Marshall's demand function

The Marshallian demand function (also Walrasian demand function ), named after the economist Alfred Marshall (or Léon Walras ), is a mathematical function in microeconomics and especially in household theory that indicates the amount of goods for a given price of goods and a given income every single good should be consumed if one wants to realize the greatest possible benefit .

Fig. 1. Example of a Marshallian demand function in the two-goods case: The quantity of good 1 is plotted on the horizontal axis, the price of good 1 on the vertical axis. The price of the not considered good 2 is kept constant in the diagram, as is your own budget . ${\displaystyle p_{2}}$ ${\displaystyle y}$

The starting point of the considerations that lead to the Marshallian demand function is the principle of utility maximization: a consumer (typically a household) independently decides on the allocation of his wealth to the consumption of different goods that are offered at certain prices. Depending on how he divides his assets, his spending plan differs. The basic idea of ​​Marshall's demand is that the consumer always chooses exactly the spending plan that he prefers to all other affordable spending plans. The Marshallian demand describes exactly this - optimal - spending plan by specifying how much of each individual good that is to be consumed under this. Because it is a function, the Marshallian demand describes this spending plan not only for any particular amount of property and any particular price of goods, but for all possible amounts of property and prices of goods.

The concept of the Marshallian demand function can be generalized. More generally, one speaks of a Marshallian inquiry correspondence (also Walrasian inquiry correspondence ). The mathematical concept of the function is exchanged for that of a correspondence , which makes it possible that a consumer with a certain wealth and at certain goods prices in economics can sometimes have not just one, but several optimal spending plans.

Non-technical introduction

Idea of ​​utility function

→ Main article : utility function (microeconomics)  and preference relation

There are various ways of modeling consumer demand for a good. Which one is appropriate depends on the assumption made about the making of the consumption decision. One could assume, for example, that consumers would randomly choose any combination of bundles of goods, regardless of how much the goods in question are worth to them; or one could imagine that a social planner would take all the assets of the consumers and assign them certain shopping carts according to their own criteria. The basic idea of ​​modern utility theory, however, is that consumers make their decision about the consumption of the amount of a certain good based on their preferences . Consumers have individual orders of preference ; Such a preference order includes the information across all possible combinations of all goods as to whether one bundle of goods is perceived as at least as desirable, as equally desirable or at most as desirable as the other bundle of goods (an example of a bundle of goods would be about "1 apple, 1 banana, 0 oranges and 2 mangos "and the individual preference order may contain the information how the bundle of goods" 2 apples, 0 bananas, 1 orange and 2 mangos "relates to the consumer under consideration).

A simpler way to express this information is to look at a simple function instead of complex orders. Under certain conditions, a utility function can be constructed that outputs any number for a given bundle of goods. This number is meaningless in itself; their significance only emerges from a comparison with the utility values ​​of other bundles of goods. This is namely obvious which bundle of goods, the consumer prefers: Comparing any two bundle of goods, then the utility value of a bundle if and strictly greater than that of a bundle if the consumer whose utility function we consider the bundle over preferred. ${\displaystyle \mathrm {A} }$ ${\displaystyle \mathrm {B} }$ ${\displaystyle \mathrm {A} }$ ${\displaystyle \mathrm {B} }$

Marshall's demand

Marshall's demand connects this thought with a related one: A sensible consumer will consume a “preferred” bundle of goods, which, with the above consideration, is tantamount to providing him with the greatest possible benefit. However, it cannot be consumed on an unlimited scale. Every consumer is subject to a so-called budget restriction , which means that he cannot consume any bundles of goods that he could not afford at the prevailing goods prices. Among those bundles of goods that he can afford, he then chooses, as mentioned, precisely that which gives him the greatest benefit. Imagine now that there are only two goods that we want to call “good 1” and “good 2” as simply as possible and that are available at prices or on the market. Then the following problem describes the consumer's utility maximization problem: ${\displaystyle p_{1}}$ ${\displaystyle p_{2}}$

${\displaystyle \max \;u(x_{1},x_{2})}$     under the constraints          and     ${\displaystyle p_{1}x_{1}+p_{2}x_{2}\leq y}$ ${\displaystyle x_{1},x_{2}\geq 0}$

with the available wealth , the demanded amount of good 1 or 2 and the utility function of the consumer. To make the problem more manageable, one assumes that the utility function is continuous . This ensures that if there is a slight change in the quantity of one or more goods in a bundle of goods, there is no sudden jump in the resulting benefit. One remark seems appropriate: because prices and income are variables in the above maximization problem, the solution to the problem will not be a concrete bundle of goods; Which goods bundle maximizes the term depends specifically on the exact goods prices and the available wealth, so that the solution will depend on these variables (the prices and the available wealth). ${\displaystyle y}$ ${\displaystyle x_{1,2}}$ ${\displaystyle u(x_{1},x_{2})}$

The optimal demand for good is 1 and it is dependent on the price of this good, the income available to the individual and the price of good 2. The latter can be seen intuitively, for example, from the fact that the utility-maximizing demand for cars is certainly also depends on whether a train ticket costs EUR 500 or EUR 5 (this does not rule out that the price may be independent of this in individual cases). Consequently, the optimization problem yields optimal values ​​for the two goods: (the Marshallian demand for good 1) and analogously (the Marshallian demand for good 2). ${\displaystyle x_{1}^{*}}$ ${\displaystyle y>0}$ ${\displaystyle x_{1}^{m}(p_{1},p_{2},y)}$ ${\displaystyle x_{2}^{m}(p_{1},p_{2},y)}$

Formal definition

Denote by demanded by a certain amount of consumer goods , and summarize the vector demand respect of all goods together. The price of every good is strictly positive, for everyone , and one can agree as the price vector of the economy. ${\displaystyle x_{i}\geq 0}$ ${\displaystyle i}$ ${\displaystyle i=1,\ldots ,n}$ ${\displaystyle \mathbf {x} =(x_{1},\ldots ,x_{n})\in \mathbb {\mathbb {R} } _{+}^{n}}$ ${\displaystyle p_{i}>0}$ ${\displaystyle i=1,\ldots ,n}$ ${\displaystyle \mathbf {p} =(p_{1},\ldots ,p_{n})\in \mathbb {R} _{++}^{n}}$

The consumer's benefit follows a continuous benefit function . The consumer has a budget of . Now consider the consumer's utility maximization problem, taking into account the budget restriction : ${\displaystyle u(\mathbf {x} )}$ ${\displaystyle y>0}$

${\displaystyle \max _{\mathbf {x} \in \mathbb {R} _{+}^{n}}u(\mathbf {x} )}$    under the secondary condition     ${\displaystyle \mathbf {p} \cdot \mathbf {x} =p_{1}x_{1}+p_{2}x_{2}+\ldots +p_{n}x_{n}\leq y}$

Definition: Be steadily , and . One denotes the correspondence defined by ${\displaystyle u(\mathbf {x} )}$ ${\displaystyle \mathbf {x} \in \mathbb {\mathbb {R} } _{+}^{n}}$ ${\displaystyle \mathbf {p} \in \mathbb {\mathbb {R} } _{++}^{n}}$ ${\displaystyle y>0}$ ${\displaystyle \mathbf {x} ^{m}:\mathbb {R} _{++}^{n+1}\twoheadrightarrow \mathbb {R} _{+}^{n}}$

${\displaystyle \mathbf {x} ^{m}(\mathbf {p} ,y)\equiv \arg \max\{u(\mathbf {x} )|\mathbf {p} \cdot \mathbf {x} \leq y\}}$,

as Marshallian demand correspondence (also Walrasian demand correspondence ).

If the maximization problem has a one-element solution set for each tuple ( i.e. a unique solution), the assignment is called a Marshallian demand function (also known as a Walrasian demand function ). ${\displaystyle (\mathbf {p} ,y)}$ ${\displaystyle \mathbf {x} ^{m}(\mathbf {p} ,y)}$

A correspondence is a set valued function . While a function in the narrower sense assigns a single element from the target set to each element from the definition area (in this case the set of goods bundles), a correspondence assigns each element from the definition area a subset of the target set. The Marshallian demand function can thus be understood as a special case of demand correspondence , in which every tuple is assigned an exactly one-element subset of the target set. ${\displaystyle (\mathbf {p} ,y)}$

Other spellings for the definition of the Marshallian demand correspondence are also used. It's trivial about

${\displaystyle {\begin{aligned}\mathbf {x} ^{m}(\mathbf {p} ,y)&\equiv \arg \max\{u(\mathbf {x} )|\mathbf {p} \cdot \mathbf {x} \leq y\}\\&=\left\{\mathbf {x} \in B_{\mathbf {p} ,y}\left|\left(\forall \mathbf {x} '\in \mathbb {R} _{+}^{n}\right):u(\mathbf {x} ')>u(\mathbf {x} )\Rightarrow \mathbf {p} \cdot \mathbf {x} '>y\right.\right\}\end{aligned}}}$

With

${\displaystyle B_{\mathbf {p} ,y}\equiv \{\mathbf {x} \in \mathbb {R} _{+}^{n}\left|\mathbf {p} \cdot \mathbf {x} \leq y\right.\}}$

the permitted amount (budget amount). In words: The Marshallian demand for a given price system and a given household wealth corresponds exactly to the amount of those permissible bundles of goods which have the property that all bundles of goods with strictly greater use would be so expensive that their consumption would violate the budget restriction.

