Question

Please explain this for me

6) Explain the concept of consumer optimum.

Answer 1

A consumer optimum represents a solution to a problem facing all individuals -- maximizing the satisfaction (utility) from consuming different goods and services subject to the constraint of household income and product prices. This problem can be described as follows:

max U = f(X,Y)

s.t. P_x(X) + P_y(Y) < I

In this problem, the objective function is unobservable leading to the use of assumptions about consumer preferences and diagrammed through the use of indifference curves. From our understanding of the utility function and utility surface we can derive the slope of an indifference curve as:

the Marginal Rate of Substitution = MRS^xy = MU_x/MU_y

However, the all variables and parameters in the budget constraint are observable and thus in defining our consumer optimum, we assume that this optimum lies on this constraint. This budget constraint can be written in several ways. First we can write it as a budget set 'B':

B = {X,Y e R² | X,Y > 0; P_x(X) + P_y(Y) < I₀}

This budget set represents all combinations of the two goods that are attainable to the consumer given his level of income and the the market-determined prices of these goods. Second, we can write it as a budget constraint expressed as an exact equality in intercept-slope form:

Y = I₀/P_y - (P_x/P_y)X

The slope of this budget constraint is a relative price (the price of good-x relative to the price of good-y) where a change in any price, either in absolute or relative terms, will lead to a rotation of this constraint. Both this budget constraint and budget set are shown in figure 1.

figure 1, A Consumer Optimum

The Budget Constraint IC IC IC 0

In this diagram, we can note that many bundles of 'x' and 'y' on IC₀ are within this budget set and thus attainable. However, any point in the interior of the budget set represents an inefficient use of income. Point V on this same indifference curve does represent an efficient use of income however, the consumer can do better. At this point the slope of the budget constraint is greater than the slope of the indifference curve...

P_x/P_y > MU_x/MU_y

or

MU_x/P_x < MU_y/P_y

At this point the marginal utility per dollar spent on good-x is less that the marginal utility per dollar spent on good-y. This consumer can increase his level of satisfaction by reallocating his income to buy more of good-y (thus MU_y will decrease given our assumption of diminishing marginal utility) and buy less of good-x (MU_x increases). This reallocation of income can be seen as a movement along the budget constraint from point V to point R. It is at point R that the consumer has found an optimum on IC₁ where:

MU_x/MU_y = P_x/P_y
or
MRS^xy = P_x/P_y

This is our condition for a consumer optimum. Note that any bundle on IC₂, although providing a greater level of satisfaction, lies entirely beyond the budget set and thus could never include a solution to the problem facing the consumer.