Using the CES function given as u = (ax" + byP)F a) Calculate the indirect utility function b) State the properties of the indirect utility function and the expenditure function.
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- 1. Prove if the indirect utility function is quasiconvex The indirect utility function: V(p,w) = w[P1^(p/p-1) + P2^(p/p-1)]^((1-p)/p)Write down the equation for the budget constraint and then graph it for .. Px = $100 Py = $25 Income = $5,000.003. U(X,Y)= X¹/2Y1/2 M = $36: Py = $1; Px is initially = $1; the price of Good X increases to Px = $9. 4. U(X,Y)= X1/2Y1/2 M = $64; Py = $1; Px is initially = $16; the price of Good X decreases to Px = $1.
- 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?Suppose a consumer has a monthly income of m = 100 which she spendson two commodities: french fries (x1) and beef jerky (x2). The price offrench fries is p1 = 2 and the price of beef jerky is p2 = 5.(a) Write down the consumer’s budget constraint (equation).(b) What is the maximal consumption of french fries (this is calledthe real income in french fries).(c) Find the maximal consumption of beef jerky (real income interms of beef jerky).6. Find the indirect utility function and the expenditure function for the Cobb-Douglas utility function.
- Please use Kuhn Tucker 5. A consumer's utility function is: u(x) = x1+ /®2 (c) Verify the properties of the indirect utility function. (d) Calculate the expenditure function and verify its properties.Suppose Maria would spend $1500 per year on travel, even if her annual income were zero. As her income rises, she would spend 30% of each additional dollar of income on travel. a. Write a mathematical equation that describes the functional relation between Maria travel spending (T) and her income (Y). b. Draw a graph from the equation developed in part a above.Total Utility derived from the consumption of a product to a consumer is given by U = (20-0.50)'Q where Q is the number of units consumed Using a graphical representation, show what is the nature of variation in the utility with the increasing consumption of the product? Is it in line with the law of diminishing marginal utility? Explain.
- 4.13 CES indirect utility and expenditure functions In this problem, we will use a more standard form of the CES utility function to derive indirect utility and expenditure functions. Suppose utility is given by U(x, y) = (x° +y®)'/8 [in this function the elasticity of substitution o = 1/(1 – 6)]. a. Show that the indirect utility function for the utility function just given is V = I(p, + p,)¬/", where r = 8/(ò – 1) = 1 – 0. b. Show that the function derived in part (a) is homogeneous of degree zero in prices and income. c. Show that this function is strictly increasing in income. d. Show that this function is strictly decreasing in any price. e. Show that the expenditure function for this case of CES utility is given by E = V(p', + p,)''". f. Show that the function derived in part (e) is homogeneous of degree one in the goods' prices. g. Show that this expenditure function is increasing in each of the prices. h. Show that the function is concave in each price.4. Find the marginal rate of substitution in consumption for the utility function and determine its value at x = 2, y = 3 U (x, y) = 5x² + x²y + y2+6yConsider the two bundles A and B. Note: each bundle is a pair of values, (c1, c2).Calculate the expenditure for the each of the two bundles. Put your answers in theblue highlighted cells for Q5.