Microeconomic Theory: A Mathematical Approach, James M. Henderson and Richard E. Quandt. New York: McGraw-Hill Book Co., Inc., Pp. xii Advanced Search. This content is only available as a PDF. Issue Section: Book Reviews. Microeconomic Theory: A Mathematical Approach, James M. Henderson and Richard E. Quandt. New York: McGraw-Hill Book Co., Inc., Pp. xii, This content is only available as a PDF. Download all figures. 56 Views. 0 Citations. Capet Marcel, "Henderson (James H.), Quandt (Richard E.) - Microeconomic theory: a mathematical approach," Revue Économique, Programme National.
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Mar 17, By Capet Marcel; Henderson (James H.), Quandt (Richard E.) - Microeconomic theory: a mathematical approach. _Quandt]_Microec(etgabentisttus.ga).pdf - Ebook download as PDF File .pdf), Text File .txt) or Henderson and Quandt discuss microeconomics with the help of mathe The basic concepts of microeconomic theory are developed with the aid of. , James Mitchell Henderson, Richard E. Quandt McGraw-Hill, Microeconomic Theory: A. Mathematical Approach file download etgabentisttus.ga STANFORD Microconomic thory
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Related articles in Google Scholar. Citing articles via Google Scholar. Price Discovery in Agricultural Futures Markets: Credit and Land Contracting: A Test of the Theory of Sharecropping. Evidence from Kenya. Robbins was also aware of the attacks of non-orthodox economists on the psychological assumptions of economics Robbins, , p.
However, he emphasized that these assumptions were not a fundamental part of the theory, thus implying that economic theory could be built without these subjective notions Robbins, , pp. His new definition of economics was in the same spirit, stressing that economics is a study of the relationship between ends and means not of the ends and means themselves.
Thus the study of ends is beyond the scope of economics. Robbins, , p. This was the first step towards an appar- ently psychology-free economic science. Furthermore, Robbins realized that the idea of a psychology-free economic science was incompatible with the exist- ence of the concept of economic man which was based on psychological assumptions. The attempt to redefine the concept of utility was another part of the anti- psychologist movement. In a subsequent book, Robbins attempted to expel any hedonistic connotation from the concept.
He pointed out that utility is not utility in the sense of psychological hedonism, but rather a neutral quality of being the object of desire-hedonist or otherwise Robbins, , p. Robbins provided the methodological justification of a value-free, positive economic science, and therefore assisted the development of an economic theory apparently independent of hedonism.
The replacement of the cardinal approach of the margin- alists with the ordinal approach can be seen in the same context. Hicks thought that the subjective, utilitarian-hedonistic approach of marginalist economists did not constitute a legitimate scientific method.
Thus his basic effort was the replacement of marginalist utility theory with an objective, positive theory of choice. As one would expect, the introduction of a neutral theory of choice tended to minimize the hedonistic image of microeconomics. But S. Hicks, then, undertook a purge in order to eliminate all the hedonistic elements from economic theory.
His first target was the very notion of utility. Hicks, , p. He conceived of indifference curves as showing the combi- nation of goods for which the individual is indifferent. They portray preferences and that is why sometimes they are called preference-based indiffer- ence curves. It has to be noticed, however, that the explanation of lower and higher indifference curves again involves the notion of utility level.
The other motive for the construction of his theory was the critique of the hedonistic or utilitarian bias of orthodox economics by heterodox economists. In his most important work, he emphasized the non- hedonistic nature of modern theory in contrast to the marginalist one.
Samuelson, however, was not perfectly happy with the state of modern con- sumer theory. His starting point was that the individual behaves as he behaves.
However, for the construction of RPT the above point is not enough; three axioms are also needed. The transitivity axiom is also known as the weak axiom, which Samuelson-with the assistance of H.
Houthakker-was to substitute with the strong axiom stronger transitivity some years later see Samuelson, , pp. Samuelson constructed the familiar notion of an indifference curve from his RPT, allegedly without using psychological hedonistic terms.
