Econometrics

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Econometrics

• Statistical analysis of economic data
• Basic method of econometric analysis is multiple
  regression, which expresses the dependent
  variable as a linear function of multiple
  dependent variables and an error term (e).
• Continuing with our labor supply example, lets
  say we wanted to measure an individuals labor
  supply (L) as a function of his age (A), net
  wages (W) and how many children s/he had (C).

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Multiple Regression

• Model takes the form of
• L ß1A ß2C ß3W e
• So, if ß1 gt0, what does that mean about the
  effect of age on labor supply?
• Likewise, what does ß2 0 mean about the effect
  of the number of children on labor supply?

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So linear regression fits a line to the gathered
data, so as to observe general trends
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Statistical significance

• Obviously, the line is only an estimation, and as
  such is prone to error.
• A regression coefficient is said to be
  statistically significant if the standard error
  is small relative to the value of the
  coefficient.
• Statistical significance provides a benchmark for
  economic inference.

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Pitfalls of Econometrics

• Omitted variable bias omitting relevant
  variables from the regression can lead to bias in
  the estimated regression coefficients, so you may
  not get the true coefficients.
• Related to this is the problem of finding
  proxies when relevant variables cant be
  quantified. Labor supply is influenced by work
  ethic, but this cant be directly measured.
• May need to distinguish between types of people
  (18 year old women may have different labor
  supply patterns than 45 yr. old men.

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Do numbers lie?

• As you may have noticed, econometrics involves a
  lot of technical work. Because of this, it is
  possible to manipulate the regression model until
  the scientist gets the results that he wants.
  This is bad science, but it happens.
• Ultimately, it is important to take all results
  with a grain of salt, until you have seen the
  study and concluded that it is the result of good
  econometric work.
• Numbers dont lie, but they can be made to say
  what you want to say.

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So, some conclusions

• We have viewed three methods of empirical
  analysis interviews, experiments, and
  econometrics.
• All have some good points, and some bad points.
• Ultimately, it is not possible for empirical work
  to prove anything with 100 certainty. But if the
  methods are sound, we can be reasonably sure that
  what we are saying is correct.

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Tools of Welfare Analysis 1 Indifference Curves
and Budget Constraints

• For graphical simplicity, lets assume an economy
  with two goods donuts and coffee.
• An individual derives utility from the two goods.
  An indifference curve shows all the possible
  combinations of goods that generate utility level
  U0 .
• The entire collection of indifference curves is
  referred to as the indifference map.

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An Indifference Map
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Marginal Rate of Substitution

• The slope of the indifference curve at any given
  point shows the rate at which an individual will
  willingly exchange goods. This rate at which the
  individual is willing to give up good x for good
  y is what economists refer to as the marginal
  rate of substitution (MRSyx).

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MRS A graphical example
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MRS continued

• The above example is one of a decreasing marginal
  rate of substitution As the individual gets more
  of one good (holding the amount of the other
  constant), he is more willing to trade that good
  to get the other.

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The budget constraint

• The budget constraint (budget line) is the line
  that shows the feasible levels of consumption
  given the individuals income. So, if income
  equals 10, and the prices of donuts and coffee
  are 2 and 1 respectively, we get the following
  budget line

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Changes in prices and income

• If the price of coffee were to increase from 1
  dollar to 2 dollars, while donuts remained 2 a
  piece, the relative price of coffee has
  increased.
• Similarly, if income fell to 8, the relative
  prices do not change, but the budget constraint
  shifts inward, as the ability to consume both
  goods decreases. The new budget constraint is
  parallel to the old one.

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Income Change Relative Price Change
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Optimal Behavior

• The individual optimizes by choosing the bundle
  of goods that gives the most utility subject to
  the budget constraint.
• Graphically, this bundle is represented by
  finding the indifference curve that is tangent to
  the budget constraint.

