© 1999 - 2004
Douglas A. Ruby
A Consumer Optimum
Tutorial: A Consumer Optimum
The optimization problem facing the consumer can be stated as:
max U = f(X,Y)
PxX + PyY < I
Answers are below
- Drag the green triangle along the horizontal axis to represent
different levels of consumption of 'Apples' and 'Tea'. Watch the numeric
changes in the level of utility and try to find the bundle which
maximizes this level of utility. Once you have identified this bundle,
Click on the 'Indifference Curves' button to visually understand this
- Click on the number boxes for consumer Income
or the Price of X (Apples) and repeat the above step.
Experiment with different values for consumer income and the price of Apples
identifying other optimum consumption bundles.
- Press the Indifference Curves button to see the IC's.
- Press the Reset button to start over.
A. Given a level of consumer income equal to $120.00 and the price of
both goods equal to $2.00, try to detemine the combinations of apples and tea
that will maximize the consumer's level of utility (pay attention to the
Utility (U) value on the screen). How many units of apples and tea are consumed? Comment on
this combination of goods. Is this combination possible given the consumer's level of income?
What is the level of utility?
B. Press Reset and then change the consumer's level of income to $160. Repeat question 'A'.
Is the consumer better or worse off with this $40 change in Income? By how much?
C. Press Reset and now increase the price of apples (good-X) to $4.00. Repeat question 'A'.
Discuss your results with respect to the combination of two goods consumed. Is the consumer better or worse off?
A. The consumer will choose equal amounts of apples and tea (30 units each). This result occurs because the
contribution to utility from each good is the same:
U = AXαYβ, α = β = 0.50)
and the price of the two goods are equal (PX = PY = $2.00. Solving for a Consumer Optimum,
MUX / MUY = [0.5050X-0.5Y0.50 /
0.5050X0.5Y-0.50] = [Y/X] = ($2.00/$2.00) = (PX/PY)
[Y/X] = 1 ... Y = X!
The level of utility in this case is:
U = 50(30)0.5(30)0.5 = 1500.
B. When consumer income increases, the budget constraint shifts outward. The ratio of prices has not
changes nor has the nature of the utility function. In this case, the consumer still consumes equal amounts of
both goods (40 units each) and utility has increased to 2000 (a $40 change in income has led to a 500 unit change in utility --
this ratio [ΔU/ΔI]is the marginal utility of income). The consumer is better off being able to consume more of both
C.An increase in the price of one good will rotate the budget constraint inward. The price of apples is now twice that of tea
(PX / PY = $4 / $2) and yet the contribution to consumer satisfaction is still equal. A consumer optimum will exist where:
[Y/X] = (4/2) or Y = 2X
The new consumer optimum is at 15 units of apples and 30 units of tea. The consumer is worse off since an increase in
the price of a good reduces purchasing power for the consumer.