Law of demand

As a law of demand ( English law of demand ) is called in economics a frequently used theorem , which states in its simplest version, that the demand for a normal good decreases when its price increases. A good is called normal if an increase in income means that more of the good is in demand.

Formal definition and derivation

theorem

Let the Marshallian demand for a good depend on a price vector and the individual income . (The Marshallian demand results from the utility maximization problem of the household and indicates the quantity of goods - depending on the goods prices - that is required to achieve the highest possible level of utility with a given income .) ${\displaystyle x_{i}(\mathbf {p} ,y)}$ ${\displaystyle i}$ ${\displaystyle \mathbf {p} =(p_{1},\ldots ,p_{n})}$ ${\displaystyle y}$ ${\displaystyle y}$

Law of demand: Let i be a normal good, that is, let : ${\displaystyle \partial x_{i}(\mathbf {p} ,y)/\partial y>0}$

${\displaystyle {\frac {\partial x_{i}(\mathbf {p} ,y)}{\partial p_{i}}}<0}$

proof

The theorem follows directly from Slutsky's equation, according to which

${\displaystyle \underbrace {\frac {\partial x_{i}(\mathbf {p} ,y)}{\partial p_{j}}} _{\mathrm {Gesamteffekt} }=\underbrace {\frac {\partial x_{i}^{h}(\mathbf {p} ,{\overline {u}})}{\partial p_{j}}} _{\mathrm {Substitutionseffekt} }\underbrace {-x_{j}(\mathbf {p} ,y){\frac {\partial x_{i}(\mathbf {p} ,y)}{\partial y}}} _{\mathrm {Einkommenseffekt} }}$

(For an explanation, refer to the article Slutsky equation .) In the case of own price i = j , this immediately shows that the income effect is negative (according to the assumption of normality). The substitution effect is also negative, however, since Hicks's demand for a good always falls in the price of this good. This follows from Shephard's lemma : Because of too . But since the output function is concave, this is a partial derivative . ${\displaystyle x_{i}^{h}(\mathbf {p} ,{\overline {u}})=\partial e(\mathbf {p} ,{\overline {u}})/\partial p_{i}}$ ${\displaystyle \partial x_{i}^{h}(\mathbf {p} ,{\overline {u}})/\partial p_{i}=\partial ^{2}e(\mathbf {p} ,{\overline {u}})/\partial p_{i}^{2}}$ ${\displaystyle e}$ ${\displaystyle \leq 0}$

Hence the overall effect is also negative, which has been shown.

Law of compensated demand

definition

Law of Compensated Demand: Consider any two price tuples and , where from has resulted from a Slutsky-compensated price change. Then the Marshallian demand function fulfills the law of compensated demand if and only if: ${\displaystyle \mathbf {p} ^{0}=\left(p_{1}^{0},\ldots ,p_{n}^{0}\right)}$ ${\displaystyle \mathbf {p} ^{1}=\left(p_{1}^{1},\ldots ,p_{n}^{1}\right)}$ ${\displaystyle \mathbf {p} ^{1}}$ ${\displaystyle \mathbf {p} ^{0}}$ ${\displaystyle \mathbf {x} (\mathbf {p} ,y)}$

${\displaystyle \left(\mathbf {p} ^{1}-\mathbf {p} ^{0}\right)\cdot \left(\mathbf {x} ^{1}-\mathbf {x} ^{0}\right)\leq 0}$.

Relation to the theory of revealed preferences

Preliminary consideration

Consider that in the initial situation, given the goods prices and household income, a customer chooses an optimal bundle of goods . Now the price fall of good i from on , resulting in a new Preistupel results. At the same time, an omniscient planner modifies the household income in such a way that the best bundle of goods available for the household before the price change is just as affordable even after the price reduction (Slutsky compensation). ${\displaystyle \mathbf {p} ^{0}=\left(p_{1}^{0},\ldots ,p_{n}^{0}\right)}$ ${\displaystyle y^{0}}$ ${\displaystyle \mathbf {q} ^{0}=\left(q_{1}^{0},\ldots ,q_{n}^{0}\right)}$ ${\displaystyle p_{i}^{0}}$ ${\displaystyle p_{i}^{1}}$ ${\displaystyle \mathbf {p} ^{1}}$ ${\displaystyle \mathbf {q} ^{0}}$

