Household optimum

As household Optimum (also budgetary balance or consumption Optimum ) is referred to in the microeconomic household theory that consumption decision of an individual, which it most available of all prefers .

The optimization process is based on the following assumptions:

1. Individuals have preferences with regard to possible combinations of goods, that is, they can basically decide whether they prefer one combination of goods to another or whether they are indifferent (analysis using indifference curve systems ).
2. The individuals make their consumption decisions on the basis of a limited budget (instrumentalization via the budget line ).

Mathematically, the household balance is a maximization under constraints.

Determination of the household optimum

Indifference curves

→ Main article : Indifference curve

Indifference curves: combination of goods A is worse than combination B of goods from the perspective of the individual

An indifference curve (Latin : indifferent : “not differing”) is the amount of all bundles of goods that are rated equally well by the household, which means that the household is indifferent to them. It is assumed that such indifference curves can be obtained by questioning and / or observing the household.

Under certain assumptions (see the main article: indifference curve ) there exist in the two-dimensional (or n-dimensional) real goods space an infinite number of convex indifference curves. Assuming unsaturation ( more is better ), larger bundles of goods are preferred and are therefore on higher indifference curves. Indifference curves then have a negative slope. The indifference curve with point A contains the least of all drawn curves, the curve with point D the most highly estimated combinations. There are an infinite number of others between the curves drawn.

The absolute value of the slope of an indifference curve is called the marginal rate of substitution . It indicates how many units of good 2 an individual must receive if he surrenders a (marginal) unit of good 1 and wants to maintain the original level of utility. ${\displaystyle {\tfrac {dx_{2}}{dx_{1}}}}$

Budget straight

Budget straight line: the household can afford combination D, but not E.

→ Main article : Budget restriction

Let us assume that an individual has an exogenously given income and is faced with a certain vector of prices of consumer goods that are perceived as unaffected. The budget line (also consumption possibility limit , budget restriction , balance line ) then represents all combinations of bundles of goods that the individual can just afford with his income.

If there are goods with prices and if the individual has an income of , then the budget amount is given by ${\displaystyle n>1}$ ${\displaystyle x=(x_{1},\ldots ,x_{n})}$ ${\displaystyle p=(p_{1},\ldots ,p_{n})}$ ${\displaystyle y}$

${\displaystyle B(y,p_{1},\ldots ,p_{n}):=\{x|p\cdot x\leq y\}}$.

The budget line is the subset of in which the weak inequality is satisfied with equality, i.e. i.e., where the income is fully spent. It is shown as an example in the adjacent drawing. Point A is not on the budget line, but below, i.e. This means that not all of the available budget is used for the two available goods. Point E cannot be reached: the budget is insufficient for this (the budget would have to go into debt for this). In point B nothing of good 1 is consumed, only good 2 in an amount of . The reverse applies to point C. In point D, the entire disposable income is spent and distributed between good 1 and good 2. ${\displaystyle B}$ ${\displaystyle n=2}$ ${\displaystyle y/p_{2}}$

The negative slope of the budget line expresses the fact that with a given income, more consumption of good 1 is associated with less consumption of good 2. The budgetary opportunity costs correspond to the price ratio: ${\displaystyle -{\tfrac {p_{i}}{p_{j}}}}$

The household optimum

Household optimum

If one draws the budget line in the above indifference curve system (with an infinite number of indifference curves, of which only a few are given as examples), one recognizes that the household optimum is given by a tangential point. In the household optimum, the slope of the indifference curve is equal to the slope of the budget line:

${\displaystyle {\frac {dx_{2}}{dx_{1}}}=-{\frac {p_{1}}{p_{2}}}}$

In the optimum, the marginal rate of substitution is equal to the price ratio . ${\displaystyle -{\tfrac {dx_{2}}{dx_{1}}}}$ ${\displaystyle {\tfrac {p_{1}}{p_{2}}}}$

Household optimum and utility function

Ordinal utility function

Note that the concept of a (measurable, cardinal) benefit was not required to determine the household optimum and its most important property. Vilfredo Pareto first drew attention to this. Often one wants to describe the preferences or the indifference curves by mathematical functions. To this end, an ordinal utility is introduced in the following way:

Each point in the indifference curve system is assigned a utility index that fulfills the following conditions, but is otherwise arbitrary: ${\displaystyle x}$ ${\displaystyle u(x)}$

1. Two points between which the individual is indifferent, that is, which are on the same indifference curve, receive the same utility index; each indifference curve is thus characterized by a fixed value.

2. If one combination is preferred to another, it receives a higher utility index.

The utility function defined in this way is monotonic (because of the unsaturation) and quasi-concave (because of the convexity of the indifference curves) but not necessarily concave. It is not unique because a monotonous transformation of a possible utility function describes the same indifference curve system.

