Lecture11.ppt - Cost Constraint/Isocost Line COST...

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Cost Constraint/Isocost Line
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COST CONSTRAINT C= wL + rK (m = p 1 x 1 +p 2 x 2 ) w: wage rate (including fringe benefits, holidays, PRSI, etc) r: rental rate of capital Rearranging: K=C/r-(w/r)L
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COST CONSTRAINT L K Represents society’s willingness to trade the factors of production C=wL+rK C/w C/r Slope of the isocost = K/ L= K/ L= (-) w/r (-) w/r
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EQUILIBRIUM L K Y e rK wL C
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EQUILIBRIUM We can either Minimise cost subject to e and C Y Y or Maximise output subject to e and Y C C equilibrium equilibrium
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EQUILIBRIUM L K f Y rK wL L , * rL wL C K L f L ) , ( * Minimise cost subject to output or Maximise output subject to costs Lagrangian method
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Lagrange Method Set up the problem (same as with utility maximisation subject to budget constraint). Find the first order conditions. It is easier to maximise output subject to a cost constraint than to minimise costs subject to an output constraint. The answer will be the same (in essence ) either way.
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Example: Cobb-Douglas Equilibrium Set up the problem rK wL C L AK L a a 1 * Multiply out the part in brackets rK wL C L AK L a a 1 * Derive a demand function for Capital and Labour by maximising output subject to a cost constraint. Let L* be Lagrange, L be labour and K be capital.
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Cobb-Douglas: Equilibrium Find the First Order Conditions by differentiation with respect to K, L and 0 1 1 * r L AaK K L a a 1 st FOC 0 1 * w L K a A L L a a 2 nd FOC 0 * rK wL C L 3 rd FOC
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Cobb-Douglas: Equilibrium Rearrange the 1 st FOC and the 2 nd
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