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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.

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Presentation on theme: "13. Production Varian, Chapter 31. Making the right stuff The exchange economy examined the allocation of fixed quantities of goods amongst agents Here."— Presentation transcript:

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 Input, e.g., labor, L Output, c e.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

4 Constant returns to scale Input, e.g., labor, L Output, c e.g., coconuts c = f(L) = a.L where a is a constant Here, f(.) exhibits a constant marginal product of labor, or constant returns to scale

5 Increasing returns to scale Here, f(.) exhibits increasing marginal product of labor, or increasing returns to scale Output, c e.g., coconuts c = f(L) Input, e.g., labor, L

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 For a given consumption of leisure, what is the highest number of coconuts that can be produced? Leisure, l Coconuts PPF – Production Possibility Frontier PPS – Production Possibility Set Slope = Marginal rate of transformation, MRT

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

10 Finding the MRT

11 Subsistence farming fish, f Coconuts u0u0 At optimum, MRS = MRT PPF Autarky: Production and consumption decisions are made without trade Exactly analogous to the utility maximization problem

12 Example: Production and no trade PPF given by 500 =c 2 +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 p f and p c Each production choice is like an endowment fish, f Coconuts Slope = -p l /p c Budget Set Exactly analogous to profit maximization

14 Profit maximization The market value of a chosen endowment point is v(c,l) = p c c + p l l Value is constant along iso-profit lines p c c + p l l = k or c = k/p c – (p l /p c )l So choosing largest budget set is the same as maximizing market value, or profit

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

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

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 =c 2 +4f 2 Prices p c =p f =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 Edgeworth box Endowment Lemons, l Coconuts Agent A Agent B Extent of productive inefficiency: A produces too many coconuts B produces too many lemons

23 Aggregate production possibilities If a total of l 0 lemons are produced, what is the largest number of coconuts that can be produced? Lemons, l Coconuts Agent A Agent B l0l0 c0c0 This point must be on the aggregate PPF A’s production of lemons B’s production of lemons B’s production of coconuts A’s production of coconuts

24 Some algebra Let c A (l A ) be the largest number of coconuts A can produce if he picks l A lemons. Let c B (l B ) be the largest number of coconuts B can produce if he picks l B lemons. We want to solve: Max c A (l A ) + c B (l B ) s.t. l A + l B = l 0 (l A,l B )

25 Algebra and geometry But this means Max c A (l A ) + c B (l 0 - l A ) Solution: c’ A (l A ) = c’ B (l 0 - l A ) = c’ B (l B ) lAlA lAlA lBlB A’s marginal cost B’s marginal cost l0l0 Efficient allocation of production

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

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 Lemons, l Coconuts Agent A Aggregate PPF Agent B

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

30 Equilibrium Prices p l and p c 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 Lemons, l Coconuts Agent A Aggregate PPF Agent B Excess demand for lemons Excess supply of coconuts

32 Price adjustment At these prices, there is –excess demand for lemons –excess supply of coconuts Lowering p c /p l does two things –Reduces demand for lemons –Increases production of lemons

33 Equilibrium prices At equilibrium, MRT A = MRT B = MRS A = MRS B Lemons, l Coconuts Agent A Aggregate PPF Agent B Pareto set

34 Example: finding equilibrium Person B PPF given by 500= 4c B S2 +f B S2 u B (c B,f B )= min{c B,f B } Person A PPF given by 500=c A S2 +4f A S2 u A (c A,f A )= min{c A,f A } Find equilibrium prices (p c,p f ), production (c A S,f A S ) and (c B S,f B S ), and consumption (c A,f A ), and (c B,f B )

35 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

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

38 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 coconuts B makes only lemons

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

40 Pinning down the equilibrium prices lemons coconuts Endowment

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? lemons coconuts lemons coconuts Agent A Bad at both, but better at making coconuts Agent B Good at both, but better at making lemons

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


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