Corner solution
Encyclopedia
A corner solution is a special solution to an agent
's maximization problem in which the quantity of one of the arguments in the maximized function is zero
. The more usual solution will lie in the non-zero interior at the point of tangency between the objective function and the constraint. For example, in consumer theory
the objective function is the indifference-curve
map (the utility function) of the consumer. The budget line is the constraint. In the usual case, constrained utility is maximized on the budget constraint with strictly positive quantities consumed of both goods. For a corner solution, however, utility is maximized at a point on one axis where the budget constraint intersects the highest attainable indifference curve at zero consumption for one good with all income used for the other good. Furthermore, a range of lower prices for the good with initial zero consumption may leave quantity demanded unchanged at zero, rather than increasing it as in the more usual case.
Alternatively stated, a corner solution is a solution to a minimization or maximization problem where the non-corner solution is infeasible, that is, not in the domain. Instead, the solution is a corner solution on an axis where either x or y is equal to zero. For instance, from the example above in economics, if the maximal utility of two goods is achieved when the quantity of goods x and y are (−2,5), and the utility is subject to the constraint x and y are greater than or equal to 0 (one cannot consume a negative quantity of goods) as is usually the case, then the actual solution to the problem would be a corner solution where x = 0.
A corner solution is an instance where the 'best' solution (i.e. maximizing profit, or utility, or whatever value is sought) is achieved based not on the market-efficient maximization of related quantities, but rather based on brute-force boundary conditions. Such a solution lacks mathematical elegance, and most examples are characterized by externally forced conditions (such as "variables x and y cannot be negative") that put the actual local extrema outside the permitted values.
Agent (economics)
In economics, an agent is an actor and decision maker in a model. Typically, every agent makes decisions by solving a well or ill defined optimization/choice problem. The term agent can also be seen as equivalent to player in game theory....
's maximization problem in which the quantity of one of the arguments in the maximized function is zero
0 (number)
0 is both a numberand the numerical digit used to represent that number in numerals.It fulfills a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures. As a digit, 0 is used as a placeholder in place value systems...
. The more usual solution will lie in the non-zero interior at the point of tangency between the objective function and the constraint. For example, in consumer theory
Consumer theory
Consumer choice is a theory of microeconomics that relates preferences for consumption goods and services to consumption expenditures and ultimately to consumer demand curves. The link between personal preferences, consumption, and the demand curve is one of the most closely studied relations in...
the objective function is the indifference-curve
Indifference curve
In microeconomic theory, an indifference curve is a graph showing different bundles of goods between which a consumer is indifferent. That is, at each point on the curve, the consumer has no preference for one bundle over another. One can equivalently refer to each point on the indifference curve...
map (the utility function) of the consumer. The budget line is the constraint. In the usual case, constrained utility is maximized on the budget constraint with strictly positive quantities consumed of both goods. For a corner solution, however, utility is maximized at a point on one axis where the budget constraint intersects the highest attainable indifference curve at zero consumption for one good with all income used for the other good. Furthermore, a range of lower prices for the good with initial zero consumption may leave quantity demanded unchanged at zero, rather than increasing it as in the more usual case.
Alternatively stated, a corner solution is a solution to a minimization or maximization problem where the non-corner solution is infeasible, that is, not in the domain. Instead, the solution is a corner solution on an axis where either x or y is equal to zero. For instance, from the example above in economics, if the maximal utility of two goods is achieved when the quantity of goods x and y are (−2,5), and the utility is subject to the constraint x and y are greater than or equal to 0 (one cannot consume a negative quantity of goods) as is usually the case, then the actual solution to the problem would be a corner solution where x = 0.
A corner solution is an instance where the 'best' solution (i.e. maximizing profit, or utility, or whatever value is sought) is achieved based not on the market-efficient maximization of related quantities, but rather based on brute-force boundary conditions. Such a solution lacks mathematical elegance, and most examples are characterized by externally forced conditions (such as "variables x and y cannot be negative") that put the actual local extrema outside the permitted values.