a) The utility maximization problem is as follows:
Maximize u(x1,x2) =2x1x2 +1
Subject to : x1p1 +x2p2 = m
b) the lagrange expression of the above maximization problem is
L= 2x1x2 +1 +(m - x1p1- x2p2)
the first order conditions are
2x2 - p1=0 2x2=p1 .....(1)
2x1 - p2=0 2x1=p2 ......(2)
m-x1p1 - x2p2=0 ......(3)
dividing (1) by(2) x2/x1= p1/p2 x2p2=x1p1......(4)
Putiing (4) in (3) m-x1p1-x1p1 = 0 2x1p1=m x1=m/2p1
x2= m/2p2
c) demand functions are x1= m/2p1 and x2=m/2p2
if p1=1 , p2=2 and m=20
x1=20/(2x1) = 10 and x2=20/(2x2)=5
6. CHOICE (10 POINTS) You consume only 2 goods: 11 and 12. Your utility function for...
d @ See page 78 05 Question (2 points) In addition to finding the optimal bundles given prices and income, utility maximization can be used to find individual demand functions at any prices and income. Setting up the problem and solving it are the same, except that the prices of each good and the income will be left in variable form (economists call these parameters or exogenous variables). 1st attempt See Hint Consider a utility function that represents preferences over...
3) Jim likes to consume ice-cream and strawberries according to the perfect complements utility function u(21,22) = min {2.01,22}. Currently his income is m = 15 and pı = P2 = 1. Suppose that P2 increases to pl = 2. Which of the following is true about the optimal choice ? The substutition effect is positive The income effect is equal to the total effect The substitution effect is equal to the total effect The total effect is zero None...
A consumer has the following preferences u(11, 12) = log (11) + 12 Suppose the price of good 1 is pı and the price of good 2 is P2. The consumer has income m. (a) Find the optimal choices for the utility maximization problem in terms of P1, P2 and m. Denote the Lagrange multiplier by 1. (b) How do the optimal choices change as m increases? What does the income offer curve (also called the income expansion path) look...
1. Consider a case where utility of Ali from two goods I, and 2, is given by (21,22) = x1*29. Good 1 has price Pa, good 2's price is P2 and Ali has money m. (6 points) a) What is Ali's marginal utility from consuming good 1, what about good 2? Hint: Take the natural logarithm of Ali's utility function first. (1 point) b) What is Ali's demand function for good 1, what about good 2? (1 point) c) Consider...
Suppose Alex’s preferences are represented by u(x1,x2) = x1x32. The marginal utilities for this utility function are MU1 = x23 and MU2 = 3x1x22. (a) Show that Alex’s utility function belongs to a class of functions that are known to be well-behaved and strictly convex. (b) Find the MRS. [Note: find the MRS for the original utility function, not some monotonic transformation of it.] (c) Write down the tangency condition needed to find an optimal consumption bundle for well-behaved preferences....
6) A pure-exchange economy has n consumers and two goods. The aggregate excess demand functions for goods 1 and 2, defined for all strictly positive price vectors p (pi, p2), are given by Z,(p) = I'l n-A and Z. (p)-n n-B where A and B are real numbers. Assume that 2p2 P1 these excess demand functions are derived from each consumer i maximizing a strictly monotonic utility function subject to the budget constraint р.Х. DN a) Find all values of...
1. (10 points) Market demand Part 1 There are two consumer goods, X1 and 22. Consumers all have income given by m, and a utility function U (x1, x2) = aln(x1) + ln(x2). The price of the two goods are pı and p2. (a) Find the individual demand functions for Xı and 22. (b) The parameter a differs across consumers. Type A consumers have a = 1. Type B consumers have a = 2. If there is one type A...
Question 1 (20 points). The utility function of the consumer is u(x1, x2) = x1 + x2. a) Let pı = 2 ,P2 = 20 and m = 24. Calculate the optimal quantity demanded of good 1 and 2. (7 points) b) Let p1 = 1,P2 = 4 and m = 100. Calculate the optimal quantity demanded of good 1 and 2. (6 points) c) Let P1 = 1, p2 = 4 and m = 4. Compared to point b),...
Suppose you have a total income of I to spend on two goods x1 and x2, with unit prices p1 and p2 respectively. Your taste can be represented by the utility function u left parenthesis x subscript 1 comma x subscript 2 right parenthesis equals x subscript 1 cubed x subscript 2 squared (a) What is your optimal choice for x1 and x2 (as functions of p1 and p2 and I) ? Use the Lagrange Method. (b) Given prices p1...
Consider the following utility function over goods 1 and 2, plnx1 +3lnx2: (a) [15 points] Derive the Marshallian demand functions and the indirect utility function. (b) [15 points] Using the indirect utility function that you obtained in part (a), derive the expenditure function from it and then derive the Hicksian demand function for good 1. (c) [10 points] Using the functions you have derived in the above, show that i. the indirect utility function is homogeneous of degree zero in...