EC487 Advanced Microeconomics, Part I: Lecture 1 Leonardo Felli 32L.LG.04 29 September, 2017
Course Outline Microeconomic Theory ◮ Lecture 1: Production Theory and Profit Maximization. ◮ Lecture 2: Cost Minimization and Aggregation. ◮ Lecture 3: Pure Exchange Economy and Walrasian Equilibrium. ◮ Lecture 4: Existence of Walrasian Equilibrium and Welfare Theorems. ◮ Lecture 5: Walrasian Equilibrium with Production and Externalities. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 29 September, 2017 2 / 55
Course Outline (cont’d) Games of Complete Information ◮ Lecture 6: Static games and Nash Equilibrium. ◮ Lecture 7: Nash Theorem and Duopoly. ◮ Lecture 8: Dynamic Games, Subgame Perfection and Bargaining Games. ◮ Lecture 9: Repeated Games. ◮ Lecture 10: The Folk Theorem. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 29 September, 2017 3 / 55
Admin ◮ My coordinates: 32L.4.02, x7525, L.Felli@lse.ac.uk ◮ PA: Emma Taverner, 32L.1.28, E.Taverner@lse.ac.uk. ◮ Office Hours: Friday 11:00-13:00 or by appointment (e-mail L.Felli@lse.ac.uk). ◮ Course Material: available at: http://econ.lse.ac.uk/staff/lfelli/teaching Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 29 September, 2017 4 / 55
Textbooks ◮ Andreu Mas-Colell, Michael Whinston and Jerry Green, Microeconomic Theory , Oxford: Oxford University Press, 1995. ◮ Martin J. Osborne and Ariel Rubinstein, A Course in Game Theory M.I.T. Press, 1994. ◮ Drew Fudenberg and Jean Tirole, Games Theory , Cambridge: M.I.T. Press, 1991. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 29 September, 2017 5 / 55
Examining ◮ This part of the course (together with the Microeconomics part of EC451) will be officially examined in a two hours exam that will take place in the Lent term exam week (2-5 January 2018). ◮ Last year lent term exam is available with the course material and solutions will be posted on week 10 of the current term. ◮ Exclusively for your own benefit there will be a weekly assignment that will be discussed in the following class, starting from week 2. ◮ Again, only for your benefit, two of these assignments will be graded: Assignment 4 due on week 5 and Assignment 10 due on week 11. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 29 September, 2017 6 / 55
Producer Theory ◮ With respect to what you have seen in EC451, we consider now the other side of the market: the production side. ◮ Individual agent = the firm. ◮ The firm ⇐ ⇒ production activity . ◮ The firm produces outputs by using inputs (both measured in terms of flow amounts per unit time). ◮ Both outputs and inputs are services and commodities. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 29 September, 2017 7 / 55
Production Plan ◮ In particular let ◮ y o j = quantity of commodity j produced by the firm as output, ◮ y i j = quantity of commodity j used as input, ◮ z j = y o j − y i j net output/input depending on whether the sign of z j is positive/negative. ◮ Production plan = vector of net outputs and/or inputs of all available commodities z 1 z 2 z = . . . z L Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 29 September, 2017 8 / 55
Net Inputs and Outputs ◮ Without loss of generality we assume that: ◮ the first h commodities are net inputs ◮ while the remaining L − h commodities are net outputs. ◮ Define: x 1 = − z 1 , . . . , x h = − z h , y 1 = z h +1 , . . . , y L − h = z L Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 29 September, 2017 9 / 55
Production Plan and Possibility Set ◮ A production plan is: − x 1 . . . − x h z = y 1 . . . y L − h Definition (PPS) The Production Possibility Set Z ⊂ R L (PPS) is the set of all technologically feasible production plans: the set of all vectors of inputs and outputs that are technologically feasible. The PPS Z provides a complete description of the technology identified with the firm. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 29 September, 2017 10 / 55
