Chapter 2, Problem 2.9P

Hi, I need some help with the attached question. It is taken from an optimisation problem set in my intermediate maths for economists course.

A utility function is called separable if it can be written as U(x, y) = U (x) + U,(y), where U > 0, U" < 0, and U1, U, need not be the same function. a. What does separability assume about the cross-partial derivative U„? Give an intuitive discussion of what word this condition means and in what situations it might be plausible. b. Show that if utility is separable then neither good can be inferior. c. Does the assumption of separability allow you to con- clude definitively whether x and y are gross substitutes or gross complements? Explain. d. Use the Cobb-Douglas utility function to show that separability is not invariant with respect to monotonic transformations. Note: Separable functions are examined in more detail in the Extensions to this chapter.

(b) It is thought that a consumer measures the utility, u(x, y), of possessing a quantity, z, of apples and a quantity, y, of oranges by the formula u(x, y) = rªyla, where a is a constant. Suppose that apples cost p dollars each and oranges q dollars each. If the consumer has M dollars to spend on apples and oranges, use the method of Lagrange multipliers to find the quantity of each that he will buy in terms of a, p, q and M. [You are not required to justify the use of the method of Lagrange multipliers here.] For equally priced apples and oranges when he has one dollar to spend, he is actually observed buying one apple and two oranges. What value of a does this imply?

For each of the following scenarios, indicate whether the relationship between the two variables is positive or negative, as well as which line on the previous graph has a slope that reflects this type of relationship.   Hint: The X-axis and Y-axis on the graph are not labeled intentionally. You need to substitute the variables from each scenario for the horizontal and vertical axis. For example, in the first scenario, X-axis should be labeled “Income" and Y-axis should be labeled "Education". Relationship options: Positive, Negative Line: A, B, C, D   True or False: Line A has a slope of zero.

Is the function v(x1,x2)=x21x32 a monotonic transformation of u(x1,x2) ? Select one: a. No b. Yes

only typed solution

Suppose the demand function for product X is estimated by the equation: QDX= -2PX + 0,5PY - 0,2PZ + 1,2I. In there:QDX is the quantity demanded of product XPX is the price of product XPY is the price of product YPZ is the price of product ZI is the average income of consumersa/ Please make an argument to determine the relationship between two products X and Y, X and Z.b/ Please make an argument to determine whether product X is an ordinary good or a low-end good

a good is normal, then an increase in the price of the good will lead to which of the following to be true for this good? (Assume that there are only two goods, the individual's preferences lead to well-behaved preferences with strictly convex indifference curves and an interior solution for all budgets). Let SE = substitution effect, IE = income effect)   (a) The magnitude of the IE for this good must be larger than the magnitude of the SE   (b) The magnitude of the SE for this good must be larger than the magnitude of the IE   (c) The good could be a Giffen good   d) The good must be an ordinary good   ( (e) None of the above

When filing an income tax return, one can claim a deduction for charitable contributions. Let's simplify the income tax system and assume that the tax is proportional to the level of taxable income (income after deductions). (a) Suppose we increase the marginal income tax rate. An economic adviser claims that the effect of this tax change on the amount of charitable contributions is uncertain. Plot a simple graph that illustrates this situation (it involves the choice between charitable contributions and other spending), use it to explain why the effect is uncertain and explain what it depends on. (b) Now suppose that we instead increase the marginal tax rate but also provide a transfer that makes the bundle that used to be optimal before tax increased just affordable. According to the economic adviser this policy change will make people contribute more to charity. Why?