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Optimal consumption
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So, now that weve covered the basics, lets move
on to Welfare Economics

• Lets start with a simple two-person, two-good
  exchange economy. Adam and Eve live in a world
  where fig leaves and apples are the only goods,
  and there is a fixed supply of each. Both Adam
  and Eve begin with some endowment of apples and
  fig leaves, and trade is possible.
• The device we use for analyzing this economy is
  known as the Edgeworth Box.

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Figure 3.1
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Some things to note in The Edgeworth Box

• At point O, Adams consumption of both goods is
  0, and Eve gets everything. At O, Eve gets
  nothing, and Adam gets everything.
• Every point in the box exhausts all resources
  Whatever Adam doesnt consume, Eve does.
• As we move North and East in the box, Adams
  consumption increases
• As we move South and West, Eves consumption
  increases. At point v, Adams consumption of
  apples is line Ox, and his consumption of fig
  leaves is line Ou. Eves consumption of apples is
  line Oy, and her fig leaf consumption is
  represented by line Ow.

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• Now, if Adam and Eve have conventionally shaped
  indifference curves (decreasing MRSs), we can
  draw Adams indifference map, with utility
  increasing as we move northeast.
• Similarly, we can see that Eves utility
  increases as we move Southwest in the Edgeworth
  box. In this sense her indifference curves are
  flipped around.
• If we draw both Adam and Eves indifference maps,
  it would look like this

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Figure 3.2
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• Lets say that we have an arbitrary initial
  allocation, at point g in the Edgeworth box. This
  provides some level of goods to both Adam and
  Eve, and hence gives them utility levels (which
  we will denote Ag and Eg, respectively).
• From this point, we can ask the question, is it
  possible to make Adam and/or Eve better off
  through trade?
• One possibility is point h, which moves Eve along
  her indifference curve, (so shes indifferent)
  and makes Adam better off. Point p is even better
  for Adam, and Eve is still indifferent.

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Figure 3.3
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Pareto efficiency

• Note that at point p, it is no longer possible to
  make Adam or Eve better off without making the
  other worse off. We refer to such a point as
  Pareto efficient. An allocation is Pareto
  efficient if there is no way to increase one
  persons utility without lowering someone elses.
  
• Pareto efficiency is a benchmark for measuring
  the desirability of an allocation. If an
  allocation is not Pareto efficient, then it is
  possible to make at least one person better off
  through trade without making the other person
  worse off.

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Pareto Improvement

• Closely related to Pareto efficiency is the
  notion of a Pareto improvement. From a given
  point g, a new allocation is a Pareto improvement
  if it makes one person better off without harming
  the other person.
• Returning to our example, allocation h is a
  Pareto improvement on g, and Pareto efficient
  point p is a Pareto improvement on h.
• By definition, there can be no Pareto improvement
  on a Pareto efficient point.

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• Is p the only Pareto efficient point in the
  Edgeworth box? No. For instance, starting from g,
  we can find more Pareto-optimal points through
  different combinations of trades.
• For instance, we can choose to make trades such
  that Adam is indifferent, and Eve is better off

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Figure 3.4
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• Or, we could make trades that make both Adam and
  Eve better off

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Figure 3.5
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And on and on.

• Recall that we chose initial allocation g
  arbitrarily. Had we chosen a different initial
  allocation, we could have repeated the process
  and found even more Pareto efficient allocations.
  In fact, there is an large number of Pareto
  efficient allocations. The locus of all of these
  Pareto efficient points is referred to as the
  contract curve.

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Figure 3.7 The Contract Curve
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The Contract Curve Cont.

• At all points on the contract curve, Adam and
  Eves indifference curves are tangent to each
  other. Mathematically, that means that the slopes
  of the indifference curves are equal at all
  Pareto efficient points.
• Recall from earlier that the slope of a persons
  indifference curve is his marginal rate of
  substitution.

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Pareto Efficiency requires that
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Production economy

• In pure exchange economy, assumed supplies of
  commodities were fixed.
• Now consider scenario where quantities can
  change.
• The production possibilities curve shows the
  maximum quantity of figs that can produced with
  any given quantity of apples.

