EBK MICROECONOMICS
5th Edition
ISBN: 9781118883228
Author: David
Publisher: YUZU
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Chapter 3, Problem 3.25P
To determine
To evaluate:
The utility function exhibiting the property of diminishing,
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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.
A consumer has the following utility function: Ulx, y) = xy -y, *21 where x and y represents the quantities consumed of goods X and Y.
y 20
What will be the substitution and income effects for X and Yassuming that the consumer attempts to maintain the same level of utility achieved
before price of Y increased (that is, when price of Y was $1)?
SEx= +0.5 IEx = -0.5
SE, = -0.25 IE-
= -0.25
SEx= +0.293 IE = -0.293
SEy = -0.414 IE, = +0.414
SEr= +0.25 IE
SE, = -0.75 IE, = -0.75
= -0.25
SEx= +0.414 IEx = -0.414
SEy = -0.293 IE, = -0.207
Income = $3 Px= $1, Py= $2
It is common for supermarkets to carry both generic (store-label) and brand-name (producer-label) varieties of sugar and other products. Many consumers view these products as perfect substitutes, meaning that consumers are always willing to substitute a constant proportion of the store brand for the producer brand. Consider a consumer who is always willing to substitute 4 pounds of a generic store brand for 2 pounds of a brand-name sugar. Do these preferences exhibit a diminishing marginal rate of substitution? Assume that this consumer has $24 of income to spend on sugar, and the price of store-brand sugar is $1 per pound and the price of producer-brand sugar is $3 per pound. How much of each type of sugar will be purchased? How would your answer change if the price of store-brand sugar was $2 per pound and the price of producer-brand sugar was $3 per pound?
Chapter 3 Solutions
EBK MICROECONOMICS
Ch. 3 - Prob. 1RECh. 3 - Prob. 2RECh. 3 - Prob. 3RECh. 3 - Prob. 4RECh. 3 - Prob. 5RECh. 3 - Prob. 6RECh. 3 - Prob. 7RECh. 3 - Prob. 8RECh. 3 - Prob. 9RECh. 3 - Prob. 10RE
Ch. 3 - Prob. 11RECh. 3 - Prob. 3.1PCh. 3 - Prob. 3.2PCh. 3 - Prob. 3.3PCh. 3 - Prob. 3.4PCh. 3 - Prob. 3.5PCh. 3 - Prob. 3.6PCh. 3 - Prob. 3.7PCh. 3 - Prob. 3.8PCh. 3 - Prob. 3.9PCh. 3 - Prob. 3.10PCh. 3 - Prob. 3.11PCh. 3 - Prob. 3.12PCh. 3 - Prob. 3.13PCh. 3 - Prob. 3.14PCh. 3 - Prob. 3.15PCh. 3 - Prob. 3.16PCh. 3 - Prob. 3.17PCh. 3 - Prob. 3.18PCh. 3 - Prob. 3.19PCh. 3 - Prob. 3.20PCh. 3 - Prob. 3.21PCh. 3 - Prob. 3.22PCh. 3 - Prob. 3.23PCh. 3 - Prob. 3.24PCh. 3 - Prob. 3.25PCh. 3 - Prob. 3.26P
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- Law of equi marginal utility is an important law of cardinal utility analysis. Explain this law with the help of its assumptions. Also explain the mechanism that how the total utility will be maximum at a point when the marginal utilities of both the goods become equal. Furthermore, there is a relationship between total and marginal utilities where they both pass through different stages when the consumer continues his or her consumption regularly. Describe this case brieflyarrow_forwardLaw of equi marginal utility is an important law of cardinal utility analysis. Explain this law with the help of its assumptions. Furthermore, there is a relationship between total and marginal utilities where they both pass through different stages when the consumer continues his or her consumption regularly. Elaborate.arrow_forwardEren’s two main hobbies are taking vacations overseas (V) and eating expensivemeals (M). His utility function is given as: U(V,M) = V2MLast year, the average price of taking a vacation overseas was US$200 and the averageprice of an expensive meal is $50. However, due to supply problems in Onions, theaverage price of an expensive meal rose to $75. The average price of a vacation did notchange. His income, which is $1500, did not change. Calculate for the equivalent variation (EV) for the price change.arrow_forward
- -1/e Consider the utility funetion 2 (x1.x, ) = ( ax" + (1- a)x,). Ifp= 0, the elasticity of substitution is equal to 1. 1 The elasticity of substitution for CES utility functions is 1+p True Falsearrow_forwardEren’s two main hobbies are taking vacations overseas (V) and eating expensivemeals (M). His utility function is given as: U(V,M) = V2MLast year, the average price of taking a vacation overseas was US$200 and the averageprice of an expensive meal is $50. However, due to supply problems in Onions, theaverage price of an expensive meal rose to $75. The average price of a vacation did notchange. His income, which is $1500, did not change. Suppose that the Department of Welfare wants to know how much should begiven to Eren to offset his change un utility due to the price increase of an expensivemeal. Calculate the compensative variation (CV).arrow_forwardEren’s two main hobbies are taking vacations overseas (V) and eating expensivemeals (M). His utility function is given as: U(V,M) = V2MLast year, the average price of taking a vacation overseas was US$200 and the averageprice of an expensive meal is $50. However, due to supply problems in Onions, theaverage price of an expensive meal rose to $75. The average price of a vacation did notchange. His income, which is $1500, did not change. Calculate the change in consumer surplus from consuming the expensivemeals considering the price change (Hint: you need to compare his optimalconsumption bundle before and after the price change to get the change in CS).arrow_forward
- ASAP 1. Consider a consumer with utility u(x1,x2) = .5lnx1 + .5lnx2 (b) Now, consider an equivalent representation of the above utility functionx51 x52 Does this utility function have the "increasing difference property"? Howabout the "strict increasing difference" property?arrow_forwardComing on to the main solution, we have been given a perfect substitute utility function. Remember, if any utility function Ujs in the form: U = ax + by, then it means that goods X and Y are perfect dubstitutes of each other. The utility cusive in this case is represented by a down word sloping straight line. At every point on the curve, the MRS remains constant, unlike a Convex Utility curve, in which regular MRS reduces from left to eight Note: When a consumer. faces perfect dubstitute utility function, then he/she Consumes at the corner point. It depends upon the ratio between Marginal Utilities of particular good and its respective police. мох Px For eg. if. if, мор Pr > U(X,Y)= 4x+ 3 Y MUX = du(x, y) 4 ১U(X, = ах +XR³ x 4 = 1/2/3/1 = 1. 4 мож Py мох Px - =1.33 Corner Points ) 1 U=ax+by ^ cosner then only x good is consumed Ju(x,x) = 3 ۵۷ points then only Y-good is consumed. MUY = JU(X,Y)arrow_forward
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