He was concerned to note that his theory did not need the concept of economic man or the notion of utility. After Samuelson, subsequent economists worked on the construction of a general choice theory free from any psychological connotations Arrow and Debreu are the first examples that come t o mind; see, for instance, Arrow and Debreu, , and Debreu, In the neo-classical framework, the behaviour of the consumer is considered rational when it conforms to the axioms of the theory of choice.
Thus, by the s, the vast majority of theorists believed that microeconomic theory in general and consumer theory in particular was a neutral theory without value-laden concepts. The works of a number of economists, but mainly of Pareto, Robbins, Hicks and Samuelson, were responsible for this.
These economists did not invent a new theory but sug- gested that the same principles can be constructed without psychological assumptions.
In my view the success of these attempts to expel hedonism from microeconomic theory is questionable. How- ever, the important point here is that most economists thought that there was a paradigm shift from a hedonistic-based theory to a positive, scientific theory.
The influence, then, of hedonism, at least on the surface, was minimal. This was the case with the new, revolutionary in com- parison to the classicals Keynesian ideas which became part of the accepted body of knowledge and subsequently of textbook material as early as in the beginning of the s see, for instance, Cairncross, , pp. The majority of textbooks use the terms utility and satisfaction or pleasure as synonymous. This happens in elementary textbooks such as McConnell, , p.
For instance, one can read in Henderson and Quandt: If we have a utility function of the form then since the utility function is continuous 1 is satisfied by an infinite number of combinations of 41 and Imagine that the consumer derives a given level of satisfaction Uo from 5 units of 41 and 3 units of Henderson and Quandt, , p. Other widely used intermediate and sometimes advanced texts follow the same approach see, for instance, Koutsoyiannis, , p.
It is clear that these works are much closer to the marginalist line of thought than to developments of more recent decades.
Apart from identifying utility with satisfaction as the marginalists did, many contemporary textbooks explicitly refer to the concept of economic man. Although it is not usually stated explicitly, economic man is often viewed as selfish. Thus, although few authors would explicitly adopt egoistic agents, the stereotype is that the satisfaction of the individual depends on what he consumes and nothing else see also Winter, , p.
The model of economic man is present in D. The attempt by orthodox theorists to reduce all characteristics of economic man into one-rationality-is not very popular among textbook writers, who still conceive him explicitly as a satisfaction maximizer. Exactly the same definition is given by Koutsoyiannis , p. In addition, the analysis of consumer equilibrium that can be found in the majority of texts is conducted in terms of satisfaction or utility levels.
The con- sumer is viewed as having a given budget constraint which is represented by a straight line or a linear equation. The maximum of consumer satisfaction or utility is attained at the point of tendency of the budget line with the highest indifference curve maximization of a utility function.
Many textbooks follow this approach without worrying too much about using concepts which were used by the marginalists years ago see, for instance, Laidler, , p. There are cases, however, where the modern developments are taken into account this is the case among more advanced texts.
However, the term is connected with the notions of satisfaction or utility which it was originally meant to replace. Exactly the same line of explanation is followed by Ferguson and Gould, and Koutsoyiannis, where the concept of constant level of satisfaction is used t o explain compensation Ferguson and Gould, , p. In many cases RPT is mentioned only in a brief footnote see, for instance, Miller, , p. Generally, textbooks connect modern choice theories with marginalist analysis without giving a hint that the purpose of those theories was to free economics from the psychological assumptions of the marginalists.
The decision to use utility analysis or direct preference analysis is simply a matter of convenience. Lancaster, , pp. In addition, some textbooks define economics along the old marginalist lines economics is defined as a calculus of pleasure and pain.
It must be noted that such definitions are rare in textbooks; what this shows, however, is that the explicit influence of hedonism-orientated marginalist economists is still present even in so a basic thing as the definition of economics see for instance, Solmon, , p. The above examples demonstrate that the influence of the hedonistic frame- work on many textbooks is considerable.