One Input One Output Example of one input x and one output y production plan � − x � z = . y ✻ z 2 = y Z ✲ z 1 = x Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 29 September, 2017 11 / 55
Short-run PPS ◮ Sometime is possible to distinguish between: ◮ immediately technologically feasible production plans Z ( x l , . . . , x h ); ◮ and eventually technologically feasible production plans Z . ◮ Consider − x 1 � Z ( x 1 ) = z = − x 2 � x 1 = x 1 � y ◮ For example if the input x 1 is fixed at the level x 1 then we can define a short-run or restricted production possibility set. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 29 September, 2017 12 / 55
Input Requirement Set and Isoquant ◮ A special feature of a technology is the input requirement set: � − x � � � � x ∈ R h V ( y ) = ∈ Z � + y � the set of all input bundles that produce at least y units of output. ◮ We define also the isoquant to be the set: � � � x ∈ V ( y ) and x �∈ V ( y ′ ) , ∀ y ′ > y � x ∈ R h Q ( y ) = � + all input bundles that allow the firm to produce exactly y . Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 29 September, 2017 13 / 55
Two Inputs One Output ✻ x 1 V ( y ) Q ( y ) ❖ ✲ x 2 Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 29 September, 2017 14 / 55
Production Function Consider a technology with only one output Definition Production function in the case of only one output: �� − x � � f ( x ) = sup ∈ Z y ′ y ′ the maximal output associated with the input bundle x . Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 29 September, 2017 15 / 55
Technologically Efficient In general, we can define the technologically efficient production plan z as: Definition � − x � The general production plan z = ∈ Z is technologically y efficient , if and only if there does not exist a production plan � − x ′ � z ′ = ∈ Z such that z ′ ≥ z ( z ′ i ≥ z i ∀ i ) and z ′ � = z . y ′ If z is efficient it is not possible to produce more output with a given input or the same output with less input (sign convention). Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 29 September, 2017 16 / 55
Example: Cobb-Douglas Technology ◮ We definite the Cobb-Douglas technology as 1 x β f ( x 1 , x 2 ) = x α 2 , α > 0 , β > 0 or − x 1 � � y ≤ x α 1 x β Z = − x 2 � 2 y ◮ with isoquant: � � � ( x 1 , x 2 ) ∈ R 2 1 x β � y = x α Q ( y ) = � + 2 ◮ and input requirement set: � � � ( x 1 , x 2 ) ∈ R 2 1 x β � y ≤ x α V ( y ) = � + 2 Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 29 September, 2017 17 / 55
Cobb-Douglas Isoquant ✻ x 1 V ( y ) Q ( y ) ❖ ✲ x 2 Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 29 September, 2017 18 / 55
Example: Leontief Technology ◮ We definite the Leontief technology as f ( x 1 , x 2 ) = min { ax 1 , bx 2 } , a > 0 , b > 0 or − x 1 � Z = − x 2 � y ≤ min { ax 1 , bx 2 } � y ◮ with isoquant: � � � ( x 1 , x 2 ) ∈ R 2 Q ( y ) = � y = min { ax 1 , bx 2 } � + ◮ and input requirement set: � � � ( x 1 , x 2 ) ∈ R 2 V ( y ) = � y ≤ min { ax 1 , bx 2 } � + x 1 = y x 2 = y ◮ where efficiency imposes a , b Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 29 September, 2017 19 / 55
Leontief Isoquants ✻ x 2 = a b x 1 x 2 ✻ ✲ x 1 Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 29 September, 2017 20 / 55
Example: Perfect Substitutes ◮ We definite the technology where inputs are perfect substitutes as f ( x 1 , x 2 ) = ax 1 + bx 2 , a > 0 , b > 0 or − x 1 � Z = − x 2 � y ≤ ax 1 + bx 2 � y ◮ with isoquant: � � � ( x 1 , x 2 ) ∈ R 2 Q ( y ) = � y = ax 1 + bx 2 � + ◮ and input requirement set: � � � ( x 1 , x 2 ) ∈ R 2 V ( y ) = � y ≤ ax 1 + bx 2 � + Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 29 September, 2017 21 / 55
Perfect Substitutes Isoquants ✻ x 2 ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ ❭ x 2 = y b − a ❭ b x 1 ❭ ❭ ❭ ❭ ❭ ❡ ❭ ✻ ❭ ❡ ❭ ❭ ❭ ❭ ❡ ❭ ❭ ❡ ❭ ❭ ❭ ❡ ❭ ❭ ❭ ❡ ❭ ❭ ❡ ❭ ❭ ❡ ❭ ✲ x 1 Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 29 September, 2017 22 / 55
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