General properties

Existence and compactness

The Marshallian demand correspondence is not empty and has a compact value.

To see that the demand correspondence is not empty, it suffices to show that the budget set is compact . Because according to Weierstrass's extreme value theorem , a continuous function over a compact set always has a minimum and a maximum value, that is, the above utility maximization problem has at least one solution for all . As a subset of the , the (non-empty) budget set is compact if and only if it is bounded and closed ( Heine-Borel theorem ). That is the case: it is limited because, given the strict positivity of prices, it is always and at the same time for all and for all ; and it is closed because it is defined by weak inequalities. Both properties also follow directly from Berge's maximum theorem, which will be discussed in more detail below under "Continuity properties". ${\displaystyle B_{\mathbf {p} ,y}}$ ${\displaystyle (\mathbf {p} ,y)}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle x_{i}\leq y/p_{i}}$ ${\displaystyle x_{i}\geq 0}$ ${\displaystyle i=1,\ldots ,n}$ ${\displaystyle \mathbf {x} \in B_{\mathbf {p} ,y}}$

Convexity and performance

1. Let the utility function be quasi-concave . Then the Marshallian demand correspondence is convex-valued.
2. Let the utility function be strictly quasi-concave . Then the Marshallian demand correspondence is one element for all , in other words: it is a function . ${\displaystyle (\mathbf {p} ,y)\in \mathbb {R} _{++}^{n+1}}$ ${\displaystyle \mathbf {x} ^{m}(\mathbf {p} ,y)}$

Regarding these two properties, it should be noted that the order of preference underlying a quasi-concave utility function is convex; to (2.) that the order of preference underlying a strictly quasi-concave utility function is strictly convex. Note that for (1.) and (2.) it is not sufficient to presuppose the convexity (or strict convexity) of the preference order. Conversely, the (strict) convexity of implies that every representative utility function is (strictly) quasi-concave. However, there is not a real-valued representation for every (strictly) convex order of preference. For example, to take up the famous example of Debreu (1959), lexicographical preference orders are strictly convex, but cannot be represented by a utility function. However, it is possible to introduce the concepts introduced here on the basis of orders of preference, so that a representation function is no longer important. The proof of (1) is based on the consideration of two bundle of goods , . From the definition of Marshall's demand it follows that . Designate this level of utility with . For a quasi-concave utility function, by definition, it also applies to all . Moreover , because and according to the definition of Marshall's demand. Hence is . From this and with it it finally follows that . So is convex. To (2): (proof by contradiction :) Consider again two bundle of goods , . Again, by definition . Strict quasi-concavity implies for everyone - a contradiction. ${\displaystyle u}$ ${\displaystyle \succsim }$ ${\displaystyle u}$ ${\displaystyle \succsim }$ ${\displaystyle u}$
${\displaystyle \mathbf {x} ,\mathbf {x} '\in \mathbf {x} ^{m}(\mathbf {p} ,y)}$ ${\displaystyle \mathbf {x} \neq \mathbf {x} '}$ ${\displaystyle u(\mathbf {x} )=u(\mathbf {x} ')}$ ${\displaystyle u_{0}}$ ${\displaystyle \mathbf {x} ''\equiv \alpha \mathbf {x} +(1-\alpha )\mathbf {x} '}$ ${\displaystyle u(\mathbf {x} '')\geq u_{0}}$ ${\displaystyle \alpha \in [0,1]}$ ${\displaystyle \mathbf {p} \cdot \mathbf {x} ''\leq y}$ ${\displaystyle \mathbf {p} \cdot \mathbf {x} \leq y}$ ${\displaystyle \mathbf {p} \cdot \mathbf {x} '\leq y}$ ${\displaystyle \mathbf {x} ''\in B_{\mathbf {p} ,y}}$ ${\displaystyle u(\mathbf {x} '')\geq u_{0}}$ ${\displaystyle \mathbf {x} ''\in \mathbf {x} ^{m}(\mathbf {p} ,y)}$ ${\displaystyle \mathbf {x} ^{m}(\mathbf {p} ,y)}$
${\displaystyle \mathbf {x} ,\mathbf {x} '\in \mathbf {x} ^{m}(\mathbf {p} ,y)}$ ${\displaystyle \mathbf {x} \neq \mathbf {x} '}$ ${\displaystyle u(\mathbf {x} )=u(\mathbf {x} ')}$ ${\displaystyle u(\alpha \mathbf {x} +(1-\alpha )\mathbf {x} ')>u(\mathbf {x} ')}$ ${\displaystyle \alpha \in (0,1)}$

homogeneity

The Marshallian demand correspondence is homogeneous of degree zero in , that is, for all and for all . ${\displaystyle (\mathbf {p} ,y)}$ ${\displaystyle \mathbf {x} ^{m}(\alpha \mathbf {p} ,\alpha y)=\mathbf {x} ^{m}(\mathbf {p} ,y)}$ ${\displaystyle (\mathbf {p} ,y)\in \mathbb {R} _{++}^{n+1}}$ ${\displaystyle \alpha >0}$

It therefore makes no difference for the consumption decision if both wealth and all goods prices rise or fall by the same factor. This also rules out the fact that the currency in which assets and prices are invoiced does not matter. The property follows because , that is, the budget amount remains the same when modified by . Of course, the solution to the maximization problem remains unaffected by the simultaneous change in assets and prices. ${\displaystyle B_{\alpha \mathbf {p} ,\alpha y}=\{\mathbf {x} \in \mathbb {R} _{+}^{n}|\alpha \mathbf {p} \cdot \mathbf {x} \leq \alpha y\}=\{\mathbf {x} \in \mathbb {R} _{+}^{n}|\mathbf {p} \cdot \mathbf {x} \leq y\}=B_{\mathbf {p} ,y}}$ ${\displaystyle \alpha >0}$

Continuity properties

1. The Marshallian inquiry correspondence is uppermost.
2. If the Marshallian demand correspondence is one-element for all and consequently a function, then this is continuous . ${\displaystyle (\mathbf {p} ,y)\in \mathbb {R} _{++}^{n+1}}$

The properties follow directly from the maximum sentence (theorem of Berge), for which reference is made to a footnote. The central prerequisite for its applicability is the continuity of the budget correspondence given, whereby a correspondence is designated as continuous if it is both upper and lower (for definition see footnote). These two properties, in turn, can be shown one after the other for budget correspondence. ${\displaystyle (\mathbf {p} ,y)\rightarrow \!\!\!\!\!\!\!\mapsto B(\mathbf {p} ,y)}$

Isolation properties

The Marshallian demand correspondence is closed and also has a closed graph.

In principle, it would be sufficient to show the closed value, because every top-level and closed-value correspondence also has a closed graph. The final value results (as already outlined above) from Berge's theorem (see footnote).

