Show that a consumer's utility maximising bundle equates the marginal rate of substitution to the ratio of the price goods Utility maximizing...
A consumer's utility bundle is the bundle of consumption preference between which the consumer is indifferent (for graphical economics purposes, it is a matter of indifference to the consumer whether shirts or food are consumed by their budget).
(The ~ represents a subscript notation while ^ represents superscript notation.)
A consumer wants as an objective to maximize utility from the bundles within the constraints of their budget. Budget constrains maximization, bundles and utility. The budget constraint is represented graphically through the equation:
I - p~x(x) + p~y(y)
The marginal rate of substitution, or MRS, is the rate at which the consumer is willing to give up one item in the utility bundle, in other words, the rate at which they are willing to give up food for shirts or shirts for food (or any item of the more practical utility bundles that exist). MRS is represented graphically by this equation:
MRS = p~x/p~y
Maximization of the consumer's utility bundle is graphically represented in the indifference curve where the slope of the constraining budget line equals the slope of the indifference curve. The indifference curve represents maximum utility in utility bundles and is represented graphically by this equation:
u = p^x(p^y)
MRS as the slope of the indifference curve is represented graphically by this equation:
MRS - p~x/p~y