Important sections of microeconomic theory are presented in a way that is explicitly based on hedonistic ideas. Thus, although economists such as Pareto, Robbins, Hicks and Samelson thought that they had freed economics from psychological hedonistic influence, con- temporary textbooks indicate otherwise.
At this point it has to be mentioned that Hicks and Samuelson considered their theories revolutionary in com- parison to the old subjective, marginal utility theory. Although it can be maintained that the introduction of Keynesian ideas had a greater significance because it clearly disputed the old theories, one cannot help drawing parallels with the introduction of the allegedly psychology-free theory of choice. In spite of this, the textbooks did not take into account the full significance and change in terminology as they did with the Keynesian ideas.
The first cites pedagogic reasons. In particular, it can be argued that textbook writers attempt to simplify theory in order to make it easier for students.
Thus it is better to use concepts which are easier. Another explanation could be that according to a widespread methodological belief, the validity of the assumptions is not important for the construction of economic theories. Thus, it does not really matter if the assumptions of textbooks are based on hedonistic ideas. In particular, the explicit hedonistic influence on text- books can be seen as a reflection of the criticisms of the alleged neutrality of the modern theories of choice by various theorists such as N.
Georgescu- Roegen, J. Robinson, E. Mishan, W. Kroebel-Rier, B. Loasby, S. I tend to believe that the last explanation is more plausible than the first two. First explanation: pedagogic reasons It can be maintained that simplification purposes are responsible for the presence in textbooks of hedonistic concepts. An initial observation which undermines this explanation is that, as we saw, hedonistic concepts are not used only in introductory textbooks, but also in intermediate ones.
This is a first indication that hedonistic ideas are not used for simplifica- tion purposes. Another argument which supports this is the existence of few elementary textbooks where consumer theory is presented solely in terms of a general choice theory. An example of such a textbook is V. After pre- senting the main points of consumer theory according to Jevons, Marshall and Edgeworth, Walsh goes on to emphasize that this theory has been replaced by a theory of choice.
In contrast to many texts, Walsh discusses indifference curves in terms of preferences not in terms of utility see Walsh, The fact that the text has been a popular student textbook for many years indicates that choice theory can be presented simply enough to be understood by first-year university students.
Further argument against the simplification hypothesis is that the great majority of writers explicitly advocate a positivist economic method- ology. This implies that value-laden terms are not permitted even if they can simplify theories for methodological reasons see, for instance, Lipsey, Marshall was always very careful about utility analysis.
First of all, most parts of his work which can be identified in modern texts were only mentioned in appendices, and he was always anxious to emphasize that they belong to Hedonics but not to economics Marshall, , p. It is clear that he put much less emphasis on utility analysis than other contemporary theorists.
Layard and walters micro theory pdf
Furthermore, as an indication of his unique approach, the Marshallian theory of value, unlike that of other marginal utility theorists, was a combination of marginal utility and costs of production see Loasby, , p.
This again indicates the hedonistic bias of contemporary texts. Although the distorted Marshallian influence hypothesis is relevant to a number of textbooks, it is still not adequate to explain fully the above- mentioned discrepancy. Examples of texts which adopt a historical approach to the discussion of consumer theory provide evidence for this.
Thus textbook writers who are conscious of the Marshallian tradition but also of its differences from choice theory feel the need to explain modern choice theories by refering to hedonistic related concepts.
The case of Koutsoyiannis is indicative: there is a reference to Marshall when cardinal utility theory is discussed and references to Hicks and Samuelson when indifference and revealed preferences are pre- sented. Still, however, indifference and preference are explained by using the concept of utility or satisfaction see Koutsoyiannis, , pp.