It is assumed that the benefit from is equal to the from . Since the household chose the bundle of goods and not in the price system , if the weak axiom of revealed preferences (WARP) is valid, the bundle of goods must have been at least as expensive at prices as it would otherwise have been for the household at time 0 strictly would have been better to choose. Formally: ${\displaystyle \mathbf {q} ^{0}}$ ${\displaystyle \mathbf {q} ^{1}}$ ${\displaystyle \mathbf {p} ^{0}}$ ${\displaystyle \mathbf {q} ^{0}}$ ${\displaystyle \mathbf {q} ^{1}}$ ${\displaystyle \mathbf {q} ^{1}}$ ${\displaystyle \mathbf {p} ^{0}}$ ${\displaystyle \mathbf {q} ^{0}}$ ${\displaystyle \mathbf {q} ^{1}}$

1) . ${\displaystyle \mathbf {p} ^{0}\cdot \mathbf {q} ^{0}\leq \mathbf {p} ^{0}\cdot \mathbf {q} ^{1}}$

Conversely, WARP can also be used in analogy to see that with the price system the household must have at least a weak incentive to prefer the bundle of goods to the bundle - otherwise it would not have chosen. The bundle of goods can not be more expensive than the bundle at prices , that is ${\displaystyle \mathbf {p} ^{1}}$ ${\displaystyle \mathbf {q} ^{1}}$ ${\displaystyle \mathbf {q} ^{0}}$ ${\displaystyle \mathbf {q} ^{1}}$ ${\displaystyle \mathbf {q} ^{1}}$ ${\displaystyle \mathbf {p} ^{1}}$ ${\displaystyle \mathbf {q} ^{0}}$

2) . ${\displaystyle \mathbf {p} ^{1}\cdot \mathbf {q} ^{1}\leq \mathbf {p} ^{1}\cdot \mathbf {q} ^{0}}$

Adding 1) and 2) now delivers immediately

${\displaystyle \mathbf {p} ^{0}\cdot \mathbf {q} ^{0}+\mathbf {p} ^{1}\cdot \mathbf {q} ^{1}\leq \mathbf {p} ^{0}\cdot \mathbf {q} ^{1}+\mathbf {p} ^{1}\cdot \mathbf {q} ^{0}\quad \Leftrightarrow \quad \left(\mathbf {p} ^{1}-\mathbf {p} ^{0}\right)\cdot \left(\mathbf {q} ^{1}-\mathbf {q} ^{0}\right)\leq 0}$,

what was to be shown.

Equivalence to WARP

In the preliminary consideration it is shown that the weak axiom of revealed preferences implies the validity of the law of compensated demand. It can be shown that the reverse direction also applies here.

Equivalence of WARP and the law of compensated demand: Let the Marshall's demand function be homogeneous of degree zero and satisfy Walras's law . Then the weak axiom of revealed preferences suffices if and only if the law of compensated demand is satisfied. ${\displaystyle \mathbf {x} (\mathbf {p} ,y)}$ ${\displaystyle \mathbf {x} (\mathbf {p} ,y)}$

literature

• Richard Cornes: Duality and modern economics. Cambridge University Press, Cambridge u. a. 1992, ISBN 0-521-33601-5 .
• Geoffrey A. Jehle and Philip J. Reny: Advanced Microeconomic Theory. 3rd ed. Financial Times / Prentice Hall, Harlow 2011, ISBN 978-0-273-73191-7 .
• Andreu Mas-Colell, Michael Whinston, and Jerry Green: Microeconomic Theory. Oxford University Press, Oxford 1995, ISBN 0-195-07340-1 .
• Nolan H. Miller: Notes on Microeconomic Theory. online ( Memento from December 15, 2011 in the Internet Archive ) (PDF; 1 MB), p. 65, accessed on January 2, 2015. [Here p. 23 ff.]
• Hal Varian : Intermediate Microeconomics. A modern approach. 8th edition. WW Norton, New York and London 2010, ISBN 978-0-393-93424-3 .

Individual evidence

1. ↑ Hal Varian, Intermediate Microeconomics. A modern approach . 8th edition, 2010, p. 147; Geoffrey A. Jehle / Philip J. Reny, Advanced Microeconomic Theory . 3rd edition, 2011, p. 56.
2. ↑ See only Geoffrey A. Jehle / Philip J. Reny, Advanced Microeconomic Theory . 3rd ed., 2011, pp. 53–56.
3. ^ Richard Cornes, Duality and modern economics , 1992, p. 64 f .; Andreu Mas-Colell / Michael Whinston / Jerry Green, Microeconomic Theory , 1995, pp. 28-30.
4. ↑ The following presentation of the derivation follows Cornes 1992, p. 64.
5. ↑ See, also for proof, Mas-Colell / Whinston / Green 1995, p. 30 f.