Econ 522 – Dan Quint – Lecture 2 (Sept 6 2007)
Preliminaries: additional papers on syllabus now on electronic reserve through library,
follow link from syllabus online; adding people to class; apologies if today’s lecture
scares anyone.
Today: day 1 of 2 of a very quick review of lots of material you hopefully learned in
Econ 301 (Consumers, Producers, and General Equilibrium), plus a numerical example of
solving for General Equilibrium.
“Standard Model”
Consumers
Preferences
 Consumers have well-defined and coherent preferences
o Ability to rank any two alternatives – either A is better than B, B is better
than A, or you like them both equally
o If A is better than B and B is better than C, then A is better than C
 Preferences are stable
 We don’t worry about where they come from
o Fascinating question from a marketing point of view – how do you get
people to like your product? But for our purposes, we will assume people
are born knowing what they like
o Implicit assumption: preferences independent of the order in which the
choices are presented, what other options are available, and so on
Utility
 With finite alternatives, this is enough to ensure that a utility function exists
o A utility function maps alternatives to numbers, with u(x) > u(y) if and
only if x is preferred to y
o With finite alternatives, just assign a utility of 0 to the one you like least, 1
to the one you like next-least, and so on
o With infinite alternatives, need an additional technical condition, which
will generally be met
 A utility function is any function whose value is higher at bundles that the
consumer prefers
 Suppose our economy consists of apples and oranges, and I always value each
apple as being just as good as two oranges. Then the function ua,o) = 2a + o
represents my preferences. But so does u’(a,o) = 200a + 100o, and u’’(a,o) = 50 +
sqrt(a + 0.5 o), and u’’’(a,o) = f(2a + o), where f is any strictly-increasing function
 Since utility functions are not uniquely determined, it’s meaningless to think
about simply adding them up and maximizing “overall utility” – I could just
multiply my own utility function by a million, in order to become more important
in the overall calculation of who gets what
Choice
 Given a set of alternatives (different bundles of goods), a consumer is assumed to
choose the one that gives him the highest utility
 Now suppose the consumer has some money (say, his income), and each good in
the economy has a price. His set of alternatives is just all the different bundles he
can afford – his “budget set”. (In some models, consumers are endowed with
goods, not money, and can sell these to buy other goods.)
 The consumer’s problem: maximize u(x,y,z) subject to (x,y,z) being in the budget
set, or subject to p1 x + p2 y + p3 z <= w
 Consumers are price-takers – they don’t bargain, their decisions don’t impact the
price of each good in a way they consider – they just take their budget and the
price of each good as a given, and choose the best bundle they can afford
 Continuous (divisible) goods, indifference curves (draw them)
 If preferences are locally non-satiated – the consumer would always prefer to
have a little more of something – he will use up his entire budget
 Assuming goods are available in continuous quantities, then at the consumer’s
optimal choice, the marginal rate of substitution – the ratio at which he’s
indifferent about substituting one good for another – must be equal to the ratio of
the prices if he consumes both at his optimum (draw it)
 Otherwise, the consumer could benefit by selling a little bit of one good and using
the money to buy a little bit of the other
 Or, to put it another way, the marginal benefit of a little bit more of any good,
divided by the price of that good, must be the same across all the goods you
consume
 We can think of the problem in two ways: maximizing utility subject to the
budget constraint, or choosing a target utility level and trying to get there as
cheaply as possible
 How do consumers respond to a change in prices? Consider a drop in the price of
good 1. There are two effects:
o Substitution effect – good 1 is relatively cheaper, so you use more of it to
get to the old level of utility as cheaply as possible
o Income effect – as the price of good 1 drops, you are effectively wealthier,
so you can afford a higher level of utility
 Substitution effect is always to substitute toward the good that is getting relatively
cheaper; but the income effect can go in either direction (as you get wealthier, you
might demand more BMWs and less Mac and Cheese)
 Normal goods – goods you demand more of with more wealth
 Inferior goods – goods you demand less of with more wealth
 Substitutes – goods you consume more of when the price of the other one goes up.
Beer and wine might be substitutes – as the price of wine goes up, you consume
less wine and more beer
 Complements – goods you consume less of when the price of the other goes up.
