Expense function

The cost function - in microeconomics ( consumption theory ) is the function of the dependence of the minimum consumer expenditures on the prices of goods and the required (minimum) value of utility or the volume of goods with a given utility. This is a monetary estimate of Hixian demand .

Formal Definition

The dual task of the consumer is to choose such a set of benefits $h$ ${\ displaystyle h}$ so that its utility is not less than a given utility (the utility of a given set $x$ ${\ displaystyle x}$ ), and total expenses $p h$ ${\ displaystyle ph}$ were minimal ( $p$ ${\ displaystyle p}$ - vector of prices for benefits). The solution to this problem is $h(p,x)$ ${\ displaystyle h (p, x)}$ - Hicks demand . Expense function $e(p,x)$ ${\ displaystyle e (p, x)}$ called the dependence of the cost of acquiring a set $h(p,x)$ ${\ displaystyle h (p, x)}$ from $p$ ${\ displaystyle p}$ and $x$ ${\ displaystyle x}$ (or from $p$ ${\ displaystyle p}$ and $u(x)$ ${\ displaystyle u (x)}$ ), i.e:

$e(p,x)=ph(p,x)$ ${\ displaystyle e (p, x) = ph (p, x)}$

Properties

Under some weak assumptions (neoclassical continuous preferences of the consumer), the cost function is a continuous function along the vector $p$ ${\ displaystyle p}$ - concave (convex upward), homogeneous first degree and non-decreasing function. In addition, it can be shown that if the set $x$ ${\ displaystyle x}$ "No worse" set $y$ ${\ displaystyle y}$ in the sense of a loose relationship of preference then $e(p,x)\geqslant e(p,y)$ ${\ displaystyle e (p, x) \ geqslant e (p, y)}$ .

Hicks demand is equal to the partial derivative of the cost function of prices ( Shepard's lemma ).

Expense function

Formal Definition

Properties

See also

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