).
The compensating variation (CV) represents the amount of additional income the consumer needs to reach the same utility level as before the price change.
To calculate the CV, we need to compare the consumer's initial utility level with their utility level after the price change.
The initial utility level can be calculated by substituting X1 = 20 and X2 = 80 into the utility function:
U(X1, X2) = X1^(1/2) * X2^(1/2) = 20^(1/2) * 80^(1/2) = 4 * 8 = 32
The utility level after the price change can be calculated by substituting X1 = 10 and X2 = 80 into the utility function:
U(X1, X2) = X1^(1/2) * X2^(1/2) = 10^(1/2) * 80^(1/2) = 2 * 8 = 16
The compensating variation (CV) is the difference between the two utility levels:
CV = Initial Utility - New Utility = 32 - 16 = 16
Therefore, the compensating variation is 16.
Step 4: Calculate Equivalent Variation (EV).
The equivalent variation (EV) represents the amount of income the consumer would need to be as well off as they are after the price change, with the initial prices.
To calculate the EV, we need to compare the consumer's utility level after the price change with their utility level at the initial prices.
The utility level after the price change can be calculated as we did in step 3:
U(X1, X2) = X1^(1/2) * X2^(1/2) = 10^(1/2) * 80^(1/2) = 2 * 8 = 16
The utility level at the initial prices can be calculated by substituting X1 = 20 and X2 = 80 into the utility function:
U(X1, X2) = X1^(1/2) * X2^(1/2) = 20^(1/2) * 80^(1/2) = 4 * 8 = 32
The equivalent variation (EV) is the difference between the two utility levels:
EV = New Utility - Initial Utility = 16 - 32 = -16
Therefore, the equivalent variation is -16.
In conclusion, the compensating variation is 16, indicating that the consumer would need an additional income of 16 to be as well off as before the price change. The equivalent variation is -16, meaning that the consumer would be willing to forgo an income reduction of 16 to be as well off as after the price change.