In the following, a “direct” proof for the existence of a closed graph is outlined. Consider a sequence im with the limit value and a sequence im with the limit value . Be further for all . To show: . According to the definition of the Marshallian demand is for all and because of for all therefore also in the limit value . So is . (Proof by contradiction :) Suppose that . Then, by definition , there would be a with which . So there would be a suitable environment around as well as a suitable environment around that for everyone . And because of that there would also be a with (steadiness and strict positivity of prices). It then follows that for sufficiently large , so that . At the same time it follows that for sufficiently large . In summary: for sufficiently large . But that contradicts the assumption that . So is what was to be shown. ${\displaystyle \left\{(\mathbf {p} _{k},y_{k})\right\}_{k\in \mathbb {N} }}$ ${\displaystyle \mathbb {R} _{++}^{n+1}}$ ${\displaystyle (\mathbf {p} ,y)\in \mathbb {R} _{++}^{n+1}}$ ${\displaystyle \left\{(\mathbf {x} _{k})\right\}_{k\in \mathbb {N} }}$ ${\displaystyle \mathbb {R} _{+}^{n}}$ ${\displaystyle \mathbf {x} \in \mathbb {R} _{+}^{n}}$ ${\displaystyle \mathbf {x} _{k}\in \mathbf {x} ^{m}(\mathbf {p} _{k},y_{k})}$ ${\displaystyle k\in \mathbb {N} }$ ${\displaystyle \mathbf {x} \in \mathbf {x} ^{m}(\mathbf {p} ,y)}$
${\displaystyle \mathbf {p} _{k}\cdot \mathbf {x} _{k}\leq y_{k}}$ ${\displaystyle k\in \mathbb {N} }$ ${\displaystyle \mathbf {x} _{k}\in \mathbb {R} _{+}^{n}}$ ${\displaystyle k\in \mathbb {N} }$ ${\displaystyle \mathbf {p} \cdot \mathbf {x} \leq y}$ ${\displaystyle \mathbf {x} \in B_{\mathbf {p} ,y}}$
${\displaystyle \mathbf {x} \notin \mathbf {x} ^{m}(\mathbf {p} ,y)}$ ${\displaystyle \mathbf {x} ^{\text{alt}}\in B_{\mathbf {p} ,y}}$ ${\displaystyle u(\mathbf {x} ^{\text{alt}})>u(\mathbf {x} )}$ ${\displaystyle U_{0}}$ ${\displaystyle \mathbf {x} }$ ${\displaystyle U_{\text{alt}}}$ ${\displaystyle \mathbf {x} ^{\text{alt}}}$ ${\displaystyle u(\mathbf {x} '')>u(\mathbf {x} ')}$ ${\displaystyle (\mathbf {x} ',\mathbf {x} '')\in U_{0}\times U_{\text{alt}}}$ ${\displaystyle \mathbf {x} ^{\text{alt}}\in B_{\mathbf {p} ,y}\Rightarrow \mathbf {p} \cdot \mathbf {x} ^{\text{alt}}\leq y}$ ${\displaystyle \mathbf {x} ''\in U_{\text{alt}}}$ ${\displaystyle \mathbf {p} \cdot \mathbf {x} ''<y}$ ${\displaystyle \left\{(\mathbf {p} _{k},y_{k})\right\}_{k\in \mathbb {N} }\rightarrow (\mathbf {p} ,y)}$ ${\displaystyle \mathbf {p} _{k}\cdot \mathbf {x} ''\leq y_{k}}$ ${\displaystyle k\in \mathbb {N} }$ ${\displaystyle \mathbf {x} ''\in B_{\mathbf {p} _{k},y_{k}}\cap U_{\text{alt}}}$ ${\displaystyle \left\{(\mathbf {x} _{k})\right\}_{k\in \mathbb {N} }\rightarrow \mathbf {x} }$ ${\displaystyle \mathbf {x} _{k}\in U_{0}}$ ${\displaystyle k\in \mathbb {N} }$ ${\displaystyle u(\mathbf {x} '')>u(\mathbf {x} _{k})}$ ${\displaystyle k\in \mathbb {N} }$ ${\displaystyle \mathbf {x} _{k}\in \mathbf {x} ^{m}(\mathbf {p} _{k},y_{k})}$ ${\displaystyle \mathbf {x} \in \mathbf {x} ^{m}(\mathbf {p} ,y)}$

Walras Law

Let the order of preference underlying the utility function not be saturated locally. Then the Marshallian demand satisfies the Walras law , that is, it holds . ${\displaystyle \mathbf {p} \cdot \mathbf {x} ^{m}=y}$

The property of local unsaturation is a common requirement that is placed on preference orders. To put it bluntly, it means that each bundle of goods can always be modified minimally in such a way that the resulting bundle of goods is strictly preferred over the original bundle. For the formal definition, reference is made to a footnote.

(Proof by contradiction :) If indeed for any , then it follows from the unsaturation requirement that there must be another bundle of goods in the vicinity , with which also and at the same time . But then there can be no solution to the utility maximization problem, contrary to the assumption. ${\displaystyle \mathbf {p} \cdot \mathbf {x} '<y}$ ${\displaystyle \mathbf {x} '\in \mathbf {x} ^{m}(\mathbf {p} ,y)}$ ${\displaystyle \mathbf {x} '}$ ${\displaystyle \mathbf {x} ''\in \mathbf {x} ^{m}(\mathbf {p} ,y)}$ ${\displaystyle \mathbf {p} \cdot \mathbf {x} ''<y}$ ${\displaystyle u(\mathbf {x} '')>u(\mathbf {x} ')}$ ${\displaystyle \mathbf {x} '}$

Local unsaturation is obviously a weaker requirement for the order of preference than strict monotony . Because every strictly monotonically increasing utility function is based on a strictly monotonous order of preferences, the above requirement for the validity of Walras' law is thus trivially fulfilled for a strictly monotonically increasing utility function.

Analytical determination

Necessary and sufficient optimality conditions

Assuming that the utility function is continuously differentiable , the Karush-Kuhn-Tucker method (KKT method) provides the necessary conditions for the above utility maximization problem. Designate

${\displaystyle {\mathcal {L}}(\mathbf {x} )=u(\mathbf {x} )+\lambda (y-\mathbf {p\cdot x} )}$.

as a long-term function of the utility maximization problem.

KKT theorem applied to the utility maximization problem:

1. Be continuously differentiable . Then the following applies: If there is a feasible solution to the utility maximization problem, then there is necessarily also one such that the following conditions (KKT-conditions) are fulfilled: ${\displaystyle u}$ ${\displaystyle \mathbf {x} ^{*}}$ ${\displaystyle \lambda ^{*}\geq 0}$

i) ${\displaystyle \mathbf {p} \cdot \mathbf {x} ^{*}\leq y}$

ii) for all (filled with equality whenever ) ${\displaystyle \left.{\frac {\partial u(\mathbf {x} )}{\partial x_{i}}}\right|_{\mathbf {x} ^{*}}\leq \lambda p_{i}}$ ${\displaystyle i=1,\ldots ,n}$ ${\displaystyle x_{i}^{*}>0}$

iii) ${\displaystyle \lambda ^{*}(\mathbf {p} \cdot \mathbf {x} ^{*}-y)=0}$

2. Be continuously differentiable, quasi-concave and be the gradient for all . Then: Satisfy and the conditions (1) (i) - (iii), then is a solution to the utility maximization problem. ${\displaystyle u}$ ${\displaystyle \nabla u(\mathbf {x} )\neq \mathbf {0} }$ ${\displaystyle \mathbf {x} \in \mathbb {R} _{+}^{n}}$ ${\displaystyle \mathbf {x} ^{*}\in \mathbb {R} _{+}^{n}}$ ${\displaystyle \lambda ^{*}\geq 0}$ ${\displaystyle \mathbf {x} ^{*}}$

3. Be continuously differentiable and concave. Then: Satisfy and the conditions (1) (i) - (iii), then is a solution to the utility maximization problem. ${\displaystyle u}$ ${\displaystyle \mathbf {x} ^{*}\in \mathbb {R} _{+}^{n}}$ ${\displaystyle \lambda ^{*}\geq 0}$ ${\displaystyle \mathbf {x} ^{*}}$

Remarks:

• If one compares (1) with the common formulation of a non-linear program, it is noticeable that no so-called constrained qualification (often categorized under the term “regularity condition” in German usage) is required. The reason is that this is always fulfilled in the utility maximization problem. If we convert the complete utility maximization problem into standard form, it reads under the constraints and for . So all constraints are linear. Thus, using a common corollary of the KKT theorem, the requirements for the applicability of the KKT conditions (1) (i) - (iii) are met. ${\displaystyle \max u(\mathbf {x} )}$ ${\displaystyle n+1}$ ${\displaystyle g_{1}(\mathbf {x} )=\mathbf {p} \cdot \mathbf {x} -y\leq 0}$ ${\displaystyle g_{1+j}(\mathbf {x} )=-x_{j}\leq 0}$ ${\displaystyle j=1,\ldots ,n}$
• It has already been shown above that the Marshallian demand is not empty (see the section General Properties ). Thus, there always exists one that satisfies KKT conditions (1) (i) - (iii). ${\displaystyle \mathbf {x} ^{*}}$
• The gradient condition under (2) has a very low threshold; all that is required is that some good provides a strictly positive marginal utility.

Interpretation of the optimality conditions

Fig. 2. Maximizing utility in the two-goods case, inner solution. The red-colored area is the budget amount that is limited by the budget line. All the quantity combinations that meet the budget restriction with equality are located on this . ${\displaystyle p_{1}x_{1}+p_{2}x_{2}\leq y}$

Fig. 3. Maximizing utility in the two-goods case, marginal solution.

Fig. 4. The construction of the Marshallian demand functions for fixed income in the two-goods case.

