Handout A: Microeconomic Foundation of Macroeconomics
This handout is concerned with microeconomics foundation for
macroeconomics. In particular we focus on the Theory of
Consumer Choice, or consumer theory for short. The goal is to
understand how a consumer makes choices with a given
preference and budget constraint.
Budget Constraint
Consider two goods: Apple (A) and Banana (B).
Denote their quantities by
and
; prices by
and
The budget constraint is
where
denotes the income. Equation (1) can be rewritten as
( )
(
)
We can draw the budget line using (2). We put
vertical axis, and
on horizontal axis.
on the
The intercept of the budget line is____________________
The slope of the budget line is ____________________
The maximum amount of
is
1
The maximum amount of
is
The budget line looks like
Exercise 1:
After falls, the intercept (rises falls) and the absolute value
of slope (rises falls). So the budget line shifts (inward outward)
Exercise 2:
What happens to the maximum amount of
after
falls?
Exercise 3:
What happens to the budget line when Y rises?
Preference
The preference of the consumer is captured by his utility
function
. Denote the marginal utility (MU) by
2
measures the change in utility when
changes by a small
amount. We assume utility rises when either more apples or
more bananas are consumed, i.e.,
Moreover we assume decreasing marginal utility (DMU)
DMU says that as more and more apples (bananas) are
consumed, the additional utility brought by the last apple
(banana) becomes less and less. In other words as more goods
are consumed total utility rises but at decreasing rate.
Intuitively, DMU is due to the fact that people prefer diversity.
Indifference Curve
A consumer’s preference can also be described by indifference
curve. By definition, an indifference curve shows the bundles of
consumption that make the consumer equally happy, or
equivalently, bundles of goods that give the consumer the same
level of utility.
Consider a given level of utility ̅:
̅
Taking total differentiation of (6) yields
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̅
The last equality in (7) is due to the fact that ̅ is fixed when we
move along a given indifference curve. Equation (7) can be
rewritten as
The slope of the indifference curve is
. Equation (8)
shows that the slope is (positive negative), so indifference curve
is (downward upward) sloping. That means the consumer must
consume more bananas as he consumes fewer apples in order to
maintain the level of happiness.
Formally we call the slope of indifference curve
the
marginal rate of substitution (MRS). MRS measures how many
apples the consumer requires to be compensated for a one-unit
reduction in banana consumption. Put differently MRS measures
how many apples a consumer is willing to trade for one unit of
banana. Next we show MRS has certain property.
Suppose
falls, DMU (5) implies that
(rises falls)
To make consumer equally happy,
must (rise fall) when
falls. This change in implies that
(rises falls).
Thus equation (8) indicates that the absolute value of MRS (rises
falls) when
falls and
rises. In other words, the
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indifference curve gets (steeper flatter) when
rises. Now we reach the below conclusion
falls and
The indifference curve is convex (bowed inward).
Intuitively, the indifference curve is convex because people are
more willing to trade away goods that they have in abundance
and less willing to trade away goods of which they have little.
When a person has many apples, he is willing to trade, say, 10
apples for 1 banana. In contrast when he has many bananas, he
is willing to trade 1 apple for 10 bananas.
Optimal Choice
The goal of a consumer is to find the optimal bundle of apples
and bananas (optimum) given his preference and budget.
Mathematically the consumer tries to solve the below constraint
optimization problem:
We can transform this constraint optimization problem into an
unconstraint optimization problem by substituting (2) into the
objective function:
(
)
Taking derivative of (9’) with respect to
and letting the
derivative equal zero lead to the first order condition (FOC):
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Equation (10) gives hints about how to find optimum using
graphs. The answer is the optimum is located where the budget
line is tangent to the indifference curve. To see this,
Step 1: the slope of budget line is___________
Step 2: the slope of indifference curve is_____________
Step 3: at the tangent point, the two slopes ______________
The graph looks like:
Exercise: show that utility is not maximized and can be
increased when (10) is violated. More explicitly, consider the
case where
Step1: utility decreases by ______ when one unit of apple is
given up.
Step 2: on market, one unit of apply can be traded for ___banana.
Step 3: the amount of banana in step 2 can increase utility
by____
Step 4: the reduction of utility in step 1 is (more less) than the
increase of utility in step 3. So overall the utility (rises falls) if
the consumer trades apple for banana.
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Homework: show how to increase utility if
Therefore the FOC (10) has a simple interpretation: at the
optimum, the marginal utility per dollar spent on good A should
equal the marginal utility per dollar spent on good B. If not, the
consumer can increase utility by spending less on the good that
provides lower marginal utility per dollar and more on the good
that provides higher marginal utility per dollar.
