204_summer_2009_lecture_8

204_summer_2009_lecture_8 - University of Toronto...

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Unformatted text preview: University of Toronto Department of Economics ECO 204 Summer 2009 Sayed Ajaz Hussain Lecture 8 1 Ajaz Hussain. Department of Economics Last Time ¡ Some interesting production functions ¡ Graphing iso ‐ quant curves ¡ Marginal rate of technical substitution ¢ Interpretation and application ¡ Modeling some technologies ¡ (Long Run) Cost minimization problem (CMP) ¢ Optimal choice ¢ See Appendix A for summary of some cases ¢ Sensitivity analysis of optimal choices 2 Ajaz Hussain. Department of Economics Today ¡ Optimized Cost Function ¢ Sensitivity analysis ¡ (Long run) cost function analysis ¢ Choice of Technology ¢ AC ¢ Returns to scale ¢ A closer look at U ‐ shaped (long run) AC ¢ Functional form of the cost function ¢ linear vs. non ‐ linear ¢ Modeling technological progress ¢ Learning by doing Ajaz Hussain. Department of Economics 3 CMP: Last Time Ajaz Hussain. Department of Economics 4 Set up CMP Solve CMP Demands for Inputs Differentiate wrt P K , P L , q “Messy” Sensitivity Analysis Percentages Sensitivity Analysis Levels Elasticity “Nice” TODAY Optimal Cost CMP: Optimized Cost Function Ajaz Hussain. Department of Economics 5 Demands for Inputs Substitute demands in C = P L L + P K K Yields optimized cost function: C(q) Differentiate wrt P K , P L , q “Messy” Sensitivity Analysis Percentages Sensitivity Analysis Levels Elasticity “Nice” Cobb ‐ Douglas Technology Cost Function ¡ Minimize C = P L L + P K K subject to L α K β = q ¢ L = P L L + P K K – λ [L α K β – q] ¡ Optimal choices from lecture 7 : ¢ L = q 1/( α + β ) [( α / β )(P K /P L )] β /( α + β ) ¢ K = q 1/( α + β ) [( β / α )(P L /P K )] α /( α + β ) ¡ Substitute in C = P L L + P K K: C(q) = q 1/( α + β ) P L α /( α + β ) P K β /( α + β ) [( α / β ) β + ( β / α ) α ] 1/( α + β ) ¢ This is Cost function, C(q) ¢ You can – if you want ‐‐ solve for λ Ajaz Hussain. Department of Economics 6 Cobb ‐ Douglas Technology Cost: Sensitivity Analysis Ajaz Hussain. Department of Economics 7 Parameter q P L P K Examine impact of q, P L and P K on C(q) in percentage terms Interpretation? Example: Cobb ‐ Douglas C(q): Sensitivity Analysis Ajaz Hussain. Department of Economics 8 Parameter q P L P K Suppose: α = β = ½ Examine impact of q, P L and P K on C(q) in percentage terms Cobb ‐ Douglas C(q): Sensitivity Analysis Ajaz Hussain. Department of Economics 9 Parameter q P L P K Examine impact of q, P L and P K on C(q) in levels Cobb ‐ Douglas C(q): Sensitivity Analysis ¡ CMP Lagrangian = P...
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This note was uploaded on 05/02/2011 for the course ECO 204 taught by Professor Hussein during the Fall '08 term at University of Toronto- Toronto.

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204_summer_2009_lecture_8 - University of Toronto...

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