More specifically, in his Essays in Posi- tive Economics published in , Friedman suggests that the validity of a theory does not depend on the realism of its assumptions, but on the predic- tions that it yields: also, that it is a positive advantage if the assumptions are unrealistic see Friedman, , pp. One could justify the use of hedonism-orientated assumptions by textbook authors in terms of this meth- odological position. Samuelson, Koopmans, Rotwein , it can be maintained that it had a significant influence on economic methodology see Samuelson, ; Koopmans, , and Rotwein, On the contrary, there are examples of texts which refer to the necessity of realistic assumptions.
One of these texts is that of Stonier and Hague, who state that the ideal would be to device assumptions which come closest to reality Stonier and Hague, , p.
This may indicate that essentially choice theories presuppose the same hedonistic psychology as the marginalist theories did. The above point can be connected with a variety of other criticisms of the modern theory of choice. In particular, N. Georgescu-Roegen, J. Robinson and S. Wong have pointed out the dependence of RPT on an implicitly assumed utility function.
Kroeber-Riel has indicated that RPT still presupposes an economic agent with specific psychological characteristics Kroeber-Riel, , pp. Mishan and B. Finally, L. Boland has demonstrated that one of the foundations of choice theories, the maximization hypothesis, is basically a metaphysical hypothesis Boland, , pp.
The implicit existence of psychological assumptions supports the view that the explanation of the explicit hedonistic influence on textbooks is the implicit presence of psychological hedonism within the very theories of choice. Coconut He will download at least one coconut. It associates certain numbers with various quantities of commodities consumed.
This reformulation of the postulates of the theory of consumer behavior was effected only around the tum of the last century. Intuitively one can see that the. It is shown in Sec. Consider the simple case in which the consumer's downloads are limited to two commodities.
The effect of price and income variations on consumption levels is examined in Sec. His ordinal 1 How much a particular item on the list is liked is irrelevant. If the utility of alternative A is 15 and the utility of B is 45 i.
Henderson (James H.), Quandt (Richard E.) - Microeconomic theory : a mathematical approach
One could visualize the consumer as possessing a list of commodities in decreasing order of desirability. Demand curves are derived in Sec. The much weaker assumption that he possesses a consistent ranking of preferences is sufficient.
The theory is generalized to an arbitrary number of commodities in Sec. The basic tools. His ranking of commodities is expressed mathematically by his utility function. Two alternative but equivalent methods are employed for the determination of the individual consumer's optimum consumption level in Sec.
No account is taken of the possibility 2 of transferring consumption expenditures from one period to another. It is assumed in the remainder of the chapter that such a. A particular level of utility or satisfaction can be derived from many different combmations of Ql and Q2. If he were capable of borrowing.. At the end of the period he repeats his calculations for the next one.
There is no unique time period for which the utility function should be defined. The consumer makes his calculations for only one such period at a time. Otherwise he is dealing with a. Provision can be made for both possibilities without changing the essential points of the analysis see Sec.
The consumer's utility function is not unique see Sec. Indifference Curves. The consumer usually derives utility from variety in his diet end diversification among the commodities he consumes. Different levels of satisfaction are derived from consuming ten portions of ice cream within one hour and within one month. The level of satisfaction that the consumer derives from a particular commodity combination depends upon the length of the period during which he consumes it.
The utility function is defined with reference to consumption during p a specified eriod of time. It is assumed that j q 1. Any intermediate period is satisfactory for the static theory of consumer behavior. Consider t The term "level of satisfaction" should not mislead the reader to think in terms of a cardinal measure of utility.
Since the utility function is continuous. An indifference map is a collection of indifference curves corresponding to different levels of satisfaction. A movement from point A to point B would increase the consumption of both Ql and Q2. The term is relevant only in that a particular level of satisfaction is higher or lower than some other level. The quantities q1 and q2 are measured along the axes of Fig.
