Marshallian demand function
The Marshallian demand function (also: Walrasian demand function ), named after the economist Alfred Marshall, is a term from microeconomics, specifically the budget theory. Starting from a household that maximizes its utility function is the Marshallian demand function to those quantities of goods to which maximize the benefit of the household. These optimal amounts of goods depend on the individual prices of the goods as well as on the available assets of the household.
- 4.1 output problem
- 4.2 Indirect utility function
Definition and meaning
Construction
General construction of Marshallian demand
It is initially based on a utility maximization problem by
Is given, with a continuous, strictly increasing, differentiable and strictly quasikonkave utility function is. is the vector of nachgefragen quantities of goods and the corresponding price vector.
In the above problem is the benefits to the individual from the demand of goods it gets maximized, which can not be more than a set budget to spend ( budget constraint ). The solution of such a utility maximization problem is intended a function that indicates what amount should be requested from the respective goods in order to ensure a high level of utility with the given budget. It is therefore a function of the price vector and the available budget.
This is referred to as as given Marshallian demand and agreed
Simplified notation the example of the two-goods case
Often the concept is only on smaller number of goods -related ( for example as below two goods ). One can then write the maximization problem waiving vector representations simple as:
As is maximized and the solution of the problem will issue an optimum value for both variables. The optimal demand for good 1 is so and it is dependent on the price of this good, the income available to the individual are available, and usually also by the price of good 2 Intuitively, the latter format can for example be seen in the fact that the utility-maximizing demand for cars is certainly also depends on whether a train ticket costs 500 euros or 5 euros. Consequently, arising from the optimization problem, optimal values for the two goods: ( the Marshallian demand for good 1) and analog ( the Marshallian demand for good 2).
Also see the below sample calculation.
Solution of the utility maximization problem
Formal solution via Karush -Kuhn -Tucker theorem
Specifically, the above- mentioned general utility maximization problem can be solved using the Karush -Kuhn -Tucker conditions. Let be a solution of the maximization problem and if it was necessary. The Lagrangian function is
Then a ( Lagrange multiplier ) exists such that the following first-order conditions satisfied:
With Condition 2 is due to the strict monotonicity of the utility function satisfied with equality, which implies that the constraint is binding ( and Condition 3 is trivially satisfied ). Condition 4 is also guaranteed because there in Condition 1 for all according to condition and for all, must also apply that even. By Condition 1 shows, moreover, that for the price ratio between any two goods in the optimum, for example, is that it is just the ratio of the marginal utility - the so-called marginal rate of substitution between the two goods - must correspond to:
In addition, the marginal rate of substitution between any goods shall be one. This is clear intuitively, for because then equal could be the consumer a unit of the more expensive Good substitute for one unit of the lower without changing its level of utility - but would from the transaction additional funds freely, with whom he could in turn purchase goods ( what its benefits would increase ). But this contradicts the assumption that it has previously traded at a benefit maximum.
Simplified solution without inequality restriction
A frequently made way to simplify the optimization problem is the ex ante waiver of the restriction inequality in the constraint. The problem then
So simplified two-goods case
To ensure that only one request is actually already implemented at the beginning, which was also made up ( but would theoretically dispensable ). The justification for this step is given by the fact that the budget has indeed only the two goods 1 and 2 are available; but then it is obviously not efficient, not spending comprehensive income for the consumption of 1 and 2, if you, as here, assumed positive marginal utility of 1 and 2. The resulting Lagrangian is the same as above, however, the constraints reduce to condition 1 and 2
Transition to the indirect utility function
Substituting the Marshallian demand received back to the original utility function, we obtain a utility function that depends on the prices of goods and income. They are called indirect utility function and agreed. The indirect utility function, for a given Marshallian demand at the concrete level of utility that can be achieved at a maximum.
Properties
It can be shown that under the given conditions and under the additional for the second property, provided that the preference ordering underlying satisfies the property of unsaturation, has among others the following properties:
- Homogeneity of degree zero in. and.
- Walras 's law.
- Convex set. is a convex set.
The Marshallian demand function for a particular price vector and a specific level of income is also differentiable under the given conditions if and only if i) is twice continuously differentiable on the, ii ) holds for some () and iii ), which edged Hessian matrix at the point is.
Related to the Hicksian demand function
While the Marshallian demand as shown results from the utility maximization problem of the household and the amount of goods - depending on the prices of goods - indicating that is required to achieve the highest possible level of utility with a given income, results in the Hicksian demand from the output minimization problem of budget and are the volume of goods - depending on the freight fares - which is necessary to the lowest possible cost to achieve a given level of utility.
Between marshall shear and Hicks'scher demand, however, despite the conceptual difference is a close functional connection, reference is made to the main article on the aforesaid.
Example, in the two-goods case
Initial problem
Consider a market for apples ( Good 1 ) and bananas (Good 2), the quantities demanded is denoted by or. The price amounts to an apple, the banana one. The household budget is, and he consume only apples and bananas. The benefits of the budget follows a Cobb -Douglas utility function. Then
The utility maximization problem. It solves this problem with inequality restriction by using the Karush -Kuhn -Tucker method. The Lagrangian is
Necessary conditions for the optimum benefits are:
Condition 3 would be done with enough, but this would imply (which is the assumption of positive marginal utility of both goods contrary ) so it is expected that the budget constraint is binding (ie - the entire consumption budget is spent on the two goods ). From Condition 1 and 2 then follows by dividing
And therefore ( filled with equality) because of the binding budget condition
This results in turn.
In the household optimum so 8 apples and 6 bananas are in demand.
Indirect utility function
The indirect utility function is given in the example
For and the dark colored top expressions can potentially be used; however, must come from the expression for the first still be eliminated ( the indirect utility function should not be dependent on the optimal quantity of goods explicitly ):
This now applies
The indirect utility function is given to the prices of goods and income, the maximum possible level of utility. You can check what the outcome with the values above agreed for, and delivers accordingly, and thus finds that this
And in fact applies to the above obtained optimal quantities of goods and: