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Analyzing Robinson Crusoe's Economy: Production, Exchange, and Utility Maximization

This chapter revisits the production exchange economies through the lens of Robinson Crusoe's economy. It explores how an agent, endowed with fixed resources—their 24 hours of time—must decide between labor and leisure. The analysis covers the technologies involved in coconut production, the preferences of Crusoe, and the relationship between labor and profit maximization. We also discuss the implications of the first and second fundamental theorems of welfare economics when production elements are integrated into a general equilibrium framework.

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Analyzing Robinson Crusoe's Economy: Production, Exchange, and Utility Maximization

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  1. Chapter Thirty-Two Production

  2. Exchange Economies (revisited) • No production, only endowments, so no description of how resources are converted to consumables. • General equilibrium: all markets clear simultaneously. • 1st and 2nd Fundamental Theorems of Welfare Economics.

  3. Now Add Production ... • Add input markets, output markets, describe firms’ technologies, the distributions of firms’ outputs and profits …

  4. Now Add Production ... • Add input markets, output markets, describe firms’ technologies, the distributions of firms’ outputs and profits … That’s not easy!

  5. Robinson Crusoe’s Economy • One agent, RC. • Endowed with a fixed quantity of one resource -- 24 hours. • Use time for labor (production) or leisure (consumption). • Labor time = L. Leisure time = 24 - L. • What will RC choose?

  6. Robinson Crusoe’s Technology • Technology: Labor produces output (coconuts) according to a concave production function.

  7. Robinson Crusoe’s Technology Coconuts Production function 24 0 Labor (hours)

  8. Robinson Crusoe’s Technology Coconuts Production function Feasible productionplans 24 0 Labor (hours)

  9. Robinson Crusoe’s Preferences • RC’s preferences: • coconut is a good • leisure is a good

  10. Robinson Crusoe’s Preferences Coconuts More preferred 24 0 Leisure (hours)

  11. Robinson Crusoe’s Preferences Coconuts More preferred 24 0 Leisure (hours)

  12. Robinson Crusoe’s Choice Coconuts Production function Feasible productionplans 24 0 Labor (hours)

  13. Robinson Crusoe’s Choice Coconuts Production function Feasible productionplans 24 0 Labor (hours) Leisure (hours) 24 0

  14. Robinson Crusoe’s Choice Coconuts Production function Feasible productionplans 24 0 Labor (hours) Leisure (hours) 24 0

  15. Robinson Crusoe’s Choice Coconuts Production function Feasible productionplans 24 0 Labor (hours) Leisure (hours) 24 0

  16. Robinson Crusoe’s Choice Coconuts Production function C* L* 24 0 Labor (hours) Leisure (hours) 24 0

  17. Robinson Crusoe’s Choice Coconuts Production function C* Labor L* 24 0 Labor (hours) Leisure (hours) 24 0

  18. Robinson Crusoe’s Choice Coconuts Production function C* Labor Leisure L* 24 0 Labor (hours) Leisure (hours) 24 0

  19. Robinson Crusoe’s Choice Coconuts Production function C* Output Labor Leisure L* 24 0 Labor (hours) Leisure (hours) 24 0

  20. Robinson Crusoe’s Choice Coconuts MRS = MPL Production function C* Output Labor Leisure L* 24 0 Labor (hours) Leisure (hours) 24 0

  21. Robinson Crusoe as a Firm • Now suppose RC is both a utility-maximizing consumer and a profit-maximizing firm. • Use coconuts as the numeraire good; i.e. price of a coconut = $1. • RC’s wage rate is w. • Coconut output level is C.

  22. Robinson Crusoe as a Firm • RC’s firm’s profit is  = C - wL. •  = C - wL  C = + wL, the equation of an isoprofit line. • Slope = + w . • Intercept = .

