1. 13. Production
Varian, Chapter 31

2. 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?

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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?

9. PPS with constant returns to scale
Coconuts
MRT is constant
PPS
PPF
Leisure, l

10. Finding the MRT

11. 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

12. 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?

13. 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
fish, f
Exactly analogous to profit maximization

14. 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

15. Example: Production and trade
PPF given by 500 =c2+4f 2
Prices pc=pf=5
What c, f will producer choose?

16. Production and consumption decisions
Self-sufficiency,
or autarky, at
gives lower utility
At optimum
production,
MRT = pl/pc
Coconuts
At optimal
consumption,
MRS = pl/pc
Sales of
coconuts
Production
of coconuts
Production
of lemons
Lemons, l
Purchases of
lemons

17. 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

18. 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

19. 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?

20. 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

21. Constructing an Edgeworth box
Agent B
Lemons, l
Coconuts
Agent A
Edgeworth box
Endowment

22. Inefficient production
Agent B
Lemons, l
Coconuts
Agent A
Extent of productive inefficiency:
A produces too many coconuts
B produces too many lemons
Edgeworth box
Endowment

23. Aggregate production possibilities
If a total of l0 lemons are produced, what is the largest number of coconuts that can be produced?
Coconuts
Agent B
B’s
production
of coconuts
This point must be
on the aggregate
PPF
A’s
production
of lemons c0
B’s
production
of lemons
A’s
production
of coconuts
Agent A
Lemons, l l0

24. Max cA(lA) + cB(lB) s.t. lA + lB = l0
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)

25. Algebra and geometry But this means Max cA(lA) + cB(l0 - lA)
Solution: c’A(lA) = c’B(l0 - lA) = c’B(lB) lA
B’s marginal
cost
A’s marginal
cost
Efficient allocation
of production lA lB l0

26. Constructing the aggregate PPF
Coconuts
Aggregate PPF
Agent A
Lemons, l

27. 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

28. Production efficiency means equal MRTs
Coconuts
Agent B
Aggregate PPF
Agent A
Lemons, l

29. Production inefficiency means unequal MRTs
Coconuts
Agent B
Aggregate PPF
X, an
inefficient
bundle
Each of these
bundles produces
aggregate bundle, X
Agent A
Lemons, l

30. 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

31. Dis-equilibrium prices
Agent B
Coconuts
Aggregate PPF
Excess demand
for lemons
Excess supply
of coconuts
Agent A
Lemons, l

32. 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

33. Equilibrium prices At equilibrium, MRTA = MRTB = MRSA = MRSB
Aggregate PPF
Coconuts
Agent B
Pareto set
Agent A
Lemons, l

34. 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)

35. The solution method Find production as function of p
Using production as endowment, find consumption as function of p
Use feasibility to solve for p
Substitute p back into demand, production decisions

36. 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.

37. Comparative advantage
coconuts
coconuts
lemons
lemons
Agent A
Good at making
coconuts
Agent B
Good at making
lemons

38. Aggregate PPS A makes only coconuts, B makes both coconuts
B makes only lemons
Max # coconuts
B makes only lemons,
A makes both
lemons
Max # lemons

39. Equilibrium coconuts Equilibrium almost certainly
has each agent doing the
thing he is relatively good at
lemons

40. Pinning down the equilibrium prices
coconuts
Endowment
lemons

41. 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.

42. Comparative or absolute advantage?
coconuts
coconuts
lemons
lemons
Agent A
Bad at both, but
better at making coconuts
Agent B
Good at both, but
better at making lemons

43. Equilibrium coconuts Equilibrium still almost certainly has each agent
doing the thing he is
relatively good at
lemons