In this exercise, we explore the concept o f marginal rates of substitution (and, in part B, its relation to utility functions) further.
A: Suppose I own 3 bananas and 6 apples, and you own 5 bananas and 10 apples.
(a) With bananas on the horizontal axis and apples on the vertical, the slope of my indifference curve at my current bundle is −2, and the slope of your indifference curve through your current bundle is −1. Assume that our tastes satisfy our usual five assumptions. Can you suggest a trade tome that would make both of us better off? (Feel free to assume we can trade fractions of apples and bananas).
(b) After we engage in the trade you suggested, will our MRS’s have gone up or down (in absolute value)?
(c) If the values for our MRS’s at our current consumption bundles were reversed, how would your answers to (a) and (b) change?
(d)What would have to be true about our MRS’s at our current bundles in order for you not to be able to come up with amutually beneficial trade?
(e) True or False: If we have different tastes, then we will always be able to trade with both of us benefitting.
(f) True or False: If we have the same tastes, then we will never be able to trade with both of us benefitting.
B: Consider the following five utility functions and assume that α and β are positive real numbers:
(a) Calculate the formula for MRS for each of these utility functions.
(b)Which utility functions represent tastes that have linear indifference curves?
(c) Which of these utility functions represent the same underlying tastes?
(d)Which of these utility functions represent tastes that do not satisfy the monotonicity assumption?
e) Which of these utility functions represent tastes that do not satisfy the convexity assumption?
(f) Which of these utility functions represent tastes that are not rational (i.e. that do not satisfy the completeness and transitivity assumptions)?
(g)Which of these utility functions represent tastes that are not continuous?
(h) Consider the following statement: “Benefits from trade emerge because we have different tastes. If individuals had the same tastes, they would not be able to benefit from trading with one another.” Is this statement ever true, and if so, are there any tastes represented by the utility functions in this problem for which the statement is true?