204_summer_2009_lecture_7

204_summer_2009_lecture_7 - University of Toronto...

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University of Toronto Department of Economics ECO 204 Summer 2009 Sayed Ajaz Hussain Lecture 7 1 Ajaz Hussain. Department of Economics
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Last Time ± “Black Box” Firm ± Modeling producer technology ² Production function ² Long vs. Short Run ² Representing technology ² Interesting production functions: ² Cobb Douglas production function ² Complements production function ² Linear production function ² Quasi Linear production function ² CES production function 2 Ajaz Hussain. Department of Economics
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Today ± Some interesting production functions ± Graphing iso quant curves ± Marginal rate of technical substitution ² Interpretation and application ± Modeling some technologies ± (Long Run) Cost minimization problem (CMP) ² Optimal choice ² Sensitivity analysis 3 Ajaz Hussain. Department of Economics
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Graphing Iso Quant Curves ± Recall production plots from lecture 6 ± Contours correspond to specific production level ² “iso quant curves” ² “iso” = same ² “quant” = quantity ± To graph these: ² Pick an arbitrary output Q ² Obtain equation with: ² K as “y” variable ² L as “x” variable ² Plot in two dimensions Ajaz Hussain. Department of Economics 4 Output L K
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Some Production Functions ± We will primarily work with: Q = f(L, K) ± There can be other inputs ± Linear production function: ² Q = α L + β K ± Cobb Douglas production function: ² Q = L α K β ± Complements production function: ² Q = min( α L, β K) ± CES production function: ² Q = [ ρ L + ρ K] 1/ ρ Ajaz Hussain. Department of Economics 5 Long run Production Functions
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Example 1: Q = α L+ β K Ajaz Hussain. Department of Economics 6 ¾ Q = α β K ¾ K is “y” variable ¾ L is “x” variable ¾ Re arranging: ¾ Q = α β K → β K = Q – α L K = Q/ β –( α / β )L ¾ Intercept = Q/ β ¾ Slope = ( α / β ) Suppose: α = β = 1 Slope = __, Intercept = __ K L Iso quant Curves
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Example 2: Q = L α K β Ajaz Hussain. Department of Economics 7 K L ¾ Consider: Q = L α K β ¾ Aka: “Cobb Douglas” production ¾ Typical to assume α + β = 1 ¾ Re arranging: ¾ K β = Q / L α K = [ Q / L α ] 1/ β ¾ Example: ¾ α = 1, β = 1 ¾ K = Q/L ¾ Slope = dK /dL ¾ dK/dL = Q/L 2 Iso quant Curves
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Example 3: Q = min( α L, β K) Ajaz Hussain. Department of Economics 8 ¾ Consider: Q = min( α β K) ¾ Example: α = β = 1 ¾ Q = min(L, K) L K Output 111 121 131 311 211 222 232 322 333 343 .. .. ..
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204_summer_2009_lecture_7 - University of Toronto...

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