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

Input, e.g., labor, L

Output, ce.g., coconuts

c = f(L)

Slope = marginal product of labor, f’(L)

Here, f(.) exhibits declining marginal product of labor,or decreasing returns to scale

Constant returns to scale

Input, e.g., labor, L

Output, ce.g., coconuts

c = f(L) = a.Lwhere a is a constant

Here, f(.) exhibits a constant marginal product of labor,or constant returns to scale

Increasing returns to scale

Here, f(.) exhibits increasing marginal product of labor,or increasing returns to scale

Output, ce.g., coconuts

c = f(L)

Input, e.g., labor, L

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

• For a given consumption of leisure, what is the highest number of coconuts that can be produced?

Leisure, l

Coconuts

PPF – ProductionPossibility Frontier

PPS – ProductionPossibility Set

Slope = Marginal rateof transformation, MRT

PPS with constant returns to scale

Leisure, l

Coconuts

PPS

PPF

MRT is constant

Finding the MRT

•  

Subsistence farming

fish, f

Coconuts

u0At optimum,MRS = MRT

PPF

Autarky: Production and consumption decisions are made without trade

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

fish, f

Coconuts

Slope = -pl/pc

Budget Set

Exactly analogous to profit maximization

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

Lemons, l

Coconuts

At optimumproduction,MRT = pl/pc At optimal

consumption,MRS = pl/pc

Purchases oflemons

Sales ofcoconuts

Self-sufficiency,or autarky, at gives lower utility

Productionof lemons

Productionof coconuts

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

Constructing an Edgeworth boxAgent B

Lemons, l

Coc

onut

s

Agent A

Edgeworth box

Endowment

Inefficient production

Edgeworth box

Endowment

Lemons, l

Coc

onut

s

Agent A

Agent BExtent of productive inefficiency:A produces too many coconutsB produces too many lemons

Aggregate production possibilities

• If a total of l0 lemons are produced, what is the largest number of coconuts that can be produced?

Lemons, l

Coconuts

Agent A

Agent B

l0

c0

This point must beon the aggregatePPF

A’sproductionof lemons

B’sproductionof lemons

B’sproductionof coconuts

A’sproduction

of coconuts

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)

• Solution: c’A(lA) = c’B(l0 - lA) = c’B(lB)

lA

lA lB

A’s marginalcost

B’s marginalcost

l0

Efficient allocationof production

Constructing the aggregate PPF

Lemons, l

Coconuts

Agent A

Aggregate PPF

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 meansequal MRTs

Lemons, l

Coconuts

Agent A

Aggregate PPF

Agent B

Production inefficiency means unequal MRTs

Lemons, l

Coconuts

Agent A

Aggregate PPFX, aninefficientbundleEach of these

bundles producesaggregate bundle, X

Agent B

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

Lemons, l

Coconuts

Agent A

Aggregate PPF

Agent B

• Excess demandfor lemons• Excess supplyof coconuts

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• At equilibrium,

MRTA = MRTB = MRSA = MRSB

Lemons, l

Coconuts

Agent A

Aggregate PPF

Agent B

Pareto set

Example: finding equilibrium

• Person B• PPF given by

500= 4cBS2+fB

S2

• uB(cB,fB)=

min{cB,fB}

• Person A• PPF given by

500=cAS2+4fA

S2

• uA(cA,fA)=

min{cA,fA}

Find equilibrium prices (pc,pf),production (cA

S,fAS) and (cB

S,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

lemons

coconuts

lemons

coconuts

Agent AGood at makingcoconuts

Agent BGood at makinglemons

Aggregate PPS

lemons

coconuts

Max # coconuts

Max # lemons

A makes only coconuts,B makes both

B makes only lemons,A makes both

A makes only coconutsB makes only lemons

Equilibrium

lemons

coconuts

Equilibrium almost certainlyhas each agent doing thething he is relatively good at

Pinning down the equilibrium prices

lemons

coconuts

Endowment

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?

lemons

coconuts

lemons

coconuts

Agent ABad at both, butbetter at making coconuts

Agent BGood at both, but better at making lemons

Equilibrium

lemons

coconuts

Equilibrium still almostcertainly has each agentdoing the thing he isrelatively good at