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Robinson Crusoe's Economy Adding Production • There will be a Production Sector – Defined by some production function • And Consumers with preferences and endowments Consumer Surplus – Usually assume they are endowed with labor/leisure and (sometimes) Capital • Consumers own the production firms ECON 370: Microeconomic Theory – So all profits are distributed back to the consumers in some way • Markets and prices Summer 2004 – Rice University – We assume there are factor markets for labor and capital Stanley Gilbert – And consumer markets for the produced good(s) – And market prices for all factors and produced goods Econ 370 - Production 2 Adding Production (cont) Robinson Crusoe’s Economy: Intro • Consumers and firms decide how much to • One agent, RC, endowed w/ a fixed qty of one demand/supply of inputs/outputs based on resource, Time = 24 hrs – Profit maximization for firms • Can use time for – Utility maximization for consumers – labor (production) or – Market prices – leisure (consumption) • We are interested in: • Labor time = L – Efficiency in production – Overall economic efficiency • Leisure time = 24 – L • What will RC choose? Econ 370 - Production 3 Econ 370 - Production 4 1 Robinson Crusoe’s Technology: Graph Robinson Crusoe’s Preferences Technology: Labor produces output (coconuts) • To represent RC’s preferences: according to a concave production function – coconut is a good Coconuts – leisure is a good • Yields standard indifference map with leisure • Yields indifference curves with positive slopes if Production function plot labor (bad) Feasible production plans 0 24 Labor (hours) Econ 370 - Production 5 Econ 370 - Production 6 Robinson Crusoe’s Preferences: Graph Robinson Crusoe’s Preferences: Graph Coconuts Coconuts More preferred More preferred Labor (hours) 0 24 Leisure (hours) 24 0 Leisure (hours) Econ 370 - Production 7 Econ 370 - Production 8 2 Robinson Crusoe’s Choice: Graph Schizophrenic Robinson Crusoe • Now suppose RC is both – a utility-maximizing consumer and Coconuts – a profit-maximizing firm – And decides separately how much to produce or consume C* • Use coconuts as the numeraire good – So, price of a coconut = $1 • RC’s wage rate is w Labor (hours) • Coconut output level is C 024L* Leisure (hours) Econ 370 - Production 9 Econ 370 - Production 10 Robinson Crusoe as a Firm Isoprofit Lines: Graph • RC’s firm’s profit is π = C – wL • Isoprofit line equation is • π = C – wL ⇔ C = π + wL, in C-L space Coconuts • Slope = + w Higher profit C = π + wL (π < π < π ) • Intercept = π 1 2 3 π3 Slopes = + w π2 π1 0 24 Labor (hours) Econ 370 - Production 11 Econ 370 - Production 12 3 Profit-Maximization: Graph Profit-Maximization: Graph • At optimum: • As a firm, at wage w Robinson: – Isoprofit slope = production function slope – demands Labor L* * – That is, w = p × MPL = 1 × MPL = MRPL – Produces C coconuts * * Coconuts –Gets π* = C – wL in dividends from the firm Production function π C* 3 * π2 π Coconuts π1 * 0 24 Labor (hours) 0 L 24 Labor (hours) Econ 370 - Production 13 Econ 370 - Production 14 Utility-Maximization: Introduction Utility-Maximization: Budget Constraint • Now consider RC as a consumer endowed with $π*, who can work for $w per hour • What is RC’s most preferred consumption bundle? • Budget constraint is: Budget constraint: C = π* + wL C = π * +wL π* Coconuts 0 24 Labor (hours) Econ 370 - Production 15 Econ 370 - Production 16 4 Utility-Maximization: Preferences Utility-Maximization: Choice •Given w, – RC’s quantity supplied of labor is L* and – output quantity demanded is C* More preferred C* π* π* Coconuts Coconuts 0 24 Labor (hours) 0 L* 24 Labor (hours) Econ 370 - Production 17 Econ 370 - Production 18 Output and Factor Markets Clear Utility and Profit-Maximization: Graph • Price is determined based on the requirement that all markets clear • That is: Coconuts – quantity output supplied = quantity output demanded – quantity labor demanded = quantity labor supplied C* • If we have: – well-behaved preferences, and – Convex production function Labor (hours) • Then such a price exists 024L* Leisure (hours) Econ 370 - Production 19 Econ 370 - Production 20 5 Pareto Efficiency: Graph Pareto Efficiency: Graph Pareto Efficiency: Must have MRS = MP Common slope ⇒ relative wage rate w results in Pareto efficient outcome Coconuts Coconuts w=MRS = MP MRS ≠ MPL L Preferred consumption C* bundles Labor (hours) Labor (hours) 024Leisure (hours) 024L* Leisure (hours) Econ 370 - Production 21 Econ 370 - Production 22 Fund. Theorems of Welfare Economics Extend to Two-Good Economy • 1st Fundamental Theorem of Welfare Economics • Consider RC economy with two goods – A competitive market equilibrium is Pareto efficient if: – Coconuts and fish – consumers’ preferences are convex – Both require labor to be produced – there are no