The first derivation of the utility function according to the amount of one of the consumer goods is called the marginal utility of this good. ${\displaystyle {\tfrac {\partial u}{\partial x_{i}}}}$

It applies to the relationship (see total differential ) ${\displaystyle u(x)}$

${\displaystyle {\rm {d}}u(x_{1},x_{2})={\frac {\partial u}{\partial x_{1}}}dx_{1}+{\frac {\partial u}{\partial x_{2}}}dx_{2}}$

This results in an indifference curve ( ): ${\displaystyle {\rm {d}}u=0}$

${\displaystyle {\frac {dx_{2}}{dx_{1}}}=-{\frac {\frac {\partial u}{\partial x_{1}}}{\frac {\partial u}{\partial x_{2}}}}}$

The marginal rate of substitution of good 2 to good 1 is equal to the negative ratio of marginal utility of good 1 to good 2. Since the marginal rate of substitution in the household optimum is equal to the negative price ratio, results in the household optimum

${\displaystyle {\frac {p_{1}}{p_{2}}}={\frac {\frac {\partial u}{\partial x_{1}}}{\frac {\partial u}{\partial x_{2}}}}}$

This relationship represents Gossen's second law . Note that this relationship does not change with a monotonic transformation of , and therefore does not depend on the chosen representative of the ordinal utility function. ${\displaystyle u(x)}$

example

A household would like to determine its household optimum with an income of euros and the prices of and . The utility function is given by the Cobb-Douglas function Then: ${\displaystyle y=10}$ ${\displaystyle p_{1}=1}$ ${\displaystyle p_{2}=1}$ ${\displaystyle u(x_{1},x_{2})=x_{1}\cdot x_{2}}$

${\displaystyle {\frac {p_{1}}{p_{2}}}={\frac {\frac {\partial u}{\partial x_{1}}}{\frac {\partial u}{\partial x_{2}}}}={\frac {x_{2}}{x_{1}}}}$

and thus

${\displaystyle p_{1}x_{1}=p_{2}x_{2}\quad }$

Inserted into the budget condition you get:

${\displaystyle p_{1}x_{1}+p_{2}x_{2}=y\quad }$

and thus as a demand for good 1 in the household optimum: and accordingly for good 2: ${\displaystyle x_{1}={\tfrac {y}{2p_{1}}}=5}$ ${\displaystyle x_{2}={\tfrac {y}{2p_{2}}}=5}$

Note that a monotonous transformation of the utility function (e.g. ) for given prices and incomes leads to the same household optimum. ${\displaystyle u(x_{1},x_{2})=\ln(x_{1})+\ln(x_{2})\quad }$

Determination of the household optimum with the help of the Lagrangian

Range of benefits

Maximizing utility under secondary conditions - the household optimum

The microeconomic household theory assumes that individuals maximize their utility , whereby their consumption possibilities are limited by the level of the income (which is assumed to be fixed in this context).

This can be formally represented as follows:

${\displaystyle \max _{x_{1},\ldots ,x_{n}}U(x_{1},\ldots ,x_{n})\quad }$ under the secondary condition ${\displaystyle \quad \sum _{i=1}^{n}p_{i}x_{i}\leq y}$

${\displaystyle U}$= Utility function , = consumption quantity of the good , = price of the consumer good , = number of consumer goods, = income. This maximization problem under constraints can be solved with the following Lagrange approach: ${\displaystyle x_{i}}$ ${\displaystyle i}$ ${\displaystyle p_{i}}$ ${\displaystyle i}$ ${\displaystyle n}$ ${\displaystyle Y}$

${\displaystyle {\mathcal {L}}=U(x_{1},\ldots ,x_{n})+\lambda (y-p\cdot x)}$

The first-order conditions (necessary and sufficient under the assumptions made) for an optimal consumption allocation are:

${\displaystyle {\frac {\partial {\mathcal {L}}}{\partial x_{i}}}={\frac {\partial U(x_{1},\ldots ,x_{n})}{\partial x_{i}}}-\lambda p_{i}=0\quad \forall \,i}$

${\displaystyle {\frac {\partial {\mathcal {L}}}{\partial \lambda }}=y-\sum _{i=1}^{n}p_{i}x_{i}=0}$

The last condition is that the entire budget is actually spent. The remaining conditions state that the marginal utility weighted with its price must be the same for every consumption quantity for all goods (namely the Lagrange multiplier). The Lagrange multiplier measures the value ( shadow price ) that the individual attaches to an increase in income (in general: a marginal relaxation of the budget constraint). An optimal consumption plan thus requires that, should there actually be a marginally small increase in income, the individual does not care for which of the goods he would spend this additional income. ${\displaystyle y}$

For every two goods, the optimality conditions require that the marginal rate of substitution is equal to the price ratio: ${\displaystyle i,j}$

${\displaystyle {\frac {\partial U(x_{1},\ldots ,x_{n})}{\partial x_{i}}}/{\frac {\partial U(x_{1},\ldots ,x_{n})}{\partial x_{j}}}={\frac {p_{i}}{p_{j}}}}$.

This is a very intuitive condition: the marginal rate of substitution on the left indicates how much the individual would be willing to give up at most of good if he were to get one more unit of good in return (individual appreciation). The price ratio on the right indicates how much of the good the individual must objectively give up at given prices if he were to acquire one more unit of good . With an optimized consumption plan, individual appreciation and price ratio match. ${\displaystyle i}$ ${\displaystyle j}$ ${\displaystyle i}$ ${\displaystyle j}$

literature

• Varian, HR: Fundamentals of Microeconomics. 6th edition, Munich 2004, pp. 19–93.
• Mankiw: Fundamentals of Economics. 2nd Edition.

Individual evidence

1. ^ Jörg Beutel : Microeconomics. Oldenbourg Verlag, 2006., ISBN 978-3-486-59944-2 (accessed from De Gruyter Online). P. 53