One indifference curve passes through every point in the positive quadrant of the q 1q2 plane. Imagine that the consumer derives a given level of satisfactionU0 from 5 units of Ql and 3 units of Q2. Other commodity combinations which yield the consumer the same level of satisfaction can be discovered in a similar manner. Therefore B must correspond to a higher level of satisfaction than A. If his consumption of Ql were decreased from 5 to 4 without an increase in his consumption of Q2.
This justifies the use of the word approximate in the above argument see t Sec. Then the total differential is the equation of the tangent plane to this surface at some f: Clarendon Press.
The negative of the slope. A1 and A3 are on the same indifference curve contrary to assumption. Taking arbitrarily small increments. The slope of an indifference curve. The Rate of Commodity Substitution. Setting d U 0. The points A2 and Aa are also on the same indifference curve.
Since At and A2 are on the same indifference curve. Value and Capital 2d ed. The consumer is not assumed to be aware of the existence of marginal utilities.
A positive value for h signifies that an increase in q1 will increase the consumer's satisfaction level and move him to a higher indifference curve. Method 1. In a cardinal analysis the partial derivatives! It is immaterial whether the verbal definition is in terms of substituting Q l for Q 2 or vice versa. The amount he spends on the first commodity p1q1 plus the amount he spends on the second p2q2 equals his income y0.
In order to maximize the utility function subject to the budget constraint the consumer must find a combination of commodities that satisfies and also maximizes the utility function His problem is one of maximization. The signs as well as the ratios of marginal utilities are meaningful in an ordinal analysis. The Maximization of Utility The rational consumer desirt: He could increase his satisfaction by shifting some of his expenditure from Q2 to Q1.
Setting the first derivative of equal to zero. A-2 and A The ratio of the marginal utilities must equal the ratio of prices for a maximum. If more satisfaction could be gained by spending an additional dollar on Q1 rather than Q2. If maxima exist. Equations and together imply that indifference curves are negatively sloped. Its equation is The second derivative of the utility function is negative for these values of q1 and q2. The price line AB is the geometric counterpart of the budget constraint and shows all possible combinations of Q 1 and Q2 that the consumer can download.
Figure contains a graphic presentation of this example. The consumer can download 50 units of Q 1 if he downloads no Q 2. Expressing q2 as a function of q1 from 40 the budget constraint.
The budget constraint is His equilibrium is at point E. Movements in either direction from point E result in a diminished level of utility. If he spends all his income on one commodity. The constant slope of the price line.
Customers who bought this item also bought
He consumes only one commodity at the optimum. V is identically equal to U for those values of q1 and q2 which satisfy the budget constraint. A-3 on constrained maxima. Imagine that one wishes to compare the satisfaction a consumer derives from one hat and t See Sec. P2q2 where X is the as yet undetermined Lagrange multiplier see Sec. To maximize V.
The second-order condition for a coustrained maximum is that the relevant bordered Hessian determinant be positive: V is a function of q1. Method 2. The Choice of a Utility Index The numbers which the utility function assigns to the alternative commodity combinations need not have cardinal significance.
A-1 on expanding a determinant and Sec. As before. This proves that the first-order conditions are invariant with respect to the particular choice of the utility index The numbers that are assigned to these combinations for the purpose of showing the strength of his preferences are arbitrary in the sense that the difference between them has no meaning.
Thus 3 for the: Since the second batch is preferred to the first batch. Note that the function of a function rule is applied see Sec. The function F U is then a monotonic. The consumer is kilown to prefer the latter to the former combination. P2q2 0 iJ. The marginal utilities for different indices may be quite different. The value of a determinant does not change if a multiple of one row is added to some other row or if a multiple of a column is added to some other column.
It can be concluded that if the consumer maximizes his utility subject to the budget constraint for one given utility index. Now multiply the last column by F' and divide the first two rows by F': If F U is a monotonic transformation. This proves that the second-order condition is invariant with respect to the choice of the utility index.