  23. Isoprofit Lines Coconuts Higher profit; Slopes = + w 24 0 Labor (hours)

  24. Profit-Maximization Coconuts Production function Feasible productionplans 24 0 Labor (hours)

  25. Profit-Maximization Coconuts Production function 24 0 Labor (hours)

  26. Profit-Maximization Coconuts Production function 24 0 Labor (hours)

  27. Profit-Maximization Coconuts Production function C* L* 24 0 Labor (hours)

  28. Profit-Maximization Isoprofit slope = production function slope Coconuts Production function C* L* 24 0 Labor (hours)

  29. Profit-Maximization Isoprofit slope = production function slope i.e. w = MPL Coconuts Production function C* L* 24 0 Labor (hours)

  30. Profit-Maximization Isoprofit slope = production function slope i.e. w = MPL = 1 MPL = MRPL. Coconuts Production function C* L* 24 0 Labor (hours)

  31. Profit-Maximization Isoprofit slope = production function slope i.e. w = MPL = 1 MPL = MRPL. Coconuts Production function C* L* 24 0 Labor (hours) RC gets

  32. Profit-Maximization Isoprofit slope = production function slope i.e. w = MPL = 1 MPL = MRPL. Coconuts Production function C* Given w, RC’s firm’s quantity demanded of labor is L* Labor demand L* 24 0 Labor (hours) RC gets

  33. Profit-Maximization Isoprofit slope = production function slope i.e. w = MPL = 1 MPL = MRPL. Coconuts Production function C* Output supply Given w, RC’s firm’s quantity demanded of labor is L* and output quantity supplied is C*. Labor demand L* 24 0 Labor (hours) RC gets

  34. Utility-Maximization • Now consider RC as a consumer endowed with $* who can work for $w per hour. • What is RC’s most preferred consumption bundle? • Budget constraint is

  35. Utility-Maximization Coconuts Budget constraint 24 0 Labor (hours)

  36. Utility-Maximization Coconuts Budget constraint; slope = w 24 0 Labor (hours)

  37. Utility-Maximization Coconuts More preferred 24 0 Labor (hours)

  38. Utility-Maximization Coconuts Budget constraint; slope = w 24 0 Labor (hours)

  39. Utility-Maximization Coconuts Budget constraint; slope = w 24 0 Labor (hours)

  40. Utility-Maximization Coconuts Budget constraint; slope = w C* L* 24 0 Labor (hours)

  41. Utility-Maximization Coconuts MRS = w Budget constraint; slope = w C* L* 24 0 Labor (hours)

  42. Utility-Maximization Coconuts MRS = w Budget constraint; slope = w C* Given w, RC’s quantity supplied of labor is L* Labor supply L* 24 0 Labor (hours)

  43. Utility-Maximization Coconuts MRS = w Budget constraint; slope = w C* Output demand Given w, RC’s quantity supplied of labor is L* and output quantity demanded is C*. Labor supply L* 24 0 Labor (hours)

  44. Utility-Maximization & Profit-Maximization • Profit-maximization: • w = MPL • quantity of output supplied = C* • quantity of labor demanded = L*

  45. Utility-Maximization & Profit-Maximization • Profit-maximization: • w = MPL • quantity of output supplied = C* • quantity of labor demanded = L* • Utility-maximization: • w = MRS • quantity of output demanded = C* • quantity of labor supplied = L*

  46. Utility-Maximization & Profit-Maximization Coconut and labor markets both clear. • Profit-maximization: • w = MPL • quantity of output supplied = C* • quantity of labor demanded = L* • Utility-maximization: • w = MRS • quantity of output demanded = C* • quantity of labor supplied = L*

  47. Utility-Maximization & Profit-Maximization Coconuts MRS = w= MPL Given w, RC’s quantity supplied of labor = quantity demanded of labor = L* and output quantity demanded = output quantity supplied = C*. C* L* 24 0 Labor (hours)

  48. Pareto Efficiency • Must have MRS = MPL.

  49. Pareto Efficiency Coconuts MRS  MPL 24 0 Labor (hours)

  50. Pareto Efficiency Coconuts MRS  MPL Preferred consumption bundles. 24 0 Labor (hours)

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