Hicks demand function

In microeconomic theory and especially in household theory, the Hicks demand function (also: compensated demand function ) is a function that specifies the demand for goods as a function of their price and a certain (minimum) level of utility that is to be achieved overall.

The demand function is named after John Richard Hicks , who formalized the concept of compensated demand for the first time in 1939.

Definition and meaning

Formal representation

First of all, one starts with a problem of minimizing expenditure

${\displaystyle \min _{(x_{1},\ldots ,x_{n})\in \mathbb {R} _{+}^{n}}\sum _{i=1}^{n}p_{i}x_{i}}$ under the secondary condition ${\displaystyle u(x_{1},\ldots ,x_{n})\geq {\overline {u}}}$

is given, where is continuous , strictly monotonically increasing , differentiable and strictly quasi-concave . is the vector of the quantities of goods in demand and the associated price vector. ${\displaystyle u(\cdot )}$ ${\displaystyle \mathbf {x} =(x_{1},\ldots ,x_{n})}$ ${\displaystyle \mathbf {p} =(p_{1},\ldots ,p_{n})}$

In the problem mentioned, the total expenditure for the goods from the shopping cart is minimized, but a certain level of benefit should be preserved. The solution to such an expenditure minimization problem is intended to be a function that indicates which amount of the respective goods should be requested in order to achieve the given level of utility as cost-effectively as possible. It is therefore a function of the price vector and the established level of utility . ${\displaystyle n}$ ${\displaystyle \mathbf {x} ^{*}}$ ${\displaystyle \mathbf {x} ^{*}}$ ${\displaystyle \mathbf {p} }$ ${\displaystyle {\overline {u}}}$

What is given in this way is called Hicks's demand and it is agreed . ${\displaystyle \mathbf {x} ^{*}}$ ${\displaystyle \mathbf {x} ^{*}(\mathbf {p} ,{\overline {u}})\equiv \mathbf {x} ^{h}(\mathbf {p} ,{\overline {u}})}$

Simplified representation based on the two-goods case

The expenditure problem is reduced in the classic two-goods case

${\displaystyle \min _{x_{1},x_{2}}p_{1}x_{1}+p_{2}x_{2}}$under the secondary condition . ${\displaystyle u(x_{1},x_{2})\geq {\overline {u}}}$

So minimizing the total expenditure for the two goods , and with their current prices, or . The solution to the minimization problem are two functions and , which indicate how much of good 1 or good 2 should optimally be consumed depending on the goods prices (of all goods!) And the minimum desired level of utility. These functions are called Hicks' inquiries and write or . ${\displaystyle x_{1}}$ ${\displaystyle x_{2}}$ ${\displaystyle p_{1}}$ ${\displaystyle p_{2}}$ ${\displaystyle x_{1}^{*}(p_{1},p_{2},{\overline {u}})}$ ${\displaystyle x_{2}^{*}(p_{1},p_{2},{\overline {u}})}$ ${\displaystyle x_{1}^{h}}$ ${\displaystyle x_{2}^{h}}$

example

In the example, the price of good 1 and that of good 2 are . The consumer derives his benefit exclusively from these two goods. Its utility function is . We formulate the following optimization problem for simplification with equality restriction ( ), which is justified by the properties of the utility function. The minimization problem is: ${\displaystyle p_{1}}$ ${\displaystyle p_{2}}$ ${\displaystyle u(x_{1},x_{2})=x_{1}x_{2}^{2}}$ ${\displaystyle u(x_{1},x_{2})={\overline {u}}}$

${\displaystyle \min _{x_{1},x_{2}}p_{1}x_{1}+p_{2}x_{2}}$ under the secondary condition ${\displaystyle x_{1}x_{2}^{2}={\overline {u}}}$

The corresponding Lagrange function is . The optimality conditions are ${\displaystyle {\mathcal {L}}(x_{1},x_{2},\lambda )=p_{1}x_{1}+p_{2}x_{2}+\lambda ({\overline {u}}-x_{1}x_{2}^{2})}$

1. ${\displaystyle \partial {\mathcal {L}}/\partial x_{1}=p_{1}-\lambda ^{*}\left(x_{2}^{*}\right)^{2}=0}$
2. ${\displaystyle \partial {\mathcal {L}}/\partial x_{2}=p_{2}-2\lambda ^{*}x_{1}^{*}x_{2}^{*}=0}$ and
3. ${\displaystyle \partial {\mathcal {L}}/\partial \lambda ={\overline {u}}-x_{1}^{*}\left(x_{2}^{*}\right)^{2}=0}$