The consumer's utility function is unique except for a monotonic tran8formation. This leaves A unchanged: It follows from the invariance of the first. Demand Curves The consumer's demand curve for a commodity gives the quantity he will download as a function of its price. Demand curves can be derived from the analysis of utility maximization The first-order conditions for maximization consist of three equations in the three unknowns: Form the expression and set its partial derivatives equal to zero: As above.
Squaring is not a monetonic transformation if negative numbers are admissible. Two important properties of demand functions can be deduced: The budget constraint becomes ky0. Expression 2. Of course. Given the consumer's income and prices of commodities. To prove the second property assume that all prices and income change in the same proportion.
The first property follows from the convexity of the indifference curves: It is equally easy to demonstrate that the second-order conditions are unaffected.
If such proportionate changes leave his behavior unaltered. This i the Pigou effect. Therefore the demand curve for the price-income set kp1. It is generally assumed that demand curvee are negatively sloped: Elsewhere in this volume it is assume that the demand function is negatively sloped. Thet are several alternative definitions of consumer surplus. This proves that the demand functions are homogeneous of degree zero in prices and income.
An example is provided by ostentatious consumption: He may consequently increase his demand for commodities. The nature of price-induced changes in the quantity demandec is analyzed in detail in Sec. A rise in money income is desirable for the consumer. If all prices and the consumer's income are increased in the same proportion. This implies a relevant and empirically testable restriction upon the consumer's behavior. Both income and leisure are desirable.
Income and Leisure If the consumer's income is payment for work performed by him. The rate of substitution of income for leisure is dy Y1. W where T is the total amount of available time.
One can also derive the consumer's demand curve for income from this analysis. By definition. In the preceding sections it is assumed that the consumer derives utility from the commodities he downloads with his income.
The second-order condition is fulfilled: In order to find the magni-. W Wr and setting the derivative equal to zero. Since the offer of work is equivalent to the demand for income. Changes in prices and income will normally alter his expenditure pattern One can infer that the consumer will work 12 hours per day irrespective of the wage level.
Assume that the utility function is of the same form as in previous sections: It is therefore the consumer's offer curve for work and states how much he will work at various wage rates.
Denoting this determinant by D and the cofactor of the element in the first row and the first column by Du. P2 dA X dp2 This is accomplished by total differentiation of Eqs.
The array of coefficients formed by contains the same elements as the bordered Hessian determinant Ceteris paribus. He is thereby forced to stay on the same indifference curve. Imagine now that a price change is accompanied by an income change that. In Fig. Substitution and Income Effects. The consumer may wish to substitute Q1 for Q2 because 1 Ql has become cheaper and 2 the fall in the price of Q1 is equivalent to an increase in 0 the consumer's income.
The movement from R to S is accounted for by the substitution and the movement from S to T by the income effect. These two discrete changes correspond to rather than are the substitution effect and the income effect. The first term on the right-hand side of is the substitution effect. The original price line is AB..
The movement. Mter the change in Pt the price line is represented by AC. If the price of QI rises and the consumer's income is so adjusted that his final equilibrium point is on the same indifference curvE This proves that the sign of the substitution effect is always negative. Expanding the determinant D. The utility gained from the last dollar spent is the marginal utility of income. The income effect is. Expanding Du.
Q l is said to be an inferior good. The Slutsky equation can be derived for the specific utility function assumed in the previous examples. State the budget constraint in the general form y. This may occur if a consumer is sufficiently poor so that a considerable portion of his income is spent on a commodity such as potatoes which he needs for his s11bsistence.
Assume now that the price of potatoes falls. This is a weaker definition in the serure that it does not imply the definition given in the text above. The expression. Jp2 D i: The sign of the substitution effect is unknown in the present case.. From Eqs. This provides a rationale for the definitions. The analysis can be extended to account for the change in the demand for one commodity resulting from a change in the price of some other commodity. Two commodities are substitutes if both can satisfy the same need of the consumer.
