File Name: microeconomic theory basic principles and extensions solutions .zip
He got a spoon from a drawer by the kitchen sink, took a one-gallon, cardboard tub of chocolate ripple from the freezer, and sat at the table for more than a quarter of an hour, smiling, eating big spoonfuls of ice cream right out of the carton, until at last he was too full to swallow even one more bite. Sooner or later, the bitch would think to have some of her spies put a watch on this place, and then he would be caught. When he finished packing, the big suitcase was extremely heavy, but he had the muscles to handle it. He hated being dirty, for being dirty somehow always made him think of the whispers and the awful crawling things and the dark place in the ground. He decided he could risk taking a quick shower before he carried the food back to the clifftop house, even if that meant being naked and defenseless for a few minutes.
Consequently, no commentary is provided. All of the problems are relatively simple and instructors might choose from among them on the basis of how they wish to approach the teaching of the optimization methods in class. This is a local and global maximum. Lagrangian Method:? This is the volume of a rectangular solid made from a piece of metal which is x by 3x with the defined corner squares removed.
This would require a solution using the Lagrangian method. The optimal solution requires solving three non-linear simultaneous equations—a task not undertaken here.
But it seems clear that the solution would involve a different relationship between t and x than in parts a-c. Set up Lagrangian? Any positive value for x1 reduces y. Because x2 provides a diminishing marginal increment to y whereas x1 does not, all optimal solutions require that, once x2 reaches 5, any extra amounts be devoted entirely to x1. See, for example, Mas Colell et al.
Intuitively, because concave functions lie below any tangent plane, their level curves must also be convex. But the converse is not true. Clearly, all the terms in Equation 2. Using equation 2. All of these results can be shown by applying the various definitions to the partial derivatives of y. The primary focus is on illustrating the notion of a diminishing MRS in various contexts.
The concepts of the budget constraint and utility maximization are not used until the next chapter. Comments on Problems www. All of the functions are monotonic transformations of one another, so this problem illustrates that diminishing MRS is preserved by monotonic transformations, but diminishing marginal utility is not.
The problem also shows how such problems can be treated as a composite commodity. The purpose is to get students to think mathematically about everyday expressions.
Part c provides an introduction to the linear expenditure system. This application is treated in more detail in the Extensions to Chapter 4. Again, the case where the same good is maximum is uninteresting. A fully condimented hot dog. This would be equivalent to a lump-sum reduction in purchasing power. See the extensions to Chapter 3. Any trading opportunities that differ from the MRS at x , y will provide the opportunity to raise utility see figure.
A preference for the initial endowment will require that trading opportunities raise utility substantially. This will be more likely if the trading opportunities and significantly different from the initial MRS see figure. Mathematics follows directly from part a. Relatively simple computational problems mainly based on Cobb—Douglas and CES utility functions are included. Comparative statics exercises are included in a few problems, but for the most part, introduction of this material is delayed until Chapters 5 and 6.
Comments on Problems 4. Part b asks students to compute income compensation for a price rise and may prove difficult for them. As a hint they might be told to find the correct bundle on the original indifference curve first, then compute its www.
Instructors may wish to introduce the expenditure shares interpretation of the function's exponents these are covered extensively in the Extensions to Chapter 4 and in a variety of numerical examples in Chapter 5.
In part b there is a total quantity constraint. Students should be asked to interpret what Lagrangian Multiplier means in this case. The problem might be used to illustrate the notion of perfect complements and the absence of relative price effects for them. The manipulations here are often quite difficult for students, primarily because they do not keep an eye on what the final goal is. Numerical examples are based on the Cobb-Douglas expenditure function. Part c of the problem focuses on how relative expenditure shares are determined with the CES function.
Solutions 4. Consumption of California wine does not change when price of French wine changes. To achieve the part b utility with part a prices, this person will need more income. Indirect utility is This is not a local maximum because the indifference curves do not have a diminishing MRS they are in fact concentric circles.
Hence, we have necessary but not sufficient conditions for a maximum. In fact the calculated allocation is a minimum utility. If Mr. No matter what the relative price are i.
A higher income makes it possible to consume z as part of a utility maximum. That is, the more important x is in the utility function the greater the proportion that expenditures must be increased to compensate for a proportional rise in the price of x. So total subsidy is 5 — one dollar greater than in part c. Raising U to 3 would require extra expenditures of 4. That is, a subsidy of 0. This is a sign of low substitutability. The algebra here is very messy. For a solution see the Sydsaeter, Strom, and Berck reference at the end of Chapter 5.
