# 13. Production Varian, Chapter 31.

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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

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 fish, f 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

PPF given by 500 =c2+4f 2 Prices pc=pf=5 What c, f will producer choose?

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

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

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 box
Agent B Lemons, l Coconuts Agent A Edgeworth box Endowment

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

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

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)

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

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 Aggregate PPF X, an inefficient bundle Each of these bundles produces aggregate bundle, X Agent A 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
Agent B Coconuts 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 At equilibrium, MRTA = MRTB = MRSA = MRSB
Aggregate PPF 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 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

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.

coconuts coconuts lemons lemons Agent A Good at making coconuts Agent B Good at making lemons

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

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.