Beer and pretzels might be complements – as the price of beer goes up, you might
consume less beer and less pretzels
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Draw the graph (indifference curves, budget line), compare a tax on a single good
to a lump-sum tax on wealth
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A few utility functions to know the behavior of…
o Linear – u(x,y,z) = ax + by + cz – consumers will generally spend all their
budget on one good (whichever has the highest value per price)
o Perfect Complements – u(x,y) = min(x,y) – consumers need both products
(in a fixed ratio, here 1-to-1) to get utility – think of left shoes and right
shoes – always buy them in equal amounts
o Cobb-Douglas – u(x,y,z) = xa yb zc, with a+b+c = 1, or u(x,y,z) = a log x +
b log y + c log z. Here, consumers always spend exactly a of their budget
on x, b of their budget on y, c of their budget on z. Goods are neither
complements nor substitutes, as the price of x doesn’t affect how much y
or z I buy! (Draw it in two dimensions)
o Using one good to represent “everything else” – suppose we have beer,
pretzels, and money. Here, the utility from money represents all the other
things we could possibly spend it on except for beer and pretzels. So let
u(b,p,m) = v(b,p) + m. Price of money is always $1.
Labor and Leisure
o In some models, individuals decide how many hours to work, knowing
that more work gives them higher wages and therefore they can buy more
other stuff. In those cases, either labor is a “bad” (enters the utility
function in a negative way), or “leisure” – time spend not working – is a
good
Choosing between consumption now and consumption tomorrow
o We can think of “consumption now” and “consumption tomorrow” as just
being two goods we can consume
o Their relative prices will generally reflect the fact that if we save money
today, we can invest it and have more money tomorrow for consumption
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o So the relative price of consumption tomorrow is 1/1+r – if we spend
1/1+r less today, we invest it and it becomes $1, which we can spend on
consumption tomorrow
If we add up lots of consumers’ demand of a single good as a function of that
good’s price, we get total market demand. In a one-good world, with demand
versus price, consumer surplus is the area above the price line and below the
demand curve. That is, each point on the demand curve represents some
consumer who would be “just barely” willing to buy the product at that price; so
the distance between the price line and that point is the excess benefit (in dollars)
that person gets; adding them all up gives consumer surplus
Producers
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Producers are defined by their technology, which is their ability to turn inputs into
outputs
In some models, inputs are capital and labor, outputs are finished goods – clear
distinction between them
In others, inputs and outputs are just different goods in the economy, and a firm
may have lots of different ways to turn one into another
The firm’s production set is the set of all feasible production plans, that is, all the
different input/output combinations that are technologically feasible
We sometimes assume that the level of certain inputs is fixed in the short-term,
but can be adjusted in the long-term
Firms are price takers – take as given the prices of their inputs and outputs,
choose the production plan that maximizes profits, or revenues minus costs
That is, among all the possible technologically feasible production plans, the firm
chooses the one that maximizes the value of outputs minus the cost of inputs
Price-taking is a big assumption, since firms often have leeway over how to price
their products
o Example where the assumption makes sense – Canadian oil fields
o Bad example – Apple with the iPhone – clearly could sell more for less, or
less for more
Implicit assumptions: firms care only about maximizing profit, price-taking
behavior (which assumes no market power)
Increasing, decreasing, constant returns to scale – if a firm has increasing or
constant returns to scale, then when it maximizes profits, it has to get either 0 or
infinity – so we’ll often assume decreasing returns to scale (or increasing
marginal costs), at least at high levels of production
If a firm has one output but multiple inputs, we can think of its profit
maximization problem in two steps: first choose the level of output that will
maximize overall profits, then choose the combination of inputs that will generate
the output at the lowest possible cost
We can use this cost minimization problem to look solely at the cost to generate a
certain amount of output
Total cost, average cost, and marginal cost curves
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A price-taking firm will choose output to set price = marginal cost, which
coincides with maximizing the area of the rectangle between average cost and
price
In the long run, we expect more firms to enter into markets where firms are
earning profits, and firms to drop out of markets where they are making losses, so
in the long run, we might expect industries to evolve such that the firms in them
are earning zero profits
But these profits are total economic profits – that is, they account for opportunity
costs, the required return on capital, etc.