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Figure 3.8
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Production Economy

• For apple production to be increased, fig
  production must necessarily fall.
• The marginal rate of transformation (MRT) (of
  apples for figs (MRTaf) shows the rate at which
  the economy can transform apples to fig leafs.
• It is the absolute value of the slope of the
  production possibilities curve.
• The marginal rate of transformation can be
  written in terms of marginal costs

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Efficiency with Variable Production

• With variable production, efficiency requires

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Efficiency in Variable Production

• If this were not the case, it is possible to make
  one person better off with an adjustment in
  production.
• For instance, if Adams MRSaf 1/3 and the MRTaf
  2/3. Then 2 fig leaves could be produced by
  giving up 3 apples. But Adam would willingly give
  up 3 apples for 1 fig leaf , so society is better
  off by giving up the apples for the 2 fig leaves.
  
• Rewriting the condition for efficiency in terms
  of marginal costs, we then have as a necessary
  condition for efficiency

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Lets consider this fact

• If Adams Marginal rate of substitution was 3/1
  at point g, he would give up 3 fig leaves for one
  apple.
• If Eves marginal rate of substitution was 1/3,
  She would be willing to trade 3 apples for one
  fig leaf.
• So, we can think of several possible trades that
  both would agree to ie Adam gives 3 fig leaves
  for 2 apples from Eve, or Eve gives 3 apples for
  one fig leaf, etc

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But once the marginal rates of substitution are
equal, there is no reason for any more trades

• If Adam and Eves MRSaf are equal (lets say at
  1/1), then the only trade that could occur would
  be 1 apple for 1 fig leaf. Nobody can be made
  better off through trade when the MRSs are
  equal. So there is no Pareto-improvement to be
  found at this point. By definition, we have
  Pareto optimality.

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Welfare Theorems

• Given the concept of Pareto efficiency, it is
  worth asking whether or not a given economy can
  achieve it. The answer depends on the
  assumptions.

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Lets assume for now

• 1. Everyone in the economy acts as a perfect
  competitor nobody has any market power
• 2. Markets exist for all goods in the economy.

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First Fundamental Theorem of Welfare Economics

• Given these assumptions, the First Fundamental
  Theorem of Welfare Economics (FWT) states that an
  economy will reach a Pareto efficient outcome,
  regardless of the initial allocation.
• This implies that a competitive market economy
  can reach an efficient point without government
  intervention.

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Second Fundamental Welfare Theorem

• So, if a market economy is efficient, what role
  does a government play? Quite a small one,
  actually.
• Recall that there are a large number of Pareto
  efficient points on the contract curve. Some of
  these are more equitable than others.

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Utility possibilities.

• Either the northeast or southwest corner of the
  Edgeworth Box is Pareto efficient, but very
  unequal distribution.
• Society may care about more than Pareto
  efficiency.
• the utilities possibilities frontier (UPF),
  derived from the contract curve, shows the
  relationship between Adam and Eves utility.

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Figure 3.10
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What point is best on the UPF?

• We answer this by postulating a social welfare
  function, which incorporates societies attitudes
  regarding equity, as well as efficiency.

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Maximizing the Social Welfare Function subject to
the UPF leads to the socially optimal allocation
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In a sense then, society may have an ideal
allocation in mind

• The market, given initial allocations, may not
  reach this point. The Second Fundamental Welfare
  Theorem (SWT) states that any Pareto efficient
  point (including societys most preferred) can be
  reached through suitably altering the initial
  allocations, and letting people trade freely.

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What does this mean?

• Taken with the FWT, the SWT means that a Market
  economy can, through appropriate lump-sum
  redistribution, achieve any desired
  Pareto-optimal point.
• This is a pretty powerful result, and it would
  seem to suggest that there is only a small role
  for government. Yet, in reality this doesnt hold