Inner solution

If there is an inner optimum, i.e. for all , the first-order optimality condition applies in this according to (1) (ii) ${\displaystyle x_{i}^{*}>0}$ ${\displaystyle i=1,\ldots ,n}$

${\displaystyle \left.{\frac {\partial u(\mathbf {x} )}{\partial x_{i}}}\right|_{\mathbf {x} ^{*}}=\lambda p_{i}}$for everyone . ${\displaystyle i=1,\ldots ,n}$

If one considers the case (two-goods case), then this implies ${\displaystyle n=2}$

${\displaystyle \left.{\frac {\partial u(x_{1},x_{2})/\partial x_{1}}{\partial u(x_{1},x_{2})/\partial x_{2}}}\right|_{(x_{1}^{*},x_{2}^{*})}={\frac {p_{1}}{p_{2}}}}$.

The left side of this equation is the marginal rate of substitution (MRS) of good 1 with respect to good 2 (on an indifference curve ), the right side is the price ratio of the two goods. Fig. 2 illustrates this condition: The Marshallian demand for a given price of goods and a given income corresponds exactly to the bundle of goods on which the highest possible indifference curve (here:) still touches the budget line. At this tangential point the slope of the indifference curve - i.e. the negative limit rate of the substitution of good 1 with respect to good 2 - corresponds exactly to the slope of the budget line, which is. If this condition did not apply, the consumer could be better off by changing his consumption marginally. Would be for example ${\displaystyle (p_{1},p_{2})}$ ${\displaystyle y}$ ${\displaystyle (x_{1}^{0},x_{2}^{0})}$ ${\displaystyle I^{1}}$ ${\displaystyle -p_{1}/p_{2}}$

${\displaystyle \left.{\frac {\partial u(x_{1},x_{2})/\partial x_{1}}{\partial u(x_{1},x_{2})/\partial x_{2}}}\right|_{(x_{1}^{0},x_{2}^{0})}>{\frac {p_{1}}{p_{2}}}}$,

then it would be possible within the budget constraint to increase the consumption of good 1 by and at the same time to reduce the consumption of good 2 by . This would turn the benefit around ${\displaystyle \mathrm {d} x_{1}}$ ${\displaystyle (p_{1}/p_{2})\mathrm {d} x_{1}}$

${\displaystyle \left.{\frac {\partial u(x_{1},x_{2})}{\partial x_{1}}}\right|_{(x_{1}^{0},x_{2}^{0})}\mathrm {d} x_{1}-\left.{\frac {\partial u(x_{1},x_{2})}{\partial x_{2}}}\right|_{(x_{1}^{0},x_{2}^{0})}\cdot {\frac {p_{1}}{p_{2}}}\mathrm {d} x_{1}>0}$

enlarge. Then, however, the bundle of goods originally considered cannot have been utility-maximizing.

Edge solution

As illustrated in Fig. 3 for the two-goods case, the optimum can also be a marginal solution; here you are at the "edge" of the budget amount, in the example at this point . As a rule, the above equality condition does not apply there, as can be seen from the necessary conditions (1) (ii). In fact, this is also shown in Fig. 3 : In the optimal point found, the following applies ${\displaystyle x_{2}^{0}=0}$

${\displaystyle \left.{\frac {\partial u(x_{1},x_{2})/\partial x_{1}}{\partial u(x_{1},x_{2})/\partial x_{2}}}\right|_{(x_{1}^{*},x_{2}^{*})}>{\frac {p_{1}}{p_{2}}}}$.

In a marginal solution, this is possible because the consumer is no longer able to reduce his consumption of good 2 in order to use the assets freed up on good 1.

construction

Fig. 4 illustrates the graphical construction of the Marshallian demand in the two-goods case and under the assumption that there is an inner solution to the utility maximization problem. To make the problem graphically manageable, one first fixes and . Then you go through the effects on demand that result from different prices for good 1. In the example, a price reduction from on is considered. This initially changes the slope of the budget line so that a new, optimal bundle of goods results. This can then be transferred to the diagram below at the changed price. If we introduce this for all kinds of prices, yields the Marshallian demand function (for fixed and ) . ${\displaystyle p_{2}}$ ${\displaystyle y}$ ${\displaystyle p_{1}^{0}}$ ${\displaystyle p_{1}'}$ ${\displaystyle p_{2}}$ ${\displaystyle y}$ ${\displaystyle x_{1}^{m}(p_{1},{\overline {p_{2}}},{\overline {y}})}$

Example in the two-goods case

Be . Consider a market for apples (good 1) and bananas (good 2), the quantities of which are denoted by or . Let the price of an apple be that of a banana . The budget of the household is and he only consumes apples and bananas. The utility of the household follows a Cobb-Douglas utility function . The utility maximization problem is ${\displaystyle n=2}$ ${\displaystyle x_{1}}$ ${\displaystyle x_{2}}$ ${\displaystyle p_{1}=2}$ ${\displaystyle p_{2}=4}$ ${\displaystyle y=40}$ ${\displaystyle u(x_{1},x_{2})=x_{1}^{0{,}4}\cdot x_{2}^{0{,}6}}$

${\displaystyle \max _{(x_{1},x_{2})\in \mathbb {R} _{+}^{2}}u(x_{1},x_{2})}$under the secondary condition . ${\displaystyle p_{1}x_{1}+p_{2}x_{2}\leq y}$

So the Lagrangian is

${\displaystyle {\mathcal {L}}(x_{1},x_{2})=u(x_{1},x_{2})-\lambda (p_{1}x_{1}+p_{2}x_{2}-y)}$.

Necessary conditions for the optimal utility are (see the section "Necessary and sufficient optimal conditions"):

1. ${\displaystyle p_{1}x_{1}+p_{2}x_{2}\leq y}$
2. ${\displaystyle {\frac {\partial {\mathcal {L}}(x_{1},x_{2})}{\partial x_{1}}}={\frac {\partial u(x_{1},x_{2})}{\partial x_{1}}}-\lambda p_{1}=0{,}4x_{1}^{-0{,}6}x_{2}^{0{,}6}-p_{1}\lambda \leq 0}$(with equality if ) ${\displaystyle x_{1}>0}$
3. ${\displaystyle {\frac {\partial {\mathcal {L}}(x_{1},x_{2})}{\partial x_{2}}}={\frac {\partial u(x_{1},x_{2})}{\partial x_{2}}}-\lambda p_{2}=0{,}6x_{1}^{0{,}4}x_{2}^{-0{,}4}-p_{2}\lambda \leq 0}$(with equality if ) ${\displaystyle x_{2}>0}$
4. ${\displaystyle \lambda (p_{1}x_{1}+p_{2}x_{2}-y)=0}$and . ${\displaystyle \lambda \geq 0}$

Note that these optimality conditions are also sufficient due to the concavity of the utility function. The budget constraint will bind in the optimum, since the utility function is strictly increasing monotonically and consequently the Walras law applies. From condition 1 and 2 it follows by division

${\displaystyle {\frac {0{,}6x_{1}^{0{,}4}x_{2}^{-0{,}4}}{0{,}4x_{1}^{-0{,}6}x_{2}^{0{,}6}}}={\frac {p_{2}\lambda }{p_{1}\lambda }}\Rightarrow {\frac {3}{2}}\cdot {\frac {x_{1}}{x_{2}}}={\frac {p_{2}}{p_{1}}}\Rightarrow x_{2}=1{,}5x_{1}\cdot {\frac {p_{1}}{p_{2}}}}$.

If you put this in the changed budget condition, the result is

${\displaystyle p_{1}x_{1}=y-p_{2}x_{2}=y-p_{2}\cdot 1{,}5x_{1}\cdot {\frac {p_{1}}{p_{2}}}\Rightarrow {\color {BrickRed}x_{1}^{*}={\frac {y}{2{,}5p_{1}}}=0{,}4{\frac {y}{p_{1}}}}}$,

with which then again

${\displaystyle {\color {BrickRed}x_{2}^{*}={\frac {1{,}5}{2{,}5}}{\frac {y}{p_{2}}}=0{,}6{\frac {y}{p_{2}}}}}$

The last two expressions for and are nothing other than the respective Marshallian demand functions for good 1 and good 2. ${\displaystyle x_{1}^{*}}$ ${\displaystyle x_{2}^{*}}$

Remarks:

• The example concerns a special case in which the demand for bananas and apples only depends on the price of the respective good, but not on the price of the other good; the demand for bananas is therefore independent of the price of apples , for example . This is generally not the case. ${\displaystyle p_{1}}$
• It is noticeable that the multiplicative terms in the Marshall's queries exactly correspond to the respective exponent in the utility function. This is not a coincidence, as the following section shows.

Inserting the prices and the income into these functions shows that in the household optimum 8 apples and 6 bananas are in demand.

