Problem 3. Consider a consumer with income of M = 400 and preferences over some

Question

Problem 3. Consider a consumer with income of M = 400 and preferences over some good, X, and all other goods, Y , represented by the utility function U(X, Y ) = 100 ln X + Y . For each part of this problem, we’ll consider the impact on the consumer of a given increase in price. (a) Without doing any math, explain how you can know that compensating variation is the same as equivalent variation. (b) Compute the Marshallian demand curve. Without doing any additional math, and based purely on the Marshallian demand curve, explain how you can know that compensating variation is the same as equivalent variation. (c) Now let’s compute this consumer’s pre- and post-change Hicksian demand curves. Suppose that the price starts at $4 and goes up to $5. (i) Compute the consumers pre-change utility level. (ii) We know from lecture that the pre-change Hicksian demand curve defined by the pre-change utility level is the quantity the consumer would choose to consume, at any given price, if they were compensated just enough to get them back to their pre-change utility level. Thus, for any given price, the pre-change Hicksian demand curve must satisfy two conditions. First, the slope of the budget constraint must be equal to the MRS, and second, the new bundle must generate the same level of utility, which is to say, it must be on the original indifference curve. Using these two conditions, compute the Hicksian demand curve defined by the pre-change utility level. [Hint: This is a trick question. It should require very little math.] (iii) Without doing any additional math, graph the inverse Marshallian demand curve and both inverse Hicksian demand curves, mark the compensating and equivalent variation for the price increase from $4 to $5 and explain why they are the same.

Explanation / Answer

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a) The utility function U(X,Y) = 100In X+Y
This utility function is quasilinear function. As we know the indifference curves for quasilinear are vertically parallel to each other. This means we have an exact measure of how much an individual is better off when they move from one indifference curve to another. As the vertical difference between the two IC remains the same. And in Quasi linear utility function, quantity purchased of one good is independent of the income.(we can see this from marshallian demand ).
Since Compensating variation is the amount of money that nee to be paid to the individual to compensate him for the adverse effect of the price change.
Equivalent variation is the amount of money that needs to be taken from the individual at the original price to reduce the individuals welfare by the same amount as the price rise.
Hence both of then are equal in case of Quasi-linear preferences.
b) Marshallian demand curve
With the help of Lagrange
L= 100lnX +Y + (400- Px.X -PyY)
dL/dx = 100/x - Px
dL/dy = 1- Py
dL/d = 400-PxX-PyY
On solving these 3 equations we get Marshallian demand
X* = 100.Py/Px
Y*= 400/Py - 100
As we can see from these demands that X doesn't depend on income . The income effect only change good y the linear part of utility function.
Since the vertical distance between the indifference curves remains the same . It is also the distance or difference between the two indifference curve that is needed to move the consume to original utility level. It is the compensating variable. This happens regardless of the values of price before and after change , which changed their optimal level and utility level.
This vertical difference is also the value that consumer will be willing to pay to avoid having any price change i.e Equivalent variation.
Hence EV and CV are equal (only) in case of quasi linear utility functions.

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