= Budget Constraint = A '''budget constraint''' is a line or plane that limits possible consumer choices. <> ---- == Bivariate == A budget contraint in terms of two goods, ''x'' and ''y'', is characterized as a line. The line is defined by the equality ''xP,,x,, + yP,,y,, = m''; wherein * ''i'' is the quantity of good ''i'' itself * ''P,,i,,'' is the price of good ''i'' * ''m'' is the available income that can be spent The intercepts of this line are ''m/P,,x,,'' and ''m/P,,y,,'' (on the x- and y-axes respectively). The slope of this line is ''-P,,x,,/P,,y,,''. This is generally referred to as the '''marginal rate of transformation''' ('''MRT'''). ---- == Multivariate == A budget constraint is defined by the equality of some income constant ''m'' to the [[LinearAlgebra/VectorMultiplication#Dot_Product|dot product]] of the price and quantity vectors '''''p''''' and '''''q'''''. (Recall that the dot product of two vectors like ''[x y]'' and ''[P,,x,, P,,y,,]'' is ''xP,,x,, + yP,,y,,''.) Also, the budget constraint is [[LinearAlgebra/Orthogonality|orthogonal]] to the price vector. ---- == Non-linear Budget Sets == The above forms of budget constraints are linear. In some cases, the constraint is not constructed in this shape. * Under autarky, the budget constraint conforms to the [[Economics/ProductionPossibilityFrontier|PPF]]. * Rationing creates a piecewise budget constraint; it operates as normal up to some threshold where it becomes constant. * Progressive (and regressive) taxing create complex piecewise budget constraints. * e.g., [[TheEffectsOfThe1MillionYenCeilingInADynamicLaborSupplyModel|Abe's work on means-tested transfers]] ---- == Usage == In neoclassical consumer choice theory, the optimal strategy is to pick the bundle where the budget constraint is tangent to the [[Economics/IndifferenceCurve|indifference curve]]: ''m,,budget,, = m,,indif,,'' ''MRT,,xy,, = -P,,x,,/P,,y,, = -MU,,x,,/MU,,y,,'' ''P,,x,,/P,,y,, = MU,,x,,/MU,,y,,'' ''MU,,x,,/P,,x,, = MU,,y,,/P,,y,,'' The intuitive explanation is that the optimal choice exists where, for all goods, the marginal utility given price is equal, so there is no incentive to substitute. ---- CategoryRicottone