Marshall's demand functions for common utility functions

Utility function	Marshall's demand
Cobb-Douglas utility function (constant returns to scale ): ${\displaystyle u(x_{1},x_{2})=x_{1}^{\alpha _{1}}\cdot x_{2}^{\alpha _{2}}}$, With ${\displaystyle \alpha _{1}+\alpha _{2}=1}$	For : ${\displaystyle i=1,2}$ ${\displaystyle x_{i}^{m}(p_{1},p_{2},y)=\alpha _{i}y/p_{i}}$
CES utility function: ${\displaystyle u(x_{1},x_{2})=\left(x_{1}^{\rho }+x_{2}^{\rho }\right)^{1/\rho }}$ With ${\displaystyle \rho \neq 0}$, ${\displaystyle \rho <1}$	For : ${\displaystyle i=1,2}$ ${\displaystyle x_{i}^{m}(p_{1},p_{2},y)={\frac {p_{i}^{r-1}y}{p_{1}^{r}+p_{2}^{r}}}}$ With ${\displaystyle r={\frac {\rho }{\rho -1}}}$
Linear utility function: ${\displaystyle u(x_{1},x_{2})=x_{1}+x_{2}}$	${\displaystyle x_{1}^{m}(p_{1},p_{2},y)={\begin{cases}y/p_{1}&{\text{falls}}\quad p_{1}<p_{2}\\\xi \in [0,y/p_{1}]\quad \mathrm {beliebig} &\mathrm {falls} \quad p_{1}=p_{2}\\0&{\text{falls}}\quad p_{1}>p_{2}\end{cases}}}$ ${\displaystyle x_{2}^{m}(p_{1},p_{2},y)={\begin{cases}0&{\text{falls}}\quad p_{1}<p_{2}\\\xi \in [0,y/p_{1}]\quad {\text{beliebig}}&{\text{falls}}\quad p_{1}=p_{2}\\y/p_{2}&{\text{falls}}\quad p_{1}>p_{2}\end{cases}}}$
Leontief utility function: ${\displaystyle u(x_{1},x_{2})=\min\{x_{1},x_{2}\}}$	${\displaystyle x_{1}^{m}(p_{1},p_{2},y)=x_{2}^{m}(p_{1},p_{2},y)={\frac {y}{p_{1}+p_{2}}}}$
Stone Geary utility function: ${\displaystyle u(x_{1},x_{2})=(x_{1}-\alpha _{1})^{\beta _{1}}\cdot (x_{2}-\alpha _{2})^{\beta _{2}}}$ With ${\displaystyle \beta _{1},\beta _{2}\geq 0}$	${\displaystyle x_{1}^{m}(p_{1},p_{2},y)=\alpha _{1}+{\frac {\beta _{1}}{p_{1}}}(y-\alpha _{2}p_{2})}$ ${\displaystyle x_{2}^{m}(p_{1},p_{2},y)=\alpha _{2}+{\frac {\beta _{2}}{p_{2}}}(y-\alpha _{1}p_{1})}$

Relation to related concepts

Indirect utility function

If the Marshallian demand obtained is put back into the original utility function , one obtains a utility function that is dependent on the prices of goods and income . It is called an indirect utility function . For a given price-income configuration, the indirect utility function indicates the specific utility level that the utility-maximizing household achieves through its demand. ${\displaystyle \mathbf {x} ^{m}(\mathbf {p} ,y)}$ ${\displaystyle u(\mathbf {x} )}$ ${\displaystyle u}$ ${\displaystyle y}$ ${\displaystyle v(\mathbf {p} ,y)\equiv u[\mathbf {x} ^{m}(\mathbf {p} ,y)]}$

Hicks's demand function

Fig. 5. Correlation between the utility maximization problem considered here and the expenditure minimization problem.

→ Main article : “Connection to Marshall's demand” in the article Hicks's demand function

While the Marshallian demand, as shown, results from the utility maximization problem of the household and indicates the quantity of goods - depending on the goods prices - which is required to achieve the highest possible level of utility with a given income , Hicksian demand results from the expenditure minimization problem of the Household and indicates the quantity of goods - depending on the goods prices - that is required to achieve a given level of utility as cheaply as possible . ${\displaystyle y}$ ${\displaystyle {\overline {u}}}$

However, despite the conceptual difference, there is a close functional relationship between Marshall's and Hicks' demand, for which reference is made to the main article mentioned above.

Example in the two-goods case (continuation)

(Continuation of the example above.)

Indirect utility function

The indirect utility function is

${\displaystyle v(p_{1},p_{2},y)\equiv u\left[x_{1}^{*}(p_{1},p_{2},y),x_{2}^{*}(p_{1},p_{2},y)\right]=\left(x_{1}^{*}\right){}^{0{,}4}\cdot \left(x_{2}^{*}\right){}^{0{,}6}}$

Inserting the received Marshallian inquiries lists

${\displaystyle v(p_{1},p_{2},y)=\left({\color {BrickRed}0{,}4{\frac {y}{p_{1}}}}\right)^{0{,}4}\cdot \left({\color {BrickRed}0{,}6{\frac {y}{p_{2}}}}\right)^{0{,}6}=\left({\frac {0{,}4}{p_{1}}}\right)^{0{,}4}\cdot \left({\frac {0{,}6}{p_{2}}}\right)^{0{,}6}\cdot y}$

Given the prices of goods and income, the indirect utility function indicates the maximum possible utility level. One can check accordingly which result it delivers with the values ​​for , and agreed above . This gives ${\displaystyle p_{1}}$ ${\displaystyle p_{2}}$ ${\displaystyle y}$

${\displaystyle v\approx 6{,}73}$.

And in fact, with the optimal quantities of goods obtained above and : ${\displaystyle x_{1}^{*}=8}$ ${\displaystyle x_{2}^{*}=6}$

${\displaystyle u(x_{1}^{*},x_{2}^{*})=8^{0{,}4}\cdot 6^{0{,}6}\approx 6{,}73}$.

Hicks's demand function

In order to get from the Marshall's demand functions and the respective Hicks' demand functions, one sets the indirect utility function at any utility level and then converts the functions according to the income: ${\displaystyle x_{1}^{m}(p_{1},p_{2},y)=0{,}4(y/p_{1})}$ ${\displaystyle x_{2}^{m}(p_{1},p_{2},y)=0{,}6(y/p_{2})}$

${\displaystyle \left({\frac {0{,}4}{p_{1}}}\right)^{0{,}4}\cdot \left({\frac {0{,}6}{p_{2}}}\right)^{0{,}6}\cdot y={\overline {u}}\Rightarrow y={\frac {\overline {u}}{\left({\frac {0{,}4}{p_{1}}}\right)^{0{,}4}\cdot \left({\frac {0{,}6}{p_{2}}}\right)^{0{,}6}}}={\overline {u}}\cdot \left({\frac {p_{1}}{0{,}4}}\right)^{0{,}4}\cdot \left({\frac {p_{2}}{0{,}6}}\right)^{0{,}6}}$

This is the expense function . Using Shephard's lemma, it follows immediately ${\displaystyle e(p_{1},p_{2},{\overline {u}})}$

${\displaystyle x_{1}^{h}(p_{1},p_{2},{\overline {u}})={\frac {\partial e(p_{1},p_{2},{\overline {u}})}{\partial p_{1}}}={\overline {u}}\cdot \left({\frac {p_{1}}{0{,}4}}\right)^{-0{,}6}\cdot \left({\frac {p_{2}}{0{,}6}}\right)^{0{,}6}}$

or.

${\displaystyle x_{2}^{h}(p_{1},p_{2},{\overline {u}})={\frac {\partial e(p_{1},p_{2},{\overline {u}})}{\partial p_{2}}}={\overline {u}}\cdot \left({\frac {p_{1}}{0{,}4}}\right)^{0{,}4}\cdot \left({\frac {p_{2}}{0{,}6}}\right)^{-0{,}4}}$.

