Consider the classic consumer's choice problem of an individual who is allocatingY dollars of wealth amongst two goods. Let c1 denote the amount of good 1 that theindividual would like to consume at a price of Pı per unit and c2 denote the amountof good 2 that the individual would like to consume at a price of P2 per unit. Theindividual's utility is defined over the consumption of these two goods (only). Supposewe allow the individual's happiness to be measured by a utility function u(c1, c2) whichis increasing and strictly concave in both goods while also satisfying the Inada condition,limcı→0du(c1,c2)= limc1→0du(c1,c2)dc2= 0.The Inada conditions simply say that the slope of the utility function becomes verticalin the direction of the good that has its consumption level go to zero. By assumingincreasing and strictly concave utility in both directions, we are assuming that the shapeof the utility function is such that, holding constant the amount of one good, as theindividual consumes more of the other good, they are happier but each additional unitof consumption yields less extra happiness than the previous unit (diminishing marginalutility of consumption). These assumptions imply that the first-order partial derivativeof the utility function with respect to good one, u1(C1, C2)du(c1,c2) and good two,acidu(c1,c2)u2(C1, C2)U1(C1, C2) remains positive but gets smaller. Similarly, holding c constant, ifU2(C1, C2) remains positive but gets smaller.Ci increases,increases,exist and are positive. Moreover, holding C2 constant,ifdc2C2 1. Let the budget constraint faced by the individual be PiC1 + P2c2 =Y. In words,interpret the meaning of the mathematical budget constraint.2. The individual's problem is to choose the consumption bundle (cı, c2) optimally.Specifically,max {u(с1, С2)}C1,C2subject to (cı, c2) satisfying the budget constraint, Pic1 + P2c2 = Y. Using theMethod of Lagrange, let A be the Lagrange multiplier on the budget constraint.The the Lagrangean function can be written asL(c1, C2, A) = u(c1, c2) + A (Y – P.c1 – P,c2).-Take the first order-derivative of the Lagrangean function with respect to c1, c2 andA. Set each of them equal to zero in order provide the equations that can be usedto identify the Lagrangean function's "critical points".

Consider the classic consumer's choice problem of an individual who is allocating Y dollars of wealth amongst two goods. Let c, denote the amount of good 1 that the ndividual would like to consume at a price of P per unit and c2 denote the amount of good 2 that the individual would like to consume at a price of P2 per unit. The ndividual's utility is defined over the consumption of these two goods (only). Suppose we allow the individual's happiness to be measured by a utility function u(Cı, c2) which s increasing and strictly concave in both goods while also satisfying the Inada condition, Limcı→0 du(c1,c2) limcı→0 du(c1,c2) = 0. The Inada conditions simply say that the slope of the utility function becomes vertical n the direction of the good that has its consumption level go to zero. By assuming ncreasing and strictly concave utility in both directions, we are assuming that the shape

Consider the classic consumer's choice problem of an individual who is allocating Y dollars of wealth amongst two goods. Let c, denote the amount of good 1 that the ndividual would like to consume at a price of P per unit and c2 denote the amount of good 2 that the individual would like to consume at a price of P2 per unit. The ndividual's utility is defined over the consumption of these two goods (only). Suppose we allow the individual's happiness to be measured by a utility function u(Cı, c2) which s increasing and strictly concave in both goods while also satisfying the Inada condition, Limcı→0 du(c1,c2) limcı→0 du(c1,c2) = 0. The Inada conditions simply say that the slope of the utility function becomes vertical n the direction of the good that has its consumption level go to zero. By assuming ncreasing and strictly concave utility in both directions, we are assuming that the shape

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter3: Preferences And Utility

Section: Chapter Questions

Problem 3.15P

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Amadea spends all her income on pizza,chocolate milk, and music streams. The price of pizza is $4 per slice and price ofchocolate milk $2 per bottle. At Amadea's current consumption level, her marginalutility for pizza is 50 units per slice and for music 100 units per stream. Assuming thatAmadea is a utility maximizer, how much is (a)her current marginal utility for a bottleof chocolate milk, and (b) the price of streams she buys. Show your work. [Hint: Ch 6:Utility-maximizing formula)
a. A consumer is willing to trade 3 units of x for 1 unitof y when she has 6 units of x and 5 units of y. She isalso willing to trade in 6 units of x for 2 units of y whenshe has 12 units of x and 3 units of y. She is indifferentbetween bundle (6, 5) and bundle (12, 3). What is theutility function for goods x and y? Hint: What is theshape of the indifference curve?b. A consumer is willing to trade 4 units of x for 1 unitof y when she is consuming bundle (8, 1). She is alsowilling to trade in 1 unit of x for 2 units of y when sheis consuming bundle (4, 4). She is indifferent betweenthese two bundles. Assuming that the utility function isCobb–Douglas of the form U (x, y) = xα y β, where α andβ are positive constants, what is the utility function forthis consumer?c. Was there a redundancy of information in part (b)? Ifyes, how much is the minimum amount of informationrequired in that question to derive the utility function?
Consider the classic consumer's choice problem of an individual who is allocating Y dollars of wealth amongst two goods. Let c1 denote the amount of good 1 that the individual would like to consume at a price of Pı per unit and c2 denote the amount of good 2 that the individual would like to consume at a price of P2 per unit. The individual's utility is defined over the consumption of these two goods (only). Suppose we allow the individual's happiness to be measured by a utility function u(c1, c2) which is increasing and strictly concave in both goods while also satisfying the Inada condition, limcı→0 du(c1,c2) = limc1→0 du(c1,c2) dc2 = 0. The Inada conditions simply say that the slope of the utility function becomes vertical in the direction of the good that has its consumption level go to zero. By assuming increasing and strictly concave utility in both directions, we are assuming that the shape of the utility function is such that, holding constant the amount of one good, as the…
Mylie’s total utility from singing the same song over andover is 50 utils after one repetition, 90 utils after tworepetitions, 70 utils after three repetitions, 20 utils afterfour repetitions, 250 utils after five repetitions, and 2200utils after six repetitions. Write down her marginal utilityfor each repetition. Once Mylie’s total utility beginsto decrease, does each additional singing of the songhurt more than the previous one or less than the previousone?
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What do we mean by well-behaved preferences? How do the indifference curves look like for suchpreferences? Provide an example, in terms of a utility function, where preferences are not well-behavedand illustrate. Provide a figure with indifference curves and a budget line where there are exactly twosolutions to the utility maximization problem.
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