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Production Functions • Profit Maximization and Cost Minimization • Cost Functions • Aggregate Supply • Efficiency (1St and 2Nd FTWE)

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Production Functions • Profit Maximization and Cost Minimization • Cost Functions • Aggregate Supply • Efficiency (1St and 2Nd FTWE) Advanced Microeconomic Theory Chapter 4: Production Theory Outline • Production sets and production functions • Profit maximization and cost minimization • Cost functions • Aggregate supply • Efficiency (1st and 2nd FTWE) Advanced Microeconomic Theory 2 Production Sets and Production Functions Advanced Microeconomic Theory 3 Production Sets • Let us define a production vector (or plan) = ( , , … , ) – If, for instance, > 0, then the firm is producing positive units of good1 22 (i.e., good ∈ 2ℝ is an output). 2 – If, instead, <0, then the firms is producing negative units of good 2 (i.e., good 2 is an input). 2 • Production plans that are technologically feasible are represented in the production set . = : ( ) 0 where ( ) is the transformation function . ∈ ℝ ≤ Advanced Microeconomic Theory 4 Production Sets • ( ) can also be understood as a production function. • Firm uses units of as an input in order to produce units of 1as an output. 2 • Boundary of the production function is any production plan such that = 0. – Also referred to as the transformation frontier. Advanced Microeconomic Theory 5 Production Sets • For any production plan on the production frontier, such that = 0 , we can totally differentiate as follows� � � + = 0 � � solving = � , where � = , � � – , − measures how much the (net) output� can increase if the firm decreases the (net) output of good by� one marginal unit. Advanced Microeconomic Theory 6 Production Sets • What if we denote input and outputs with different letters? = ( , , … , ) 0 outputs = ( , , … , ) 0 inputs 1 2 ≥ where . 1 2 − ≥ • In this case, ≥ inputs are transformed into outputs by the production function, ( , , … , ), i.e., : . Advanced Microeconomic Theory− 7 1 2 − ℝ → ℝ Production Sets • Example: When = 1 (one single output), the production set can be described as ( , , … , , ): = ( , , … , ) −1 −2 −−1 • Holding the output ≤ level1 fixed,2 −1= 0, totally differentiate production function + = 0 ̅ ̅ Advanced Microeconomic Theory 8 Production Sets • Example (continued): and rearranging = � , where � = � � • measures− the additional amount̅ of input that must be used when we marginally decrease̅ the amount of input , and we want to keep output level at = . • in production theory is analogous to the in consumer� theory, ̅ where we keep utility constant,̅ = 0. Advanced Microeconomic Theory 9 Production Sets • Combinations of ( , ) that produce the same total output , i.e.,1 2 z2 ( , : ( 0, ) = } is called isoquant . 0 1 2 1 2 • The slope of the isoquant at ( , ) is . Isoquant, Remember: • 1̅ 2̅ ̅ – refers to isoquants (and production function). z1 – refers to the transformation function. Advanced Microeconomic Theory 10 Production Sets • Example: Find the for the Cobb- Douglas production function , = , where , > 0. 1 2 1 2 • The marginal product of input 1 is , = 1 2 −1 and that of input 2 is 1 2 , 1 = 1 2 −1 Advanced Microeconomic Theory 11 1 2 1 Production Sets • Example (continued): Hence, the is ( ) = = = −1 −(−1) 1 2 2 2 • For instance, for a particular−1 vector− − 1 = 1 2 1 1 , = (2,3), and = = , then ̅ 3 1 1̅ 2̅ = = 12 .5 2 i.e., the slope of the isoquant̅ evaluated at input vector = , = (2,3) is 1.5. Advanced Microeconomic Theory 12 ̅ 1̅ 2̅ − Properties of Production Sets 1) Y is nonempty: We have inputs and/or outputs. 2) Y is closed: The production set includes its boundary points. Advanced Microeconomic Theory 13 Properties of Production Sets 3) No free lunch: No production with no resources. y2 . The firm uses amounts of input Y in order to produce 1 y1 positive amounts of output . 2 Advanced Microeconomic Theory 14 2 y Properties of Production Sets 3) No free lunch: violation y2 2 . The firm produces yY∈∩ R+ positive amounts of Y good 1 and 2 ( > 0 y1 and > 0) without 1 the use of any inputs. 2 Advanced Microeconomic Theory 15 Properties of Production Sets 3) No free lunch: violation y2 . The firm produces Y positive amounts of 2 yY∈∩ R+ good 2 ( > 0) with y1 zero inputs, i.e., = 2 0 . 