Carefully write down the consumer's utility maximization problem and find Marshallian demand functions using the Lagrange method. Be especially careful listing all the consumer's constraints. Show that the demand functions you found in (a) are indeed the so- lutions of the utility maximization problem for the utility function (1). Derive the indirect utility function of the consumer. Derive the expenditure function of the consumer.

Carefully write down the consumer's utility maximization problem and find Marshallian demand functions using the Lagrange method. Be especially careful listing all the consumer's constraints. Show that the demand functions you found in (a) are indeed the so- lutions of the utility maximization problem for the utility function (1). Derive the indirect utility function of the consumer. Derive the expenditure function of the consumer.

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter3: Preferences And Utility

Section: Chapter Questions

Problem 3.9P

See similar textbooks

Similar questions

Praxilla, who lived in ancient Greece, derives utility from reading poems and from eating cucumbers. Praxilla gets 30 units of marginal utility from her first poem 27 units of marginal utility from her second poem 24 units of marginal utility from her third poem, and so on, with marginal utility declining by three units for each additional poem. Praxilla gets six units of marginal utility for each of her next three cucumbers consumed, five units of marginal utility for each of her next three cucumbers consumed, four units of marginal utility for each of the following three cucumbers consumed, and so on, with marginal utility declining by one for every three cucumber consumed. A poem costs three bronze coins hut a cucumber costs only one bronze coin. Praxilla has 18 bronze coins. Sketch Praxillas budget set between poems and cucumbers, placing poems on the vertical axis and cucumbers on the horizontal axis. Start off with the choice of zero poems and 18 cucumbers, and calculate the changes in marginal utility of moving along the budget line to the next choice of one poem and 15 cucumbers. Using this step-by-step process based on marginal utility, create a table and identify Praxillas utilityâ€”maximizing choice. Compare the marginal utility of the two goods and the relative prices at the optimal choice to see if the expected relationship holds. Hint: Label the table columns: 1) Choice, 2} Marginal Gain from More Poems, 3) Marginal Loss from Fewer Cucumbers, 4) Overall Gain or Loss, 5) Is the previous choice Optimal? Label the table rows: 1) 0 Poems and 18 Cucumbers, 2) 1 Poem and 15 cucumbers, 3) 2 Poems and 12 cucumbers, 4) 3 Poems and 9 Cucumbers, 5) 4 Poems and 6 cucumbers 6) 5 Poems and 3 Cucumbers, 7) 6 Poems and 0 Cucumbers.
Total utility is maximized in the consumption oftwo goods by equating thea. prices of both goods for the last dollar spenton each good.b. marginal utilities of both goods for the lastdollar spent on each good.c. ratios of marginal utility to the price of bothgoods for the last dollar spent on each good.d. marginal utility of one good to the price ofthe other good.
Answer the following short questions:a. Suppose that a consumer’s preferences between goods x andy are represented by the utility function u(x, y) = x^2 + 16xy + 64y^2. If these two goods have the same price, describe the optimal consumptionchoice of this consumer.b. Suppose that when the price of a good change, the incomeand substitution effects change the consumer’s demand for that goodin opposite directions.i. Is this good a normal or an inferior good? Explain.ii. Is this good a Giffen or an ordinary good? Explain.c. Is the following statement true or false? The differencebetween a monopolist’s marginal cost and its profit-maximizing price issmaller when the demand is more elastic.
Utility Maximization. Suppose that an individual has a utility function given asU =√XYwhere U is the total utility, X is the quantity of good X that is consumed, and Y is the quantity ofgood Y consumed.It follows thatMUxMUy= X/Ywhere MUx is the marginal utility of X and MUy is the marginal utility of Y.The individual’s budget constraint isI = PxX + PyYwhere I is income, Px is the price of good X, and Py is the price of good Y.Suppose that Px = 2, Py = 1, and I = 100. How much of X and Y does this individual purchase?
