LN0401_2011

LN0401_2011 - CB046/Starr LN04012011 10:21 1 Lecture Notes for April 1 2011 An elementary general equilibrium model The Robinson Crusoe economy

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CB046/Starr LN0401˙2011 March 24, 2011 10:21 1 Lecture Notes for April 1, 2011 An elementary general equilibrium model – The Robinson Crusoe econ- omy Robinson Crusoe is endowed with 168 man-hours per week. On his island there is only one production activity, harvesting oysters from an oyster bed, and only one input to this production activity, Robinson’s labor. q = F ( L ) , (2.1) where F is concave, L is the input of labor, and q is the output of oysters. Denote Robinson’s consumption of oysters by c and his consumption of leisure by R . R = 168 - L, (2.2) u ( c, R ) F p ( · ) > 0 , F pp ( · ) < 0 , ∂u ∂R > 0 , ∂c > 0 , 2 u 2 < 0 , 2 u 2 < 0 , 2 u ∂R∂c > 0 , and that F p (0) = + . Centralized allocation u ( c, R ) = u ( F ( L ) , 168 - L ) . (2.3) We now seek to choose L to maximize u : max L u ( F ( L ) , 168 - L ) . (2.4) The Frst-order condition for an extremum then is d dL u ( F ( L ) , 168 - L ) = 0 . (2.5) That is, u c F p - u R = 0 , (2.6) where u c and u R denote partial derivatives. Hence, at an optimum — a utility maximum subject to resource and technology constraint — we have u R u c = - dq dR = F p . (2.7) Restating (2.7), MRS R,c = - c R | u = constant = u R u c = - dq dR = F p = MRT R,c
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CB046/Starr LN0401˙2011 March 24, 2011 10:21 2 Equations (2.5), (2.6), and (2.7) represent conditions evaluated at the opti- mizing allocation, fulFlling (2.4). Pareto e±cient : the allocation makes technically e±cient use of produc- tive resources (labor) to produce output (that the input-output combination is on the production frontier) the mix of outputs (oysters and leisure) is the best possible among the achievable allocations in terms of achieving household utility. Equation (2.7), which shows the equality of slopes of the production function and the
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This note was uploaded on 11/30/2011 for the course ECON 311 taught by Professor Zambrano during the Fall '08 term at Cal Poly.

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LN0401_2011 - CB046/Starr LN04012011 10:21 1 Lecture Notes for April 1 2011 An elementary general equilibrium model The Robinson Crusoe economy

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