From (1) and (2) it follows or , inserted into (3), Hicks's inquiries finally follow ${\displaystyle x_{1}^{*}=x_{2}^{*}(p_{2}/2p_{1})}$ ${\displaystyle x_{2}^{*}=2x_{1}^{*}(p_{1}/p_{2})}$

${\displaystyle x_{1}^{*}(p_{1},p_{2},{\overline {u}})={\sqrt[{3}]{{\overline {u}}(p_{2}/2p_{1})^{2}}}}$ and

${\displaystyle x_{2}^{*}(p_{1},p_{2},{\overline {u}})={\sqrt[{3}]{2{\overline {u}}(p_{1}/p_{2})}}}$.

Note that Hicks's demands for the two goods are identical if the price of good 2 is just twice as high as that of good 1.

Properties of the Hicks demand function

It can be shown that under the given conditions it has the following properties, among others: ${\displaystyle \mathbf {x} ^{h}(\mathbf {p} ,{\overline {u}})}$

• Grade zero homogeneity in . ${\displaystyle \mathbf {p} }$
• Convex set . is a convex set . ${\displaystyle \mathbf {x} ^{h}(\mathbf {p} ,{\overline {u}})}$
• Monotonously falling in its own price: The derivation of Hicks's demand for a good from the price of this good,, is not positive: ${\displaystyle i}$ ${\displaystyle p_{i}}$

${\displaystyle {\frac {\partial x_{i}^{h}(\mathbf {p} ,{\overline {u}})}{\partial p_{i}}}\leq 0\quad \forall i}$

This follows from Shephard's lemma : Because of too . But since the output function is concave, this is a partial derivative . ${\displaystyle x_{i}^{h}(\mathbf {p} ,{\overline {u}})=\partial e(\mathbf {p} ,{\overline {u}})/\partial p_{i}}$ ${\displaystyle \partial x_{i}^{h}(\mathbf {p} ,{\overline {u}})/\partial p_{i}=\partial ^{2}e(\mathbf {p} ,{\overline {u}})/\partial p_{i}^{2}}$ ${\displaystyle e}$ ${\displaystyle \leq 0}$

Connection to Marshall's demand

Although Hicks's demand functions play an important role in many areas of household theory, they are not directly observable by themselves and are therefore a hypothetical construction. While Marshall's demand functions are fundamentally accessible to empirical analysis - one can, for example, observe how a person's demand for a good changes when their income or the price of goods changes - this does not apply to compensated demand functions, since their core element, the weighing of benefits remains hidden from outside observation. However, there is a close connection between Hicks 'demand and its Marshallian counterpart, which allows, for example, the derivation of Hicks' demand for a good according to its own price or another price - that is , based on partial derivatives of the Marshallian demand function calculate ( Slutsky decomposition ). ${\displaystyle \partial x_{i}^{h}(\mathbf {p} ,{\overline {u}})/\partial p_{j}\;\forall i,j}$

In fact, Marshall's and Hicks' demand functions are themselves functionally connected:

Duality of Marshallian and Hicksian demand function: Let the order of preferences of the consumers be represented and represented by a real-valued utility function that increases continuously, strictly monotonically and strictly quasi-concave . Let there be the Marshallian demand for a good , an expenditure function and an indirect utility function on the income level . Then: ${\displaystyle \mathbb {R} _{+}^{n}}$ ${\displaystyle u}$ ${\displaystyle x_{i}(\mathbf {p} ,y)}$ ${\displaystyle i}$ ${\displaystyle e(\mathbf {p} ,{\overline {u}})}$ ${\displaystyle v(\mathbf {p} ,y)}$ ${\displaystyle y}$

1. ${\displaystyle x_{i}(\mathbf {p} ,y)=x_{i}^{h}(\mathbf {p} ,v(\mathbf {p} ,y))}$
2. ${\displaystyle x_{i}^{h}(\mathbf {p} ,{\overline {u}})=x_{i}(\mathbf {p} ,e(\mathbf {p} ,{\overline {u}}))}$

See also

• Shephard's lemma

literature

• John Richard Hicks : A Reconsideration of the Theory of Value , with RGD Allen, Economica (1934)
• 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 .

Individual evidence

1. ↑ See Jehle / Reny 2011, p. 35.
2. ↑ See Mas-Colell / Whinston / Green 1995, p. 61.
3. ↑ See Mas-Colell / Whinston / Green 1995, p. 61.
4. ↑ See Jehle / Reny 2011, p. 54.
5. ↑ See Jehle / Reny 2011, p. 45.