Let the following table represents the total utility of a given consumer, in the cardinal utility approach. Q 1 2 3 4 5 Tux 8 14 18 20 20 Tuy 6 10 13 15 16 Mux Muy Mux/px Muy/py D) Assuming the consumer has any amount of money (enough budget) how many of X and Y should the consumer buy, to maximize utility? E) What is the total utility of X and Y? F) Let now price of X is 4 birr per unit and price of Y is 2 birr per unit and budget of the consumer for consumption of X and Y is 20 birr. Given budget constraint how many of X and Y should the consumer buy to maximize utility? G) What are the total utility of X and Y
Lexicographic preferences describe preferences where an economic agent prefers any amount of one good (X) to any amount of another (Y). SpeciÖcally, if o§ered several bundles of goods, the agent will choose the bundle that o§ers the most X, no matter how much Y there is. Only when there is a tie between bundles with regard to the number of units of X will the agent start comparing the number of units of Y across bundles. Show that Lexicographic preferences are transitive
Suppose the linear demand curve for shirtsslopes downward and that consumers buy 500 shirts per yearwhen the price is $30 and 1,000 shirts per year when the priceis $25.a. Compared to the prices of $30 and $25, what can you sayabout the marginal valuation that consumers place on the300th shirt, the 700th shirt, and the 1,200th shirt they mightbuy each year?b. With diminishing marginal utility, are consumers derivingany consumer surplus if the price is $25 per shirt? Explain.c. Use a market demand curve to illustrate the change in consumersurplus if the price drops from $30 to $25.
4. a. Consider a consumer with preferences defined over x and y. Demonstrate that it is possible theywould choose to consume some of both commodities when their income is I but would choose toconsume only x when their income is I’ > I. (Remember: if you can draw it without violating anyof the basic assumptions on preferences, it could happen.) b. Conversely, demonstrate that it is possible they would choose to consume only x when theirincome is I but would choose to consume x and y when their income is I’ > I.c. Finally, show it is possible that they would choose to consume only y when their income is I butwould choose to consume only x when their income is I’ > I.
You are choosing between two goods, X and Y, and your marginal utility from each is shown in the following table. Units of X MUx Units of Y MUy 1 10 1 8 2 8 2 7 3 6 3 6 4 4 4 5 5 3 5 4 6 2 6 3 Instructions: Enter your answers as a whole number. a. If your income is $9 and the prices of X and Y are $2 and $1, respectively, what quantities of each will you purchase to maximize utility? 2 units of X and 5 units of Y b. What total utility will you realize? 48 utils c. Assume that, other things remaining unchanged, the price of X falls to $1. What quantities of X and Y will you now purchase? 3 units of X and 3 units of Y d. Using the two prices and quantities for X, complete the table to derive the demand schedule (a table showing prices and quantities demanded) for X. Instructions: Start with the highest price first Price of X Quantity Demanded of X $ $
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.
please only do: if you can teach explain each partc: what does it mean? can you show graphs: For these to be optimal choices with such preferences, the indifference curve through a must lie entirely on or above the budget line associated with (p, w), and simi- larly for r' for the budget line associated with (p', w'). how do you know this:Because each of these bundles lies below the other budget line, this implies that the indifference curves must cross, which is impossible. can you show graphs: note that (3,1) is a conver combination of x and x', so for conver preferences must be weakly preferred to x (the less preferred bundle between a and a'). But then the bundle (3,5/3) must be strictly preferred z, contradicting that is optimal given the initial budget set
Hella the Greek's preferences can be described by the utility function U(x, y) = (x^1/2 + 3^1//2)^2. (a) What is the indifference curve for a utility of 49? (b) By how much does utility increase when Hella increases consumption of good y by one (small) unit, when initially U = 49 and x = 1? (c) Holding utility constant at 49, if initially x = 1, how many additional (small) units of y does Hella have to consume if her consumption of x drops by 6 (small) units?