I I www. Many of the problems are fairly easy so that students can approach the ideas involved in shifting budget constraints in simplified settings. Comments on Problems 5. Illustrates how the goods used in fixed proportions peanut butter and jelly can be treated as a single good in the comparative statics of utility maximization.
This problem shows that Giffen's Paradox cannot occur with homothetic functions. The analysis essentially duplicates Examples 5. In this case, utility is not separable and cross-price effects are important. It shows that more customary elasticities can often be calculated from share elasticities—this is important in empirical work where share elasticities are often used.
It shows how price elasticities are determined only by income effects which in turn depend on income shares. Increases in I shifts demand for x outward.
Then demand for x falls to zero. The income-compensated demand curve for good x is the single x, px point that characterizes current consumption. This would require an increase in income of: 3. Since David N. There is no substitution effect due to the fixed proportion. A change in price results in only an income effect.
As income increases, the ratio p x p y stays constant, and the utility-maximization conditions therefore require that MRS stay constant. Thus, if MRS depends on the ratio y x , this ratio must stay constant as income increases. Therefore, since income is spent only on these two goods, both x and y are proportional to income. With fixed proportions there are no substitution effects. If this person consumes only ham and cheese sandwiches, the price elasticity of demand for those must be In part a, for example, a ten percent increase in the price of ham will increase the price of a sandwich by 5 percent and that will cause quantity demanded to fall by 5 percent.
The sum equals -2 trivially in the Cobb-Douglas case. Result follows directly from part a. A generalization from the multivariable CES function is possible, but the constraints placed on behavior by this function are probably not tenable. Year 2's bundle is also revealed preferred to Year 3's for the same reason.
They are intended to give students some practice with the concepts introduced in Chapter 2, but the problems in themselves offer few economic insights. Consequently, no commentary is provided. Results from some of the analytical problems are used in later chapters, however, and in those cases the student will be directed back to this chapter. Notice that this slope becomes more negative as x increases and y decreases. Next, use the Lagrange method. Note that the solution is the same here as in Problem 2. This is the volume of a rectangular solid made from a piece of metal, which is x by 3x with the defined corner squares removed.
Я думаю, что Стратмор сегодня воспользовался этим переключателем… для работы над файлом, который отвергла программа Сквозь строй. - Ну и. Для того и предназначен этот переключатель, верно. Мидж покачала головой. - Только если файл не заражен вирусом. Бринкерхофф даже подпрыгнул. - Вирус.
- Голос послышался совсем. - Ни за. Ты же меня прихлопнешь. - Я никого не собираюсь убивать.
Какое отношение это имеет к директорскому кабинету. Мидж повернулась на вращающемся стуле. - Такой список выдает только принтер Фонтейна. Ты это отлично знаешь. - Но такие сведения секретны.
Не поддается, сэр? - с трудом произнесла. - А как же принцип Бергофского. О принципе Бергофского Сьюзан узнала еще в самом начале своей карьеры. Это был краеугольный камень метода грубой силы. Именно этим принципом вдохновлялся Стратмор, приступая к созданию ТРАНСТЕКСТА. Он недвусмысленно гласит, что если компьютер переберет достаточное количество ключей, то есть математическая гарантия, что он найдет правильный. Безопасность шифра не в том, что нельзя найти ключ, а в том, что у большинства людей для этого нет ни времени, ни необходимого оборудования.
Тебе пора отправляться домой.
Зная, чем грозит агентству Цифровая крепость, не мог же он участвовать в заговоре по ее созданию. И все же Сьюзан понимала, что остановить Хейла могут только его представления о чести и честности. Она вспомнила об алгоритме Попрыгунчик. Один раз Грег Хейл уже разрушил планы АНБ. Что мешает ему сделать это еще .
Наверное, стоит выключить ТРАНСТЕКСТ, - предложила Сьюзан. - Потом мы запустим его снова, а Филу скажем, что ему все это приснилось. Стратмор задумался над ее словами, затем покачал головой: - Пока не стоит. ТРАНСТЕКСТ работает пятнадцать часов. Пусть пройдут все двадцать четыре часа - просто чтобы убедиться окончательно.
Готово! - крикнула Соши. Все посмотрели на вновь организованный текст, выстроенный в горизонтальную линию. - По-прежнему чепуха, - с отвращением скривился Джабба.
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