The first terms on the right-hand sides of and are the substitution effects for each commodity with respect to a change in the price of the other. These are loose definitions. Q1 and Q2 are independent. Denote the substitution effect when the quantity of the ith commodity is adjusted as a result of a variation in the jth price by S.
If Q 1 and Q 2 are substitutes in the everyday sense and if compensating variations in income keep the consumer on the same indifference curve. Substitutes a. Imagine that the consumer's demand fqr tea increases at the rate of 2 cups of tea per 1-cent increase in the price of coffee.. One can infer from this that his downloads of coffee would increase at the rate of 2 cups of coffee per 1-cent increase in the price of tea.
Coffee and tea are most likely substitutes. This is a remarkable cc. This theorem is easily proved. If there are n commodities. Substituting the values of D. Hence only substitutability can occur in the present two-variable case.
But Su. Generalization to n Variables The foregoing analysis of the consumer is now generalized to the case of n commodities. Hence implies that must be positive. Multiply by P1. The generalizR The partial derivative of V with respect to A is again the budget constraint. The bordered Hessian determinants must alternate in sign: Second-order conditions must be fulfilled in order to ensure that a batch of commodities that satisfies is optimal.
The demand curves for the n commodities can be obtained by solving for the qs. Conditions can be stated alternatively as oq. The total cost of the batch [q1].
The only way in which [q1J can be revealed to be preferred to [q0] is to have the consumer download the combination [q1] in some price situation in which he could also afford to download [q0]. If his behavior conforms to certain simple axioms. In other words. Conse- quently implies the opposite of or IJ-'. The consumer's total expenditures are given by -zpoqo.
If [q0] is revealed to be preferred to [q1]. The Theory of Revealed Preference. It t The two axioms can be collapsed into a single one. Assume that there are n commodities. It was assumed in th. Consider an alternative batch of commodities [q1] that could have been downloadd by the consumer but was not.
Axiom By the same token one could question whether he even possesses an indifference map. Q This inequality asserts that the sum of all quantity changes multiplied by the corresponding price changes is nonpositive if the consumer moves along a given indifference curve.
Foundations of Economic Analysis Cambridge. This implies that the combination [q0] must not be cheaper at the [p1] prices than [q1]: A Revision of Demand Theory Oxford: Qlarendon Preas.
The Substitution Effect. Others are 1 the homogeneity of the demand functions of zeroth degree in prices and incomes Sec. Assume now that only the price of the first commodity changes. Since he is indifferent between [q0] and [q 1] and yet downloads [q0]. Then reduces to 6 1 The proof of this theorem is somewhat difficult and is not reproduced here. The combination [q 1] is downloadd at prices [p1]. See P. It can be proved from revealed-preference theory that the substitution effect is negative.
When prices are given by [p0]. See H. This is a choice with certain outcome. Great controversy has centered around the question of whether the resulting utility index is ordinal or cardinal. His decision will depend upon the chances of winning or losing in this particular lottery.
Let A represent the situation in which the consumer possesses a satisfactory automobile. It will be shown that von Neumann-Mcrgenstern utilities possess at least some cardinal properties. The previous analysis is unrealistic in the sense that it assumes that particular actions on the part of the consumer are followed by particular. B a situation in which he possesses no automobile. If the probability of a loss is very high. The Axioms. The consumer may prefer to retain his income or money with certainty..
If the price increases. Assume that the consumer prefers A to B and B to C. P respectively. Complete-ordering axiom. For the two alternatives A and B one of the following must be true: Assume that a person engages in the following game of chance: Assume now that the consumer is offered a - choice between two lottery tickets. Assume that A is preferred to B and B to C. P respectively and another the outcomes B and C with the same probabilities P and 1 P.
The player can expect to win. Axiom of complexity. The first one. If two lottery tickets. Independence axiom. The axiom asserts that there exists some probability P. If one lottery ticket offers outcomes A and C with probabilities P and 1. Assume that the consumer prefers A to B. Assume that the consumer is indifferent between A and B and that C is any outcome whatever.