Differentiability

Matrix equation of consumer demand

Because it is important for the following consideration, the abbreviated representation of the (matrix) vector products is given up for a short time and explicitly formulated whether it is a column or a row vector. and let both be column vectors. ${\displaystyle \mathbf {x} }$ ${\displaystyle \mathbf {p} }$

Consider the first order conditions

${\displaystyle {\frac {\partial {\mathcal {L}}(\mathbf {x} )}{\partial x_{i}}}=0\Rightarrow {\frac {\partial u(\mathbf {x} )}{\partial x_{i}}}=\lambda p_{i}}$ for all ${\displaystyle i=1,\ldots ,n}$ ${\displaystyle \Leftrightarrow \nabla u(\mathbf {x} )=\lambda \mathbf {p} }$

(with the gradient of the utility function) and the secondary condition ${\displaystyle \nabla u(\mathbf {x} )}$

${\displaystyle \mathbf {p} ^{T}\mathbf {x} =y}$

The total differential is formed from these conditions :

${\displaystyle U\mathrm {d} \mathbf {x} =\mathbf {p} \mathrm {d} \lambda +\lambda \mathrm {d} \mathbf {p} }$

${\displaystyle \mathbf {p} ^{T}\mathrm {d} \mathbf {x} +\mathbf {x} ^{T}\mathrm {d} \mathbf {p} =\mathrm {d} y}$

with the - Hessian matrix of the utility function, the -th element of which is given by , and converts this system into matrix notation: ${\displaystyle U}$ ${\displaystyle (n\times n)}$ ${\displaystyle (i,j)}$ ${\displaystyle \partial ^{2}u(\mathbf {x} )/\partial x_{i}\partial x_{j}}$

${\displaystyle \left({\begin{array}{cc}U&\mathbf {p} \\\mathbf {p} ^{T}&0\end{array}}\right)\cdot \left({\begin{array}{c}\mathrm {d} \mathbf {x} \\-\mathrm {d} \lambda \end{array}}\right)=\left({\begin{array}{cc}\mathbf {0} &\lambda I\\1&-\mathbf {x} ^{T}\end{array}}\right)\cdot \left({\begin{array}{c}\mathrm {d} y\\\mathrm {d} \mathbf {p} \end{array}}\right)}$

Following Barten (1966), this equation is sometimes referred to as the " fundamental matrix equation of consumer demand". Name this expression (a). Also consider the demand system

${\displaystyle x_{i}=x_{i}(\mathbf {p} ,y)}$ for all ${\displaystyle i=1,\ldots ,n}$

${\displaystyle \lambda =\lambda (\mathbf {p} ,y)}$.

Form the total differential of this as well:

${\displaystyle \left({\begin{array}{c}\mathrm {d} \mathbf {x} \\-\mathrm {d} \lambda \end{array}}\right)=\left({\begin{array}{cc}\mathbf {x} _{y}&X_{\mathbf {p} }\\-\lambda _{y}&-\lambda _{\mathbf {p} }^{T}\end{array}}\right)\cdot \left({\begin{array}{c}\mathrm {d} y\\\mathrm {d} \mathbf {p} \end{array}}\right)}$

with , , and a matrix with -th element . Name this expression (b). ${\displaystyle \lambda _{\mathbf {p} }=(\partial \lambda /\partial p_{1},\ldots ,\partial \lambda /\partial p_{n})^{T}}$ ${\displaystyle \lambda _{y}=\partial \lambda /\partial y}$ ${\displaystyle \mathbf {x} _{y}=(\partial x_{1}/\partial y,\ldots ,\partial x_{n}/\partial y)^{T}}$ ${\displaystyle X_{\mathbf {p} }}$ ${\displaystyle (n\times n)}$ ${\displaystyle (i,j)}$ ${\displaystyle \partial x_{i}/\partial p_{j}}$

(b) in (a) yields

${\displaystyle \left({\begin{array}{cc}U&\mathbf {p} \\\mathbf {p} ^{T}&0\end{array}}\right)\left({\begin{array}{cc}\mathbf {x} _{y}&X_{\mathbf {p} }\\-\lambda _{y}&-\lambda _{\mathbf {p} }^{T}\end{array}}\right)=\left({\begin{array}{cc}\mathbf {0} &\lambda I\\1&-\mathbf {x} ^{T}\end{array}}\right)}$

or - regularity of presupposed - phrased differently ${\displaystyle U}$

${\displaystyle {\begin{aligned}\left({\begin{array}{cc}\mathbf {x} _{y}&X_{\mathbf {p} }\\-\lambda _{y}&-\lambda _{\mathbf {p} }^{T}\end{array}}\right)&=\left({\begin{array}{cc}U&\mathbf {p} \\\mathbf {p} ^{T}&0\end{array}}\right)^{-1}\left({\begin{array}{cc}\mathbf {0} &\lambda I\\1&-\mathbf {x} ^{T}\end{array}}\right)\\&={\frac {1}{\mathbf {p} ^{T}U^{-1}\mathbf {p} }}\left({\begin{array}{cc}(\mathbf {p} ^{T}U^{-1}\mathbf {p} )U^{-1}-U^{-1}\mathbf {p} (U^{-1}\mathbf {p} )^{T}&U^{-1}\mathbf {p} \\(U^{-1}\mathbf {p} )^{T}&-1\end{array}}\right)\left({\begin{array}{cc}\mathbf {0} &\lambda I\\1&-\mathbf {x} ^{T}\end{array}}\right)\\&={\frac {1}{\mathbf {p} ^{T}U^{-1}\mathbf {p} }}\left({\begin{array}{cc}U^{-1}\mathbf {p} &(\mathbf {p} ^{T}U^{-1}\mathbf {p} )\lambda U^{-1}-\lambda (U^{-1}\mathbf {p} )(U^{-1}\mathbf {p} )^{T}-U^{-1}\mathbf {p} \mathbf {x} ^{T}\\-1&\lambda (U^{-1}\mathbf {p} )^{T}+\mathbf {x} ^{T}\end{array}}\right)\end{aligned}}}$

Differentiability property

1. Theorem (Katzner 1968): The system of Marshallian inquiries is continuously differentiable if and only if: ${\displaystyle \mathbf {x} ^{m}}$ ${\displaystyle (\mathbf {p} ^{0},y^{0})}$

${\displaystyle \left({\begin{array}{cc}U&\mathbf {p} \\\mathbf {p} ^{T}&0\end{array}}\right)}$

is regularly at the point . ${\displaystyle \mathbf {x} =\mathbf {x} ^{m}(\mathbf {p} ^{0},y^{0})}$

2. Lemma: The condition under (1.) is fulfilled if and only if the following applies:

${\displaystyle \left({\begin{array}{cc}U&\nabla u(\mathbf {x} )\\\left[\nabla u(\mathbf {x} )\right]^{T}&0\end{array}}\right)}$

is regular. This matrix is ​​the modified Hessian matrix of the utility function.

Classification of the history of ideas

The ordinal concept of utility on which the Marshallian demand is based goes back to the “modern” economic conception of utility in the successor of Vilfredo Pareto . Pareto (1906) constructed, taking up and continuing a concept by Edgeworth (1881), indifference curves for different goods, whereby - which would actually no longer be necessary - he sometimes still presupposes a cardinal determinability of the benefit; nevertheless, it makes clear the strict distinction between (basic) preferences and (merely representative) benefits. Unlike Edgeworth, he does not want the indifference curves to be constructed as a graphic representation of a cardinal utility function, but on the contrary, develops his utility theory only on the basis of indifference curves (based on observability). Already Pareto (1896) shows - as well as independently Fisher (1892) - that the measurability of the utility for the construction of demand functions is not necessary.

The other conceptual building block of Marshall's demand - the construction of the demand function on the basis of utility theory - can be traced back to Léon Walras . Walras developed a model as early as 1872 in which retailers try to maximize their utility, whereby the individual utility functions are independent and additive to one another. At his request, Antoine Paul Piccard (1844–1920), who, like Walras, was a professor at the University of Lausanne , finally provided a way of constructing an equilibrium function using a limited maximization problem. Starting from two marginal utility curves for two goods and and a certain positive initial endowment of at given prices, Piccard constructs an optimality condition for the consumption of and , which can be expressed in a kind of demand curve, which does not yet follow the modern conception of a demand curve, with it considered two-goods case is related. This extended model also arose in particular from Walras 'knowledge of the proportionality of marginal utility (in Walras' terminology: rereté ) and goods prices, which permeates the modern concept of demand. Marshall (1890) provides a much simpler and more direct way of obtaining a demand curve from the demand function. As with Walras, the independence and additivity of the utility function and a decreasing marginal utility are assumed. Based on this, Marshall can indeed construct the modern formulation of the demand curve as a function of goods prices and income; However, he only succeeds in doing this under the assumption of a “constant” marginal utility of income. This assumption met with criticism when the Principles were published , including from Pareto. Marshall also shared with Walras the basis of a cardinal benefit concept, the dispensability of which Pareto was later able to show. ${\displaystyle \mathrm {A} }$ ${\displaystyle \mathrm {B} }$ ${\displaystyle \mathrm {A} }$ ${\displaystyle \mathrm {A} }$ ${\displaystyle \mathrm {B} }$

Yevgeny Slutsky played a key role in putting the methodological components together . He (1915) already largely outlines the modern concept of the Marshallian demand function. A slightly more general version is provided (unaware of Slutsky's contribution) by John Hicks and RGD Allen (1934a, 1934b).

