Transcript 13. Production - Georgetown University

13. Production
Varian, Chapter 31
Making the right stuff
• The exchange economy examined the
allocation of fixed quantities of goods
amongst agents
• Here we examine the production of goods
as well
– How much of each gets produced
– Who produces what
– Does “the market” do things well?
Production functions
Output, c
e.g., coconuts
c = f(L)
Slope = marginal product of labor, f’(L)
Input, e.g., labor, L
Here, f(.) exhibits declining marginal product of labor,
or decreasing returns to scale
Constant returns to scale
Output, c
e.g., coconuts
c = f(L) = a.L
where a is a constant
Input, e.g., labor, L
Here, f(.) exhibits a constant marginal product of labor,
or constant returns to scale
Increasing returns to scale
Output, c
e.g., coconuts
c = f(L)
Input, e.g., labor, L
Here, f(.) exhibits increasing marginal product of labor,
or increasing returns to scale
Definitions
• Increasing returns to scale: production
function f(x) has increasing returns to scale
if f’’(x) > 0
• Constant returns to scale: production
function f(x) has constant returns to scale if
f’’(x) = 0
• Decreasing returns to scale: production
function f(x) has decreasing returns to scale
if f’’(x) < 0
Production terminology
• Production possibility sets: set of all
bundles that can be produced
• Production possibility frontier: set of all
bundles that can be produced such that one
good can only be increased by decreasing
another
• Marginal rate of transformation: (-1*) The
slope of the PPF
From production functions to
production possibility sets
Coconuts
Slope = Marginal rate
of transformation, MRT
PPS – Production
Possibility Set
PPF – Production
Possibility Frontier
Leisure, l
• For a given consumption of leisure, what is the highest
number of coconuts that can be produced?
PPS with constant returns to scale
Coconuts
MRT is constant
PPS
PPF
Leisure, l
Finding the MRT
Subsistence farming
Autarky: Production and consumption decisions
are made without trade
Coconuts
At optimum,
MRS = MRT
u0
PPF
fish, f
Exactly analogous to the utility maximization problem
Example: Production and no trade
• PPF given by 500 =c2+4f 2
• Utility: u(c,f) = c+f
• What c, f will producer/consumer choose?
Production and trade
• As well as producing fish and coconuts, agent can also trade f
for c at prices pf and pc
• Each production choice is like an endowment
Coconuts
Slope = -pl/pc
Budget Set
Exactly analogous to profit maximization
fish, f
Profit maximization
• The market value of a chosen endowment point is
v(c,l) = pcc + pll
• Value is constant along iso-profit lines
pcc + pll = k
or
c = k/pc – (pl/pc)l
• So choosing largest budget set is the same as
maximizing market value, or profit
Example: Production and trade
• PPF given by 500 =c2+4f 2
• Prices pc=pf=5
• What c, f will producer choose?
Production and consumption
decisions Self-sufficiency,
Coconuts
At optimum
production,
MRT = pl/pc
Sales of
coconuts
or autarky, at
gives lower utility
At optimal
consumption,
MRS = pl/pc
Production
of coconuts
Production
of lemons
Purchases of
lemons
Lemons, l
A “separation” result
• Given a PPS and market prices, an agent
should
– Choose production bundle so as to maximize
profits
• This gives him a budget
– Choose best consumption bundle, subject to this
budget constraint
A “separation” result
• Agent owns a firm that produces output
which it sells on the market
– Firm maximizes profit
– Profit goes to shareholder, ie consumer
• Consumer takes profit, uses prices to decide
consumption
• Agents with different preferences should
choose the same production point, but
different purchases with the profit
Example: Production and trade
•
•
•
•
•
PPF given by 500 =c2+4f 2
Prices pc=pf=5
What c, f should they produce?
u(c,f)=min{c,f}
What c, f should they consume?
General Equilibrium with
Production
• Now we introduce a second agent into the
economy
• There are still two goods, coconuts and lemons
• Each agent has a production possibility set
• Both agents make production and trade (i.e.,
consumption) decisions
Coconuts
Constructing an Edgeworth box
Agent B
Edgeworth
box
Endowment
Agent A
Lemons, l
Edgeworth
box
Extent of productive inefficiency:
A produces too many coconuts
B produces too many lemons
Agent B
Coconuts
Inefficient production
Endowment
Agent A
Lemons, l
Aggregate production
possibilities
• If a total of l0 lemons are produced, what is the
largest number of coconuts that can be produced?