Income Effect and Substitution Effect
Now we are ready to analyze the effect of change in price on
consumption. The total effect of price change is the sum of
income effect and substitution effect.
Income effect is the change in consumption that results when a
price change moves the consumer to a higher or lower
indifference curve.
Substitution effect is the change in consumption that results
when a price change moves the consumption along a given
indifference curve to a point with a new marginal rate of
substitution.
For example, suppose the apple price falls.
Step 1: label the tangent point (optimum 1) of budget line and
indifference curve as point 1.
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Step 2. falling apple price causes the budget line to shift (inward
outward).
Step 3. label the tangent point (optimum 2) of new budget line
and new indifference curve as point 2.
The indifference curve where point 2 is located must give
consumer higher utility than point 1. Why?
Now draw a hypothetical budget line that is parallel to the new
budget line and tangent with the old indifference curve at point 3.
The movement from point 1 to point 3 is substitution effect
because we move along a given indifference curve
The movement from point 3 to point 2 is income effect because
we move to a new indifference curve
Consider substitution effect first. The substitution effect on
apple is positive, i.e., consumer buys more apples since apples
become relatively cheap.
Now let’s turn to income effect. Falling price has the same effect
as rising income in terms of giving rise to higher purchasing
power. If we assume apple is a normal good, i.e., its
consumption rises when income rises, then the income effect on
apple is positive.
Because substitution and income effects act in the same
direction (both are positive), consumer buys more apples for
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sure after apple price falls. Hence the demand curve for apple is
downward-sloping.
Exercise: now consider how falling apple price affects banana
consumption.
Step 1: falling apple price makes banana relatively (cheap
expensive) and consumers wants to buy (more less) banana. So
substitution effect on banana is (positive negative).
Step 2: falling apple price makes consumer feels (richer poorer).
If we assume banana is a normal good, then income effect on
banana is (positive negative).
Step 3: income and substitution effects act in (same opposite)
direction for banana. So the total effect on banana can be either
positive or negative (ambiguous), depending on which effect is
bigger (dominates).
Homework An inferior good has negative income effect, i.e., its
consumption falls when income rises. A Giffen good is an
extreme inferior good for which the income effect dominates the
substitution effect. Please draw graph to show as the price of a
Giffen good falls, its consumption falls. This means the demand
curve for Giffen good is upward-sloping.
Application 1 of Consumer Theory: Labor Supply Curve
Here we want to apply the consumer theory to analyze how a
person decides to allocate time between work and leisure.
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In this case the two goods are consumption and leisure. We
assume both are normal goods.
Suppose the real wage rises. This, in effect, makes the price of
consumption to fall. Why?
Warning: you cannot think of wage as the price of leisure. You
earn wage, not pay it.
Now consider the total effect of falling consumption price.
The income effect for consumption is positive
The substitution effect for consumption is positive
So the person definitely has more consumption after his wage
rises.
On the other hand, the income effect for leisure is ______
The substitution effect for leisure is _________
So the total effect of change in wage on leisure is _________.
If we assume substitution effect dominates the income effect,
then the person will have less leisure and will work more hours
after wage rises. Under this assumption, the labor supply curve
is upward-sloping.
Empirical data, however, suggests that the labor supply curve
can be downward-sloping.
Application 2 of Consumer Theory: Intertemporal Choice
Here we focus on how a person makes decision of consumption
vs. saving over time.
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For simplicity, assume a person lives only two periods: t=1
(young) and t=2 (old).
In period 1 the person has income , he consumes and he
saves
. This person borrows if
is negative.
In period 2 the person consumes his current income plus the
saving times interest rate (or paying debt with interest rate)
where denotes the real interest rate. Note that the saving (or
debt) in period two is zero since he dies after the second period.
Formally we call the requirement that in the last period people
have zero saving or debt as No Ponzi-Game Condition.
Equation (11) can be rewritten as
where the term on the left hand side of (12) denotes the present
value of consumption and the right-hand side term the present
value of income.
Equation (12) is the intertemporal budget constraint faced by the
person. The intertemporal constraint says that consumption
should equal income in the present value sense.
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The goal of intertemporal optimization is to find optimal bundle
of and
that satisfies the constraint (12):
It follows that the FOC is
We can think of
as the intertemporal marginal rate of
substitution and
as the relative price of . At optimum
the two terms are equal. Formally equation (14) is called Euler
Equation.
Consider the effect of rising interest rate
Rising
implies falling price for
on saving
.
. Why?
The income effect on is______________
The substitution effect on
is______________
The income effect on is______________
The substitution effect on is______________
So for given the total effect on saving
is ambiguous.
The supply curve of loanable funds can be upward or downward
sloping, just like the labor supply curve.
Homework: derive and interpret (14). Show how to increase
utility if (14) is violated, say,
.
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