literature

• Anton Barten and Volker Böhm: Consumer Theory. In: Kenneth J. Arrow and Michael D. Intrilligator (Eds.): Handbook of Mathematical Economics. Vol. 2. North Holland, Amsterdam 1982, ISBN 978-0-444-86127-6 , pp. 382-429 (also online: doi : 10.1016 / S1573-4382 (82) 02004-9 ).
• Friedrich Breyer: Microeconomics. An introduction. 5th edition. Springer, Heidelberg a. a. 2011, ISBN 978-3-642-22150-7 (also online: doi : 10.1007 / 978-3-642-22150-7 ). [Chapter 4]
• Arthur S. Goldberger: Functional form and utility. A review of consumer demand theory. Westview Press, Boulder 1987, ISBN 0-8133-7489-8 .
• Donald W. Katzner: Static Demand Theory. Macmillan, New York 1970.
• David M. Kreps: Microeconomic Foundations I. Choice and Competitive Markets. Princeton University Press, Princeton 2012, ISBN 978-0-691-15583-8 .
• Andreu Mas-Colell, Michael Whinston, and Jerry Green: Microeconomic Theory. Oxford University Press, Oxford 1995, ISBN 0-195-07340-1 . [Chapter 3]
• Efe A. Ok: Real Analysis with Economic Applications. Princeton University Press, Princeton 2007, ISBN 978-0-691-11768-3 .
• Eugene Silberberg: Hicksian and Marshallian demands. In: Steven N. Durlauf and Lawrence E. Blume (Eds.): The New Palgrave Dictionary of Economics. 2nd Edition. Palgrave Macmillan 2008, doi : 10.1057 / 9780230226203.0731 (online edition).
• Mark Voorneveld: Mathematical foundations for microeconomic theory: Preference, utility and choice. Script, Stockholm School of Economics, 2009, Internet https://studentweb.hhs.se/courseweb/CourseWeb/Public/PhD501/0701/notes2.pdf , accessed on May 5, 2014.

Remarks

1. ↑ is the set of all tuples of real numbers with ; the set of all tuples of real numbers with . ${\displaystyle \mathbb {R} _{+}^{n}}$ ${\displaystyle (x_{1},\ldots ,x_{n})}$ ${\displaystyle x_{i}\geq 0}$ ${\displaystyle \mathbb {R} _{++}^{n}}$ ${\displaystyle (x_{1},\ldots ,x_{n})}$ ${\displaystyle x_{i}>0}$
2. ↑ denotes the argument of the maximum . ${\displaystyle \arg \max(\cdot )}$
3. ↑ See James C. Moore: General equilibrium and welfare economics. An introduction. Springer, Berlin a. a. 2007, ISBN 978-3-540-31407-3 (also online: doi : 10.1007 / 978-3-540-32223-8 ), p. 88.
4. ↑ Be and two metric spaces. A correspondence is called compact-valued if there is a compact subset of for all . ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle \Gamma :X\twoheadrightarrow Y}$ ${\displaystyle \Gamma (\mathbf {x} )}$ ${\displaystyle \mathbf {x} \in X}$ ${\displaystyle Y}$
5. ↑ On the following, for example, Kreps 2012, p. 53; Mas-Colell / Whinston / Green 1995, p. 50 f.
6. ↑ See for example Kreps 2012, p. 34.
7. ↑ Kreps 2012, p. 34.
8. ^ Gerard Debreu : Theory of Value. An Axiomatic Analysis of Economic Equilibrium. Yale University Press, New Haven and London 1959, here p. 72 f.
9. ↑ cf. for example James C. Moore: General equilibrium and welfare economics. An introduction. Springer, Berlin a. a. 2007, ISBN 978-3-540-31407-3 (also online: doi : 10.1007 / 978-3-540-32223-8 ), Chapter 4; Ariel Rubinstein : Lecture Notes in Microeconomic Theory. Lecture 5th Internet http://press.princeton.edu/rubinstein/lecture5.pdf , accessed on May 6, 2014.
10. ↑ ^{a b} It refers to a correspondence as oberhemistetig, if at any point the following will apply: For each open set that contains an environment exists to such that for all . A correspondence is called subhemistig if at each point the following applies: Whenever and through an opposing converging sequence in is given, then there exists a natural number and a sequence in which converges to, where for all . See Knut Sydsæter et al. a .: Further mathematics for economic analysis. 2nd ed. Financial Times / Prentice Hall, Harlow 2008, ISBN 978-0-273-71328-9 , pp. 504 f. Note that the terminology used in the literature sometimes differs from this. Occasionally the term (lower / upper) semi-continuity is used instead of (lower / upper) semi-continuity (e.g. Kreps 2012; Gerard Debreu: Theory of Value. An Axiomatic Analysis of Economic Equilibrium. Yale University Press, New Haven and London 1959 ), which collides with a related but nevertheless different definition of the term for real-valued functions (cf. for this instead of many Forster: Analysis. Part 3. 5th edition. Springer, Berlin et al. 2009, ISBN 978-3-8348- 0704-5 , p. 39 f .; Dean Corbae, Maxwell B. Stinchcombe and Juraj Zeman: An Introduction to Mathematical Analysis For Economic Theory and Econometrics. Princeton University Press, Princeton and Oxford 2009, ISBN 978-0-691-11867- 3 , p. 349). ${\displaystyle \Gamma :X\subseteq \mathbb {R} ^{l}\twoheadrightarrow \mathbb {R} ^{m}}$ ${\displaystyle \mathbf {x} ^{0}\in X}$ ${\displaystyle U}$ ${\displaystyle \Gamma (\mathbf {x} ^{0})}$ ${\displaystyle N_{\epsilon }}$ ${\displaystyle \mathbf {x} ^{0}}$ ${\displaystyle \Gamma (\mathbf {x} )\subseteq U}$ ${\displaystyle \mathbf {x} \in N_{\epsilon }\cap X}$
    ${\displaystyle \Gamma :X\subseteq \mathbb {R} ^{l}\twoheadrightarrow \mathbb {R} ^{m}}$ ${\displaystyle \mathbf {x} ^{0}\in X}$ ${\displaystyle \mathbf {y} ^{0}\in \Gamma (\mathbf {x} ^{0})}$ ${\displaystyle \{\mathbf {x} _{k}\}}$ ${\displaystyle \mathbf {x} ^{0}}$ ${\displaystyle X}$ ${\displaystyle k_{0}}$ ${\displaystyle \{\mathbf {y} _{k}\}_{k=k_{o}}^{\infty }}$ ${\displaystyle Y}$ ${\displaystyle \mathbf {y} ^{0}}$ ${\displaystyle \mathbf {y} _{k}\in \Gamma (\mathbf {x} _{k})}$ ${\displaystyle k\geq k_{0}}$
    
11. ↑ ^{a b} Be and two metric spaces . Define a continuous, compact-valued and non-empty correspondence of on and be a continuous function. Then ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle \mathbf {x} \rightarrow \!\!\!\!\!\!\!\mapsto \Gamma (\mathbf {x} )}$ ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle u:X\times Y\rightarrow \mathbb {R} }$

    ${\displaystyle \varphi (\mathbf {x} )\equiv \sup\{u(\mathbf {x} ,\mathbf {y} )|\mathbf {y} \in \Gamma (\mathbf {x} )\}}$

    a continuous function and

    ${\displaystyle \mu (\mathbf {x} )\equiv \{\mathbf {y} \in \Gamma (\mathbf {x} )|u(\mathbf {x} ,\mathbf {y} )=\varphi (\mathbf {x} )\}}$

