Advanced Microeconomics
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1 Advanced Microeconomics Ivan Etzo University of Cagliari Dottorato in Scienze Economiche e Aziendali, XXXIII ciclo Ivan Etzo (UNICA) Lecture 1: Technolgy 1 / 61
2 Overview 1 Firms behavior and technology 2 Production Plan and Technology set 3 Isoquants 4 Marginal (Physical) Products 5 Returns-to-Scale 6 The technical rate of substitution (TRS) 7 The elasticity of substitution 8 The Constant Elasticity of Substitution (CES) production function 9 Well-Behaved Technologies Ivan Etzo (UNICA) Lecture 1: Technolgy 2 / 61
3 Firms behavior and technological constraints The firms behavior is expressed by the choices made by the firms Choices are not free, there are some constraints coming from: Customers (Demand); Competitors (Market structure); Nature (Technology); Ivan Etzo (UNICA) Lecture 1: Technolgy 3 / 61
4 Technology At the moment the firm makes the production decisions it faces some technological constraints The technology describes the feasible ways to produce the outputs from inputs at a certain point in time ( t ). Inputs or factors of production: land, labor, capital, raw materials (e.g. energy) Often, different technologies will produce the same product Usually capital and labor are considered as the two main broad categories of inputs Ivan Etzo (UNICA) Lecture 1: Technolgy 4 / 61
5 Defining capital Capital goods (or Physical capital): inputs which are themselves an output (i.e. they have been produced too). Financial capital: the amount of money which is required to start (or maintain) a business. Ivan Etzo (UNICA) Lecture 1: Technolgy 5 / 61
6 The production set Technological constraints imply that a given amount of output can be produced only with certain combination of inputs. The production set represents all combinations of inputs and outputs that are technologically feasible. Which technology is best? How do we compare technologies? Ivan Etzo (UNICA) Lecture 1: Technolgy 6 / 61
7 Input bundles x i denotes the amount used of input i ( i.e. the level of input i). An input bundle is a vector of the input levels; (x 1, x 2,, x n ). E.g. (x 1, x 2, x 3,,, x n ) = (6, 0, 9,, 3). Ivan Etzo (UNICA) Lecture 1: Technolgy 7 / 61
8 Production Functions y denotes the output level. The technology s production function states the maximum amount of output possible from an input bundle. y = f (x 1, x n ) Inputs and outputs are usually measured in flow units (i.e., hours worked per week, machines hours per week units of output per week) y t = f (x 1t, x nt ) Ivan Etzo (UNICA) Lecture 1: Technolgy 8 / 61
9 Production Functions One input, one output Ivan Etzo (UNICA) Lecture 1: Technolgy 9 / 61
10 Technology Sets A production plan is an input bundle and an output level; (x 1,, x n, y). A production plan is feasible if y f (x 1, x n ) The collection of all feasible production plans is the technology set. Ivan Etzo (UNICA) Lecture 1: Technolgy 10 / 61
11 Technology Set One input, one output Ivan Etzo (UNICA) Lecture 1: Technolgy 11 / 61
12 Technology Set One input, one output Ivan Etzo (UNICA) Lecture 1: Technolgy 12 / 61
13 Technologies with Multiple Inputs What does a technology look like when there is more than one input? The two input case: Input levels are x 1 and x 2. Output level is y. Suppose the production function is y = f (x 1, x 2 ) = 2x 1/3 1 x 1/3 2 Ivan Etzo (UNICA) Lecture 1: Technolgy 13 / 61
14 Technologies with Multiple Inputs E.g. the maximal output level possible from the input bundle (x 1, x 2 ) = (1, 8) is y = f (x 1, x 2 ) = 2 1 1/3 8 1/3 = = 4 Ivan Etzo (UNICA) Lecture 1: Technolgy 14 / 61
15 Technologies with Multiple Inputs Ivan Etzo (UNICA) Lecture 1: Technolgy 15 / 61
16 Technologies with Multiple Inputs The y output unit isoquant is the set of all input bundles that yield exactly the same output level y. Ivan Etzo (UNICA) Lecture 1: Technolgy 16 / 61
17 Isoquants with Two Variable Inputs Isoquants can be graphed by adding an output level axis and displaying each isoquant at the height of the isoquants output level. Ivan Etzo (UNICA) Lecture 1: Technolgy 17 / 61
18 More isoquants tell us more about the technology. Ivan Etzo (UNICA) Lecture 1: Technolgy 18 / 61
19 More isoquants tell us more about the technology. Ivan Etzo (UNICA) Lecture 1: Technolgy 19 / 61
20 Technologies with Multiple Inputs The complete collection of isoquants is the isoquant map. The isoquant map is equivalent to the production function each is the other. E.g. y = f (x 1, x 2 ) = 2x 1/3 1 x 1/3 2 Ivan Etzo (UNICA) Lecture 1: Technolgy 20 / 61
