5. The CES function f defined for K > 0, L > 0, by f(K, L) = A(6K-P+ (1-6) L-P) - 7, where A> 0, p = 0, 0≤6 ≤ 1. Show that function f is concave if p > -1 and convex when p<-1
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- 4. Find the maximum values of the following functions and show whether they are maximized at their maximum values. (a) y= f(x)=10x – x² (b) z=10x+10y-xy-x² - y²f(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.5. 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 A 6 4 3 2 1 Tangent Line f(x) 0+ + + + + + 0 1 2 3 4 5 6 7 8 9 10 X The marginal effect of X on Y at point A is The average effect of X on Y at point A is True or False: For this graph, the marginal effect decreases as X increases. True False Scratch Line (?)
- For what value(s) of x is the function f(x) NOT continuous? A x = 0 and x = 3 C. x = -3 only Select one: = 4 2x² — 6x B. x = 3 only D. x = 3 and x = 42. Supose that a consumer has utility function U with U (₁,22) = x1426, where ₁ and ₂ are demand of commodity 1 and 2. Suppose that the price of commodity 1 and 2 is $3 and $5, respectively. This consumer has money m = 12 to spend and wants to maximize utility with constraint the money. (a) Write the Lagrangian function. (b) Find the critical point using first order condition. (c) Find the Hessian matrix to check whether the critical point is maximiser or minimiser. (d) Find the maximum utility.Define a weakly concave on the Cobb-Douglas Function
- Suppose Steve derives 3 units of utility from consuming either 1 cup of milk or 2 glasses of apple juice. It is known that the price of milk is 1 dollar per cup and the price of apple juice is 2 dollars per glass. What is Steve’s expenditure function? (a) (2/3)U (b) (1/2)U (c) (1/3)U (d) (5/3)U4. The BestElect electronics store is having a sale on November 8th on any new electronics purchased. Every customer will receive $50 off their purchase of a new electronic item. They are also offering a student discount on every item in the store for the whole month of November. If you can prove you are a student, you will receive 15% off any item. Let x be the price of any new electronic item in the store. a. Write a function P(x) that represents the price of a new electronic item on November 8th for any customer. b. Write a function D(x) that represents the price you will pay for an item after the student discount is applied during any day in November. Now, we have two functions P and D that depend on the price of the item, x. P is a rule that says "take the price of the item, then subtract $50." D is a rule that says "take the price, then take 85% of it." If you want to take advantage of both sales, a composition of functions is necessary! But is there a difference between P(D(x))…Differentiate the profit function with respect to ?1 ??? ?2.Explain intuitively why we take the derivative of the function.
- Is the function v(x1,x2)=x21x32 a monotonic transformation of u(x1,x2) ? Select one: a. No b. YesMy Moodle My Fa The function f(x, y) = (x + y) – a² – y? – 6xy has stationary points at some of the following points, (x, y). In each case identify whether the point is stationary, and if so find out if it is a maximum, minimum or saddle point. 1. The point (0,0) is 2. The point (1/2, 1/2) is 3. The point (1/2, -1/2) is 4. The point (-1/2,1/2) is 5. The point (-1/2,–1/2) is Next pa - from video in SVG Mock paper Jump to... formatJane insists on consuming 4 bowls of yogurt with 2 bananas. If the price of banana is $1 per unit and the price of yogurt is $1 per bowl, then if Jane income is $m, the slope of the Engel curve for bananas will be (a) negatíve and constant. (b) negative and increasing. (c) zero. (d) positive and constant. (e) None of the above.