    defines a non-empty, compact and top-level correspondence. See, also for proof, Dean Corbae, Maxwell B. Stinchcombe and Juraj Zeman: An Introduction to Mathematical Analysis For Economic Theory and Econometrics. Princeton University Press, Princeton and Oxford 2009, ISBN 978-0-691-11867-3 , pp. 268 f .; Ok 2007, p. 306 ff .; James C. Moore: Mathematical methods for economic theory. Vol. 2. Springer, Berlin a. a. 1999, ISBN 3-540-66242-1 , p. 280.
12. ↑ To prove the upper hemisphere cf. Ok 2007, p. 292 and Kreps 2012, p. 55 f .; for the proof of the subhumanity cf. Ok 2007, p. 299 and Kreps 2012, p. 56.
13. ↑ Be and two metric spaces. The correspondence is said to be closed if there is a closed subset of for all . See Ok 2007, p. 289. ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle \Gamma :X\twoheadrightarrow Y}$ ${\displaystyle \Gamma (\mathbf {x} )}$ ${\displaystyle \mathbf {x} \in X}$ ${\displaystyle Y}$
14. ↑ The correspondence has a closed graph if the following implication applies at every point : Be and with and arbitrary consequences and apply to all . Then is . See e.g. Ok 2007, p. 294. ${\displaystyle \Gamma :X\subseteq \mathbb {R} ^{l}\twoheadrightarrow \mathbb {R} ^{m}}$ ${\displaystyle \mathbf {x} ^{0}\in X}$ ${\displaystyle \{\mathbf {x} _{k}\}}$ ${\displaystyle \{\mathbf {y} _{k}\}}$ ${\displaystyle \mathbf {x} _{k}\rightarrow \mathbf {x} ^{0}}$ ${\displaystyle \mathbf {y} _{k}\rightarrow \mathbf {y} ^{0}}$ ${\displaystyle \mathbf {y} _{k}\in \Gamma (\mathbf {x} _{k})}$ ${\displaystyle k}$ ${\displaystyle \mathbf {y} ^{0}\in \Gamma (\mathbf {x} ^{0})}$
15. ↑ Cf., also for proof, Ok 2007, p. 295 f.
16. ↑ Based on Voorneveld 2009, p. 24; Charalambos D. Aliprantis: Problems in Equilibrium Theory. Springer, Berlin a. a. 1996, ISBN 3-540-60753-6 , p. 39 f.
17. ↑ A preference ordering is referred to as non-saturated locally, if for any and for each environment to a exists, with the . See the Article Preference Regulations . ${\displaystyle \mathbf {x} ^{0}\in X}$ ${\displaystyle \epsilon }$ ${\displaystyle N_{\epsilon }}$ ${\displaystyle \mathbf {x} ^{0}}$ ${\displaystyle \mathbf {x} '\in N_{\epsilon }}$ ${\displaystyle \mathbf {x} '\succ \mathbf {x} ^{0}}$
18. ↑ See Mas-Colell / Whinston / Green 1995, p. 53 f .; Kreps 2012, pp. 57 f., 480 ff. (For evidence); for the general proof of the theorem also Knut Sydsæter et al. a .: Further mathematics for economic analysis. 2nd ed. Financial Times / Prentice Hall, Harlow 2008, ISBN 978-0-273-71328-9 , pp. 143 f.
19. ↑ See, also for proof, Michael Carter: Foundations of mathematical economics. MIT Press, Cambridge 2001, ISBN 0-262-03289-9 , p. 577 ff. (The argument there for the validity of the regularity condition using the example of the utility maximization problem is, however, incorrect.); Peter Kall: Analysis for Economists. BG Teubner, Stuttgart 1982, ISBN 3-519-02355-5 , p. 178 (Lemma 5.20).
20. ↑ On this, for example, Mas-Colell / Whinston / Green 1995, p. 54.
21. ↑ On this and the following Goldberger 1987, p. 3 ff .; William A. Barnett and Apostolos Serletis: The Differential Approach to Demand Analysis and the Rotterdam Model. In: Daniel J. Slottje (Ed.): Quantifying Consumer Preferences. Emerald, Bingley 2009, ISBN 978-1-84855-312-5 , pp. 61–81, here p. 63 ff.
22. ^ Anton Barten: Theory en Empirie van een Volledig Stelsel van Vraagvergelijkingen. Dissertation, Netherlands School of Economics, Rotterdam.
23. ↑ See Goldberger 1987, p. 6; Barten / Böhm 1982, p. 410.
24. ↑ Donald W. Katzner: A Note on the differentiability of Consumer Demand Functions. In: Econometrica. 36, No. 2, 1968, pp. 415-418 ( JSTOR 1907498 ).
25. ↑ See Barten / Böhm 1982, p. 411.
26. ↑ See Barten / Böhm 1982, p. 411; Mas-Colell / Whinston / Green 1995, p. 95.
27. ^ Vilfredo Pareto: Manuale di economia politica. Con una introduzione alla scienza sociale. Societa editrice libraria, Milan 1906. Reference is made here to the English translation of the Manual of Political Economy. Translated by Ann S. Schwier. Augustus M. Kelley, New York 1971.
28. ^ Francis Y. Edgeworth : Mathematical Psychics. An Essay on the Application of Mathematics to the Moral Sciences. CK Paul & Co, 1881.
29. ↑ See Christian E. Weber: Pareto and the 53% Ordinal Theory of Utility. In: History of Political Economy. 33, No. 3, 2001, pp. 541-576; George J. Stigler : The Development of Utility Theory. II. In: Journal of Political Economy. 58, No. 5, 1950, pp. 373-396 ( JSTOR 1825710 ), here p. 380 f.
30. ↑ “The notions of value in use, utility, ophelimity, indices of ophelimity, etc, greatly facilitate the exposition of the theory of economic equilibrium, but they are not necessary to construct this theory. Thanks to the use of mathematics, this entire theory […] rests on no more than a fact of experience, that is, on the determination of the quantities of goods which constitute combinations between which the individual is indifferent. " (Vilfredo Pareto: Manuale di economia politica. Con una introduzione alla scienza sociale. Societa editrice libraria, Milan 1906, quoted from the English translation Manual of Political Economy. Translated by Ann S. Schwier. Augustus M. Kelley, New York 1971. ) See also Kerrie L. Mitchener: Preference and Utility in Economic Theory and the History of Economic Thought. Dissertation, University of Queensland, 2007, Chapter 6; Ghanshyam B. Mehta: Preference and Utility. In: Salvador Barberà, Peter J. Hammond and Christian Seidl (eds.): Handbook of Utility Theory. Vol. 1. Kluwer, Dordrecht u. a. 1998, ISBN 0-7923-8174-2 , pp. 1-47, pp. 2 f.
31. ^ Vilfredo Pareto: Cours d'économie politique. Rouge, Lausanne 1896.
32. ^ Irving Fisher: Mathematical investigations in the theory of value and prices. In: Transactions of the Connecticut Academy of Arts and Sciences. 9, 1892.
33. ↑ Cf. George J. Stigler : The Development of Utility Theory. II. In: Journal of Political Economy. 58, No. 5, 1950, pp. 373-396 ( JSTOR 1825710 ); Roberto Marchionatti and Enrico Gambino: Pareto and Political Economy as a Science: Methodological Revolution and Analytical Advances in Economic Theory in the 1890s. In: Journal of Political Economy. 105, No. 6, 1997, pp. 1322-1348 ( JSTOR ), here pp. 1335 f.
34. ↑ See Katzner 1970, p. 8.
35. ↑ On this Donald A. Walker: Walras, Léon (1834-1910). In: Steven N. Durlauf and Lawrence E. Blume (Eds.): The New Palgrave Dictionary of Economics. 2nd Edition. Palgrave Macmillan 2008, doi : 10.1057 / 9780230226203.1814 (online edition); Donald A. Walker: Walras's market models. Cambridge University Press, Cambridge 2005, ISBN 9780521022958 , p. 41.
36. ↑ On this more detailed William Jaffé: Léon Walras's Role in the “Marginal Revolution” of the 1870s. In: History of Political Economy. 4, No. 2, 1972, doi : 10.1215 / 00182702-4-2-379 , p. 379-405, here p. 397 f.
37. ^ Alfred Marshall: Principles of Economics. 1st edition Macmillan, 1890 (also online: https://archive.org/details/principlesecono00marsgoog ).
38. ↑ See also Peter C. Dooley: Consumer's Surplus: Marshall and His Critics. In: The Canadian Journal of Economics / Revue canadienne d'Economique. 16, No. 1, 1983, pp. 26-38 ( JSTOR 134973 ), pp. 28 ff .; specifically on the controversy with Pareto: EB Wilson: Pareto Versus Marshall. In: The Quarterly Journal of Economics. 53, No. 4, 1939, pp. 645-650 ( JSTOR 1883289 ).
39. ^ Yevgeny Slutsky: Sulla teoria del bilancio del consumatore. In: Giornale degli economisti. 1915, pp. 1-26. Reference is made here to the English translation On the Theory of the Budget of the Consumer. In: George J. Stigler and KE Boalding (Eds.): Readings in Price Theory. Irwin, Homewood 1952, pp. 27-56.
40. ↑ See Katzner 1970, p. 7.
41. ↑ John R. Hicks and RGD Allen: A Reconsideration of the Theory of Value. Part I. In: Economica. 1, No. 1, 1934, pp. 52-76 ( JSTOR 2548574 ).
42. ↑ John R. Hicks and RGD Allen: A Reconsideration of the Theory of Value. Part II. A Mathematical Theory of Individual Demand Functions In: Economica. 1, No. 2, 1934, pp. 196-219 ( JSTOR 2548749 ).
43. ↑ On the relationship between these contributions and Slutsky (1915) cf. RGD Allen: Professor Slutsky's Theory of Consumers' Choice. In: Review of Economic Studies. 3, No. 2, 1936, pp. 120-129, doi : 10.2307 / 2967502 . On the genesis and reception history of Slutsky (1915) cf. John S. Chipman and Jean-Sébastien Lenfant: Slutsky's 1915 Article: How It Came to Be Found and Interpreted. In: History of Political Economy. 34, No. 3, 2002, pp. 553-597, doi : 10.1215 / 00182702-34-3-553 .

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