Transcript Chapter 05

• Chapter 5 Choice
• Budget set + preference → choice
• Optimal choice: choose the best one can
afford. Suppose the consumer chooses
bundle A. A is optimal (A w B for any B
in the budget set) ↔ the set of
consumption bundles which is strictly
preferred to A by this consumer cannot
intersect with the budget set. (月亮形區
域)
• A optimal↔月亮形區域為空.
• A optimal →月亮形區域為空? If not,
then月亮形區域不為空, that means there
exists a bundle B such that B s A and B is
in the budget set. Then A is not optimal.
• A optimal ←月亮形區域為空? All B
such that B s A is not affordable, so for
all B in the budget set, we must have A w
B. Hence A is optimal.
• The indifference curve tangent to the
budget line is neither necessary nor
sufficient for optimality.
• Not necessary: kinked preferences
(perfect complements), corner
solution (vs. interior solution) (!!)
(intuition)
• Not sufficient: satiation or convexity
is violated
sufficient
optimum
necessary
• Not necessary: kinked preferences
(perfect complements), corner
solution (vs. interior solution) (!!)
(intuition)
• Not sufficient: satiation or convexity
is violated
• The usual tangent condition MRS1, 2= -p1/
p2 has a nice interpretation. The MRS is
the rate the consumer is willing to pay for
an additional unit of good 1 in terms of
good 2. The relative price ratio is the rate
the market asks a consumer to pay for an
additional unit of good 1 in terms of good
2. At optimum, these two rates are equal.
(主觀相對價格 vs. 客觀相對價格)
• |MRS1, 2| > p1/ p2, buy more of 1
• |MRS1, 2| < p1/ p2, buy less of 1
• We now know what the optimal choice is,
let us turn to demand since they are
related.
• The optimal choice of goods at some
price and income is the consumer’s
demanded bundle. A demand function
gives you the optimal amount of each
good as a function of prices and income
faced by the consumer.
• x1 (p1, p2, m): the demand function
• At p1, p2, m, the consumer demands x1
• Perfect substitutes: (graph)
u(x1, x2) = x1 + x2
p1 > p2: x1 = 0, x2 = m/ p2
p1 = p2: x1 belongs to [0, m/ p1] and x2 =
(m- p1 x1)/p2
p1 < p2: x1 = m/ p1, x2 = 0
• Perfect complements: (graph)
u(x1, x2) = min{x1, x2}
x1 = x2 = m/ (p1+ p2)
• Neutrals or bads: why spend money on
them?
• Discrete goods (just foolhardily compare)
• Non convex preferences: corner solution
• Cobb-Douglas: u(x1, x2) = a lnx1 + (1-a)
lnx2
|MRS1, 2| = p1/ p2, so (a/x1)/[(1-a)/x2] = p1/
p2. This implies that a/(1-a) = p1x1/ p2x2,
so x1 = am/ p1 and x2 = (1-a)m/ p2. This is
useful if when we are estimating utility
functions, we find that the expenditure
share is fixed.
• Implication of the MRS condition: at
equilibrium, we don’t need to know the
preferences of each individual, we can infer
that their MRS’ are the same. (This has an
useful implication for Pareto efficiency as we
will see later.)
• One small example: butter (price:2) and milk
(price: 1)
• A new technology that will turn 3 units of milk
into 1 unit of butter. Will this be profitable?
• Another new tech that will turn 1 unit of butter
into 3 units of milk. Will this be profitable?
• Choosing taxes: quantity tax and income
tax
• Suppose we impose a quantity tax of t
dollars per unit of x1. budget constraint:
(p1+t) x1 + p2 x2 = m
optimum: (x1*, x2*) so that (p1+t) x1* + p2
x2* = m
income tax R* to raise the same revenue:
R* = t x1*
• optimum at income tax: p1 x1’+ p2 x2’ = m
- R*, so (x1*, x2*) is affordable at the case
of the income tax. hence, (x1’, x2’) w (x1*,
x2*). (graph)
• Income tax better than quantity tax?
two caveats:
one consumer, uniform income tax vs.
uniform quantity tax (think about the
person who does not consume good 1)
tax avoidance or income tax discourages
earning
ignore supply side