Coconuts
c0
Agent B
B’s
production
of coconuts
A’s
production
of lemons
A’s
production
of coconuts
Agent A
l
B’s
production
of lemons
This point must be
on the aggregate
PPF
Lemons, l
Some algebra
• Let cA(lA) be the largest number of coconuts
A can produce if he picks lA lemons.
• Let cB(lB) be the largest number of coconuts
B can produce if he picks lB lemons.
• We want to solve:
Max cA(lA) + cB(lB) s.t. lA + lB = l0
(lA ,lB)
Algebra and geometry
• But this means
Max cA(lA) + cB(l0 - lA)
lA
• Solution: c’A(lA) = c’B(l0 - lA) = c’B(lB)
B’s marginal
cost
A’s marginal
cost
Efficient allocation
of production
lA
lB
l0
Constructing the aggregate PPF
Coconuts
Aggregate PPF
Agent A
Lemons, l
Production efficiency
• Aggregate production is efficient if it is not
possible to make more of one good without
making less of the (an) other
• All points on the aggregate PPF are efficient
• At such points, production is organized so
that the MRT is the same for both agents
Production efficiency means
equal MRTs
Coconuts
Agent B
Aggregate PPF
Agent A
Lemons, l
Production inefficiency means
unequal MRTs
Coconuts
Agent B
Each of these
bundles produces
aggregate bundle, X
Agent A
X, an
inefficient
bundle
Aggregate PPF
Lemons, l
Equilibrium
• Prices pl and pc constitute an equilibrium if:
• When each agent maximizes profits at those
prices,
• ….. and then maximizes utility,
• ….. both markets clear
– i.e, there is no excess demand or excess supply
in either market
Dis-equilibrium prices
Coconuts
Agent B
Aggregate PPF
• Excess demand
for lemons
• Excess supply
of coconuts
Agent A
Lemons, l
Price adjustment
• At these prices, there is
– excess demand for lemons
– excess supply of coconuts
• Lowering pc/pl does two things
– Reduces demand for lemons
– Increases production of lemons
Equilibrium prices
Aggregate PPF
• At equilibrium,
MRTA = MRTB = MRSA = MRSB
Coconuts
Agent B
Pareto set
Agent A
Lemons, l
Example: finding equilibrium
• Person A
• PPF given by
500=cAS2+4fAS2
• uA(cA,fA)=
min{cA,fA}
• Person B
• PPF given by
500= 4cBS2+fBS2
• uB(cB,fB)=
min{cB,fB}
Find equilibrium prices (pc,pf),
production (cAS,fAS) and (cBS,fBS),
and consumption (cA,fA), and (cB,fB)
The solution method
1. Find production as function of p
2. Using production as endowment, find
consumption as function of p
3. Use feasibility to solve for p
4. Substitute p back into demand, production
decisions
Comparative advantage
• If producer A has a lower opportunity cost
to producing good x compared to producer
B, then producer A has a comparative
advantage in producing good x.
• 2 good, 2 producer economy – each
producer has a comparative advantage in
one of the goods.
Comparative advantage
coconuts
coconuts
lemons
Agent A
Good at making
coconuts
lemons
Agent B
Good at making
lemons
Aggregate PPS
coconuts
A makes only coconuts,
B makes both
A makes only coconuts
B makes only lemons
Max # coconuts
B makes only lemons,
A makes both
Max # lemons
lemons
Equilibrium
coconuts
Equilibrium almost certainly
has each agent doing the
thing he is relatively good at
lemons
Pinning down the equilibrium prices
coconuts
Endowment
lemons
Absolute advantage
• If producer A can produce more of good x
for a given set of inputs, compared to
producer B, then producer A has an absolute
advantage in producing good x.
• A single producer may have absolute
advantage in every good.
Comparative or absolute advantage?
coconuts
coconuts
lemons
Agent A
Bad at both, but
better at making coconuts
lemons
Agent B
Good at both, but
better at making lemons
Equilibrium
coconuts
Equilibrium still almost
certainly has each agent
doing the thing he is
relatively good at
lemons