Chapter 2, Problem 2.9P

f(x1,x2) 4 2 0 -2 -4 −1 2 0 1 1 0 2 -1 X1 3-2 x2 Figure 2: Graph of the function f(x1, x2) = 2x1 = x²-x+3 Consider the following function: f(x1, x2) = 2x1 - x-x+3. - For your visual convenience, a portion of the function is depicted in Figure 2. (a) Compute the gradient vector of f. (b) Compute the Hessian matrix of f. (c) Determine all points that yield stationary values of f. (d) Use the second-order conditions to determine whether each of the stationary val- ues of f is a maximum, a minimum, or neither.

On a given evening, J. P. enjoys the consumption of cigars (c) and brandy (b) according tothe functionU(c, b) = 20c− c²+ 18b − 3b²a. How many cigars and glasses of brandy does he consume during an evening? (Cost isno object to J. P.)b. Lately, however, J. P. has been advised by his doctors that he should limit the sum ofglasses of brandy and cigars consumed to 5. How many glasses of brandy and cigarswill he consume under these circumstances?

In the 1980s, President Reagan based his tax and spending policies õñ supply side economics. The idea behind supply side economics is the marginal tax rate is so high it discourages work. Cutting the tax rate would end up increasing tax revenue. We develop a simple model of this idea to determine the restrictions on the utility function required to generate a Laffer curve. Let t denote the tax rate, w the real wage rate, and n the labor supply. The tax revenue is T = wnT where wn is labor income, which is the tax base. For convenience, assume w is constant. There is no reason for this assumption to be true, but we impose it to focus on the restrictions on the utility function to generate the Laffer curve. As the tax rateT increases, workers substitute toward leisure and away from consumption. Hence as 7 rises, wn falls and tax revenue falls for high enough tax rates. Let U, V satisfy the standard assumptions. The model is static and households are endowed with one unit of time. A…

In the 1980s, President Reagan based his tax and spending policies on supply side economics. The idea behind supply side economics is the marginal tax rate is so high it discourages work. Cutting the tax rate would end up increasing tax revenue. We develop a simple model of this idea to determine the restrictions on the utility function required to generate a Laffer curve. Let T denote the tax rate, w the real wage rate, and n the labor supply. The tax revenue is T = wNT where wn is labor income, which is the tax base. For convenience, assume w is constant. There is no reason for this assumption to be true, but we impose it to focus on the restrictions on the utility function to generate the Laffer curve. As the tax rate T increases, workers substitute toward leisure and away from consumption. Hence as T rises, wn falls and tax revenue falls for high enough tax rates. Let U,V satisfy the standard assumptions. The model is static and households are endowed with one unit of time. A…

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?

3. Marginal versus average effects Consider the nonlinear function shown by the blue curve on the following graph. The grey line is tangent to this curve at point A. Use the grey line (star point) to answer the following question. (Note: You will not be graded on your placement of this line.) 10 9 8 7 6 5 4 3 2 1 0 0 1 2 O True 3 O False A + 4 Tangent Line The marginal effect of X on Y at point A is 5 X The average effect of X on Y at point A is 6 7 8 9 f(x) True or False: For this graph, the marginal effect decreases as X increases. 10 Scratch Line

What is the second (or second order) condition for finding the maximum point of a function? O Second derivative (or rate of change of the first derivative) is negative Second derivative is positive OSecond derivative (or rate of change of the first derivative) is zero Second deriyative (or rate of change of the first derivative) is at a minimum

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