In industries without free entry, firms in the industry may be able to continue to
earn profits in the long run – these are economic rents
But this means that firms in this industry have a “vested interest” in ensuring that
the industry continues to be uncompetitive – and will be willing to spend money
to keep it that way, such as lobbying Congress and other tactics to protect their
entrenched position
General Equilibrium
 General equilibrium can be considered in a “pure exchange economy” – where
consumers are endowed with goods and trade them with each other – as well as in
a “production economy” – where consumers and firms coexist
 When there are firms that earn positive profits, it is assumed that the consumers
are also shareholders, that is, the profits of the firms are divided up (not
necessarily equally) among some of the consumers in the economy
 Basically, General Equilibrium is characterized by:
o A set of prices for all the goods in the economy
o A production plan for each firm
o A consumption plan for each consumer
 Such that
o Given the prices, each firm is maximizing profits
o Given the prices and their budgets (including any profits distributed by the
firms), each consumer is maximizing utility subject to their budget
constraint
o Every market clears – total consumer demand for each good is exactly
what the economy start with, plus however much is produced (or minus
however much is used up in production)
 With one good, downward-sloping demand, upward-sloping supply, easy to argue
how prices will adjust to clear the market (excess supply  downward pressure
on prices, excess demand  upward pressure). With multiple goods, hard to see
how equilibrium prices will be reached…
 There are fairly general conditions on an economy that will ensure that an
equilibrium exists
 Equilibrium need not be unique – the same economy could have more than one
Example of solving for General Equilibrium
Economy with one consumer, endowed with 1000 pounds of hops. He has tastes for beer
and hops, with a utility function of u(b,h) = 100 log b + h.
There is one firm, with the technology to make x gallons of beer out of x^2/8 pounds of
hops. The consumer is the only shareholder in the firm, so he receives any profits.
Since there are only two goods in the economy, what matters is the ratio of their prices,
not the levels of their prices, so let’s set the price of hops to $1 per pound.
First, consider the firm’s problem, assuming a generic price of $p per gallon of beer and a
price of $1 per pound of hops. The firm maximizes pb – h, subject to the fact that turning
h hops into b beers is technologically feasible, which requires that h >= b^2/8. Since
inputs cost money, the constraint will hold with equality, so this is the same as
maximizing pb – b^2/8. FOC is p – b/4 = 0, or b = 4p. So at prices (p,1), the firm
maximizes profits by creating 4p gallons of beer, out of (4p)^2/8 = 2 p^2 pounds of hops.
The firm sells these 4p gallons of beer for $p per gallon, earning 4p^2 in revenue; the
inputs (2p^2 pounds of hops) cost 2p^2; so the firm’s profit is 2p^2.
Now consider the consumer’s problem. The consumer maximizes 100 log b + h, subject
to b beer plus h hops being affordable, that is, pb + h <= 1000 + 2p^2. (The 1000 is the
money he’d get from selling all the hops he was endowed with; the 2p^2 is the dividend
he’ll get from the firm, since he’s the only shareholder.) Again, the consumer maximizes
utility by spending his entire budget, or setting h = 1000 – pb – 2p^2, so we can rewrite
his problem as maximizing 100 log b + 1000 – pb – 2p^2. This has FOC 100/b – p = 0,
so his demand for beer is 100/p, and he spends whatever is left on hops.
Finally, we need the markets for both beer and hops to clear. Let’s do beer first. At
prices (p,1), the consumer demands 100/p gallons, and the firm supplies 4p gallons. For
these to be equal, 100/p = 4p, or p^2 = 25, or p = 5. So a price of $5 per gallon clears the
market for beer.
Now, let’s see if the same price clears the market for hops. When p = 5, the firm makes
20 gallons of beer, using 20^2/8 = 50 pounds of hops. At prices of (5,1), the firm earns
profits of $5 X 20 - $1 X 50 = $50, which it turns over to the consumer. The consumer
then has a budget of $1050, and demands 20 gallons of beer, at $5 per gallon, so he has
$1050 – 100 = 950 left to spend on hops, so he demands 950 pounds of hops. Total
demand for hops is 950 (consumer) + 50 (producer), and total supply is 1000
(endowment). To put it another way, the consumer starts with 1000 pounds of hops, and
demands to sell 50, which the firm demands to buy. So the market for hops clears.
So the equilibrium of this economy is prices of ($5, $1), a production plan generating 20
gallons of beer out of 50 pounds of hops, and consumer demand for 20 gallons of beer
and 950 pounds of hops.