externalities in consumption or production – Will now consider production possibilities of two good • 2nd Fundamental Theorem of Welfare Economics economy – Any Pareto efficient outcome can be achieved as a competitive market equilibrium if: – the appropriate redistribution of endowments occurs – consumers’ preferences are convex – there are no externalities in consumption or production – firms’ technologies are convex • Note convex firm technology rules out increasing returns to scale Econ 370 - Production 23 Econ 370 - Production 24 6 Production Possibilities: Introduction Production Possibilities: Graph • Resource and technological limitations restrict what an economy can produce • Production possibility set - set of all feasible Production possibility frontier (ppf) output bundles Coconuts • Production possibility frontier - the outer boundary of the production possibility set Production possibility set Fish Econ 370 - Production 25 Econ 370 - Production 26 Production Possibilities: Graph Comparative Advantage: Introduction Marginal rate of transformation (MRT) = PPF’s slope • Two agents, RC and Man Friday (MF) • Assume linear production technologies • RC can produce at most 20 coconuts or 30 fish Coconuts • MF can produce at most 50 coconuts or 25 fish • RC has comparative advantage in producing fish • PPF is concave w.r.t. origin because take advantage of comparative advantages in production Fish Econ 370 - Production 27 Econ 370 - Production 28 7 Comparative Advantage: Graph Combined Graph C C • Two agents, RC and MF Economy 20 • RC can produce at most 20 C 20 coconuts or 30 fish Use RC to produce 70 fish before using MF RC RC 30 F • MF can produce at most 30 C C F 50 50 coconuts or 25 fish 50 50 Use MF to produce coconuts before using RC MF 25 F MF 25 F 30 55 F Econ 370 - Production 29 Econ 370 - Production 30 Comparative Advantage: Graph Comparative Advantage: Graph C Economy Economy 20 C C Using low opp. cost More producers with 70 producers first RC different opp. costs 30 results in a ppf that C F “smooth out” the ppf 50 50 is concave w.r.t the origin 30 55 MF 25 F F F Econ 370 - Production 31 Econ 370 - Production 32 8 Decentralized Coordination Decentralized Coordination • RC and MF run the firm(s) producing coconuts • LRC and LMF are amounts of labor purchased from and fish RC and MF • And receive the dividends distributing the profits • Firm’s profit-maximization problem is to choose C, F, L and L to • RC and MF are also consumers who can sell labor RC MF • Price of coconut = pC maxπ = pCC + pF F − wRC LRC − wMF LMF •Price of fish = pF • RC’s wage rate = wRC •MF’s wage rate = wMF Econ 370 - Production 33 Econ 370 - Production 34 Decentralized Coordination Decentralized Coordination maxπ = pCC + pF F − wRC LRC − wMF LMF Coconuts Higher profit Isoprofit line equation is constantπ = p C + p F − w L − w L C F RC RC MF MF pF Slopes = − pC which rearranges to π + w L + w L p C = RC RC MF MF − F F pC pC 142444 43444 { intercept slope Fish Econ 370 - Production 35 Econ 370 - Production 36 9 Decentralized Coordination Decentralized Coordination Coconuts Coconuts Profit-max. plan Profit-max. plan The firm’s production Competitive markets possibility set and profit-maximization p MRT = − F pC Fish Fish Econ 370 - Production 37 Econ 370 - Production 38 Decentralized Coordination Coordinating Production and Consumption • Competitive markets, profit-maximization, and • The PPF contains many technically efficient utility maximization all together cause output bundles • Which are Pareto efficient for consumers? pF MRPT = − = MRS • As before, those where pC – If both consumers and producers face the same price, • This is the condition necessary for a Pareto – Markets clear optimal economic state • Alternatively, – Given that markets clear, – Both consumers and producers face the same price Econ 370 - Production 39 Econ 370 - Production 40 10 Coordinating Production and Consumption Coordinating Production and Consumption Suppose for given prices The production bundle forms Coconuts Coconuts output bundle (F’, C’) an Edgeworth Box for the O C′ Maximized profits C′ MF two consumers, with wages and dividends forming the endowment F′ Fish ORC F′ Fish Econ 370 - Production 41 Econ 370 - Production 42 Coordinating Production and Consumption Coordinating Production and Consumption In this case, we do not have • Efficient coordination of production and Coconuts an equilibrium, since the consumption requires: O C′ MF price to consumers and dx dx producers is different. MRS = 2 (cons) = 2 (prod) = MRT dx1 dx1 • Or, could reallocate and make both individuals better off • MRS = MRT necessary for Pareto optimality ORC F′ Fish Econ 370 - Production 43 Econ 370 - Production 44 11 Coordinating Production and Consumption Coconuts C′ OMF ORC F′ Fish Econ 370 - Production 45 12.
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