1 Advanced Microeconomic Theory 16 Properties of Production Sets 4) Possibility of inaction . Firm can choose to be inactive, using no inputs, and obtaining no output as a result (i.e., 0 ). ∈ Advanced Microeconomic Theory 17 Properties of Production Sets 4) Possibility of inaction y2 . Inaction is still Set up non- sunk cost ∈ possible when firms Y 0 Y face fixed costs (i.e., y1 0 ). ∈ y2 Advanced Microeconomic Theory 18 Properties of Production Sets 4) Possibility of inaction y2 . Inaction is NOT Sunk cost 0∉Y possible when firms Y face sunk costs (i.e., y1 0 ). ∉ Advanced Microeconomic Theory 19 Properties of Production Sets 5) Free disposal: if and , then . is less efficient than : ′ ′ ∈ ≤ ∈ –′ Either it produces the same amount of output with more inputs, or less output using the same inputs. Then, also belongs to the firm’s production set. That is, ′the producer can use more inputs without the need the reduce his output: – The producer can dispose of (or eliminate) this additional inputs at no cost. Advanced Microeconomic Theory 20 Properties of Production Sets 5) Free disposal (continued) Advanced Microeconomic Theory 21 Properties of Production Sets 6) Irreversibility . Suppose that y2 and 0. Then, . ∈ y y2 ≠ y1 . -y1 “No way back” y1 -y Y − ∉ Y -y2 ∉ Advanced Microeconomic Theory 22 Properties of Production Sets 7) Non-increasing returns to scale: If , then for any 0,1 . That is, any feasible vector can be scaled ∈ down. ∈ ∈ Advanced Microeconomic Theory 23 Properties of Production Sets 7) Non-increasing returns to scale . The presence of fixed or sunk costs violates non-increasing returns to scale. Advanced Microeconomic Theory 24 Properties of Production Sets 8) Non-decreasing returns to scale: If , then for any 1. That is, any feasible vector can be scaled∈ up. ∈ ≥ Advanced Microeconomic Theory 25 Properties of Production Sets 8) Non-decreasing returns to scale . The presence of fixed or sunk costs do NOT violate non-increasing returns to scale. Advanced Microeconomic Theory 26 Properties of Production Sets • Returns to scale: – When scaling up/down a given production plan = , : . We connect with a ray from− the1 origin.2 . Then, the ratio must be maintained in 2all points along the ray.1 . Note that the angle of the ray is exactly this ratio . 2 Advanced Microeconomic Theory 27 1 Properties of Production Sets 9) Constant returns to scale (CRS): If , then for any 0. ∈ ∈ ≥ ααy,1 if > y2 . CRS is non-increasing y and non-decreasing. ααy,1 if < Y y1 Advanced Microeconomic Theory 28 Properties of Production Sets • Alternative graphical representation of K constant returns to scale: – Doubling and 2 doubles output (i.e., proportionally 1 Q=200 increase in output). Q=100 1 2 L Advanced Microeconomic Theory 29 Properties of Production Sets • Alternative graphical representation of K increasing-returns to scale: – Doubling and 2 increases output Q=300 1 more than Q=200 proportionally. Q=100 1 2 L Advanced Microeconomic Theory 30 Properties of Production Sets • Alternative graphical representation of K decreasing-returns to scale: – Doubling and 2 increases output less Q=200 than proportionally. 1 Q=150 Q=100 1 2 L Advanced Microeconomic Theory 31 Properties of Production Sets • Example: Let us check returns to scale in the Cobb- Douglas production function , = . Increasing all arguments by a common factor , we 1 2 1 2 obtain , = ( ) ( ) = + – When 1+2 = 1, we1 have 2constant returns1 2 to scale; – When + > 1, we have increasing returns to scale; – When + < 1, we have decreasing returns to scale. Advanced Microeconomic Theory 32 Properties of Production Sets • Returns to scale in different US industries (Source: Hsieh, 1995): Industry + Decreasing returns Tobacco 0.51 Food 0.91 Constant returns Apparel and textile 1.01 Furniture 1.02 Electronics 1.02 Increasing returns Paper products 1.09 Petroleum and coal 1.18 Primary metal 1.24 Advanced Microeconomic Theory 33 Properties of Production Sets Homogeneity of the Returns to Scale Production Function = 1 Constant Returns > 1 Increasing Returns < 1 Decreasing Returns Advanced Microeconomic Theory 34 Properties of Production Sets • The linear production function exhibits CRS as increasing all inputs by a common factor yields , = + = + ( , ) • The fixed proportion≡ production function , = min{ , } also exhibits CRS as , = min , = min , ( , ) � Advanced Microeconomic Theory 35 ≡ Properties of Production Sets • Increasing/decreasing returns to scale can be incorporated into a production function ( , ) exhibiting CRS by using a transformation function ( ) , = ( , ) , where >0 � • Indeed, increasing all arguments by a common factor , yields ( ) , = ( , ) = by CRS(of,)� = , = , � , � Advanced Microeconomic Theory � 36 Properties of Production Sets • Hence, – if > 1, the transformed production function , exhibits increasing returns to scale; – if < 1, the transformed production function , exhibits decreasing returns to scale; Advanced Microeconomic Theory 37 Properties of Production Sets • Scale elasticity: an alternative measure of returns to scale. – It measures the percent increase in output due to a 1% increase in the amounts of all inputs ( , ) = , ( , ) � where denotes the common increase in all inputs. – Practice : Show that, if a function exhibits CRS, then it has a scale elasticity of , =1. Advanced Microeconomic Theory 38 Properties of Production
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