Question

As defined in Chapter \(3,\) an indifference map is homothetic if any straight line through the origin cuts all indifference curves at points of equal slope: The \(M R S\) depends on the ratio \(Y / X\) a. Prove that in this case \(d X / d\)lis constant. b. Prove that if an individual's tastes can be represented by a homothetic indifference map, price and quantity must move in opposite directions; that is, prove that Giffen's paradox cannot occur

Short answer

#Answer# a) The change in X with respect to the change in the individual's income (dX/dI) is constant for a homothetic indifference map. b) For a homothetic indifference map, price and quantity must move in opposite directions, which means Giffen's paradox cannot occur.

Step-by-step solution

Part a: Prove that in this case \(d X / dI\) is constant.

First, recall that the marginal rate of substitution (MRS) is the rate at which the consumer is willing to substitute good X for good Y, which can be expressed as \(MRS = - \frac{dY}{dX}\). Now, we are given that MRS depends on the ratio Y/X. So, let's put this relationship into function form: \(MRS = - \frac{dY}{dX}= f(Y/X)\) Take the derivative of Y with respect to the individual's income (I), and let's define it as \(dY/dI\): \(\frac{dY}{dI} = g(Y/X)\) Now, we need to find the relationship between X and I. To do this, divide the MRS equation by the \(dY/dI\) equation: \(\frac{f(Y/X)}{g(Y/X)} = - \frac{dX}{dI}\) Since the left-hand side of the equation only depends on the ratio Y/X, its derivative with respect to income must be constant: \(\frac{d(-dX/dI)}{dI} = 0\) This implies that the change in X with respect to the change in the individual's income (\(dX/dI\)) is constant, as required.

Part b: Prove that for a homothetic indifference map, price and quantity must move in opposite directions.

To prove this statement, let's consider the budget constraint of the individual: \(P_X X + P_Y Y = I\) Take the total differential of the budget constraint: \(d(P_X X) + d(P_Y Y) = dI\) Rearrange the equation: \(dX = -\frac{d(P_Y Y)}{P_X} + \frac{dI}{P_X}\) Now, we have to analyze this equation to determine the relationship between price and quantity. From part a, we know that the change in X with respect to the change in the individual's income is constant. Thus, we can write: \(-\frac{d(P_Y Y)}{P_X} + \frac{dI}{P_X} = constant\) From this equation, we can observe that if the price \(P_X\) increases, the quantity consumed X must decrease to make the equation hold true. Similarly, if the price \(P_X\) decreases, the quantity consumed X must increase. This proves that price and quantity must move in opposite directions for an individual's preferences represented by a homothetic indifference map. As a result, Giffen's paradox cannot occur, which states that price and quantity can sometimes move in the same direction, contradicting the law of demand.