He pays his opponent 3 dollars in every other case. The other. Continuity axiom.
These axioms are very general. The consumer's evaluation of alternatives is transitive: The axiom asserts that the consumer is then indifferent between Lt and L2. A and B. If he faces a set of uncertain prospects i. It can be proved that a consumer who conforms to the axioms will maximize expected utility. P Ua He could be asked to reveal the value of P for which he is indifferent between B with certainty and a chance between A and C. These numbers are completely arbitrary.
Construction of the Utility Numbers. The consumer's prospects can be arranged in order of decreasing expected utility or desirability. In the special case in which a prospect has a certain rather than uncertain or dubious outcome.
If he has a fear of gambling. Consider the earlier example in which the outcomes A. Given that these outcomes have the probabilities P and 1. The consumer prefers A to B and B to C. This type of behavior is ruled out by the continuity axiom and the axiom of complexity. Thus the utility numbers associated with various outcomes are an ordinal utility index and provide a correct ranking.
This ca. Since the consumer is an expected-utility maximizer. In order to derive a utility index. P Uc for some P. This can be illustrated with reference to the example used above. The consumer's choice between more complicated alternatives can be predicted on the basis of these utility numbers. Ordinal utility functions have been demonstrated to be unique! The reader may check that the transformation is monotonic. The rational consumer would prefer a The results obtained from the present cardinal utility index might change under some monotonic transformations.
It is no longer tme that any monotonic transformation of a utility index in the present sense can also serve as a utility index. They are derived by presenting him with mutually exclusive choices. Interpersonal comparisons of utility are still impossible.
They are derived from the consumer's risk behavior and are valid for predicting his choices as long as he maximizes expected utility. This follows from the fact that the relative magnitudes of differences between utility numbers are invariant with respect to linear transformations. It can be meaningfully asserted that one object weighs seven times as much as another.
Let the utilities of three alternatives be UA Summary Nineteenth-century economic theorists explained the consumer's behavior on the assumption that utility is measurable.
Utility numbers differ from measures of weight.
Such comparisons do not imply. In the above example Uc This restrictive assumption was abandoned around the turn of the last century. Von Neumann-Morgenstern utilities possess some. Ul In contrast to the traditional theory of the consumer. It is not meaningful to assert that the consumer prefers C " seven times as much" as A. The consumer's demand curve for a commodity can be derived from his first-order conditions for 'utility maximization.
In diagrammatic terms. The ratio of the marginal utilities must equal the price ratio for a maximum. The equilibrium conditions are similar to those which hold for the selection of an optimal commodity combination.
If a particular function describes appropriately the consumer's preferences. Demand curves are single-valued and homogeneous of degree zero in prices and income: If the price of a commodity changes. The effect of a given price change can be analytically decomposed into a substitution effect. This ranking is described mathematically by the consumer's ordinal utility function.
The consumer's utility function is not unique. The consumer's reaction to price and ineome changes can be analyzed in terms of substitution and income effects. Since his income is limited. The theory can be generalized to an arbitrary number of commodities.
Contains a proof of the existence of indi: His indifference curves can be derived. A nonmathematical discussion of the von Naumann-Morgenstern utility index.
The results are obtained by presenting the consumer with hypothetical price-income situations and observing his choices.The consumer makes his calculations for only one such period at a time. A-1 to solve for dq. An isorevenue line is the revenue counterpart of an isocost line and is defined as the locus of output combinations that will earn a specified revenue.
Schwier and A. As mentioned above, the drive towards neutralization meant that only one assumption was necessary for the construction of choice theories: rationality. The entrepreneur has an option not recognized by the calculus. Important sections of microeconomic theory are presented in a way that is explicitly based on hedonistic ideas.
A fall in the price has increased the quantity bought. Second-order conditions require that the marginal productivities of both inputs be decreasing.
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