21 Technologies with Multiple Inputs Ivan Etzo (UNICA) Lecture 1: Technolgy 21 / 61
22 Technologies with Multiple Inputs Ivan Etzo (UNICA) Lecture 1: Technolgy 22 / 61
23 Technologies with Multiple Inputs Ivan Etzo (UNICA) Lecture 1: Technolgy 23 / 61
24 Technologies with Multiple Inputs Ivan Etzo (UNICA) Lecture 1: Technolgy 24 / 61
25 Technologies with Multiple Inputs Ivan Etzo (UNICA) Lecture 1: Technolgy 25 / 61
26 Technologies with Multiple Inputs Ivan Etzo (UNICA) Lecture 1: Technolgy 26 / 61
27 Technologies with Multiple Inputs Ivan Etzo (UNICA) Lecture 1: Technolgy 27 / 61
28 Technologies with Multiple Inputs Ivan Etzo (UNICA) Lecture 1: Technolgy 28 / 61
29 Technologies with Multiple Inputs Ivan Etzo (UNICA) Lecture 1: Technolgy 29 / 61
30 Technologies with Multiple Inputs Ivan Etzo (UNICA) Lecture 1: Technolgy 30 / 61
31 Technologies with Multiple Inputs Ivan Etzo (UNICA) Lecture 1: Technolgy 31 / 61
32 Technologies with Multiple Inputs Ivan Etzo (UNICA) Lecture 1: Technolgy 32 / 61
33 Technologies with Multiple Inputs Ivan Etzo (UNICA) Lecture 1: Technolgy 33 / 61
34 Technologies with Multiple Inputs Ivan Etzo (UNICA) Lecture 1: Technolgy 34 / 61
35 Technologies with Multiple Inputs Stop Ivan Etzo (UNICA) Lecture 1: Technolgy 35 / 61
36 Cobb-Douglas Technologies A Cobb-Douglas production function is of the form E.g. y = Ax a 1 1 x a 2 2, x an n y = x 1/3 1 x 1/3 2 with n = 2, A = 1, a 1 = 1/3, a 2 = 1/3 Ivan Etzo (UNICA) Lecture 1: Technolgy 36 / 61
37 Cobb-Douglas Technologies Ivan Etzo (UNICA) Lecture 1: Technolgy 37 / 61
38 Fixed-Proportions Technologies A fixed-proportions production function is of the form y = min{a 1 x 1, a 2 x 2, a n x n } E.g. y = min{x 1, 2x 2 } with n = 2, A = 1, a 1 = 1, a 2 = 2 Ivan Etzo (UNICA) Lecture 1: Technolgy 38 / 61
39 Fixed-Proportions Technologies y = min{x 1, 2x 2 } Ivan Etzo (UNICA) Lecture 1: Technolgy 39 / 61
40 Perfect-Substitutes Technologies A perfect-substitutes production function is of the form y = a 1 x 1 + a 2 x a n x n E.g. with n = 2, a 1 = 1, a 2 = 3 y = x 1 + 3x 2 Ivan Etzo (UNICA) Lecture 1: Technolgy 40 / 61
41 Perfect-Substitutes Technologies y = x 1 + 3x 2 Ivan Etzo (UNICA) Lecture 1: Technolgy 41 / 61
42 Marginal (Physical) Products The marginal product of input i is the rate-of-change of the output level as the level of input i changes, holding all other input levels fixed. MP i = y x i The marginal product of input i is diminishing if it becomes smaller as the level of input i increases. That is, if MP i = ( ) y = 2 < 0 x i x i x i x 2 i Remember the law of diminishing marginal product. Ivan Etzo (UNICA) Lecture 1: Technolgy 42 / 61
43 Returns-to-Scale Marginal products describe the change in output level as a single input level changes. Returns-to-scale describes how the output level changes as all input levels change in direct proportion (e.g. all input levels doubled, or halved). Ivan Etzo (UNICA) Lecture 1: Technolgy 43 / 61
44 Returns-to-Scale Constant returns-to-scale The technology described by the production function f (x) exhibits constant returns-to-scale if, for any input bundle (x 1, x n ), f (kx 1, kx 2, kx n ) = kf (x 1, x 2, x n ) i.e. the production function f (x) is homogeneous of degree 1. E.g. (k = 2) doubling all input levels doubles the output level. Increasing returns-to-scale Decreasing returns-to-scale f (kx 1, kx 2, kx n ) > kf (x 1, x 2, x n ) f (kx 1, kx 2, kx n ) < kf (x 1, x 2, x n ) Ivan Etzo (UNICA) Lecture 1: Technolgy 44 / 61
45 Returns-to-Scale A single technology can locally exhibit different returns-to-scale. Ivan Etzo (UNICA) Lecture 1: Technolgy 45 / 61
46 Examples of Returns-to-Scale Perfect-Substitutes production function y = a 1 x 1 + a 2 x a n x n y = a 1 (kx 1 ) + a 2 (kx 2 ) + + a n (kx n ) = k (a 1 x 1 + a 2 x a n x n ) Constant RtS = ky Ivan Etzo (UNICA) Lecture 1: Technolgy 46 / 61
47 Examples of Returns-to-Scale Perfect-Complements production function y = min{a 1 x 1, a 2 x 2, a n x n } min{a 1 (kx 1 ), a 2 (kx 2 ), a n (kx n )} Constant RtS k (min{a 1 x 1, a 2 x 2, a n x n }) Ivan Etzo (UNICA) Lecture 1: Technolgy 47 / 61
48 Examples of Returns-to-Scale Cobb-Douglas production y = x a 1 1 x a 2 2 x n an (kx 1 ) a 1 (kx 2 ) a2 (kx n ) an = k a 1+ +a n y Thus: if n i=1 a i = 1 Constant RtS if n i=1 a i > 1 Increasing RtS if n i=1 a i < 1 Decreasing RtS Ivan Etzo (UNICA) Lecture 1: Technolgy 48 / 61
49 Return to Scale and Marginal Product A technology can exhibit increasing returns-to-scale even if all of its marginal products are diminishing. Why? E.g. y = x 2/3 1 x 2/3 2 MP x1 = y = 2 x 1 3 x 1/3 1 x 2/3 2 MP 1 = 1 2 x x 4/3 1 x 2/3 2 < 0 MP x2 = MP x1 < 0 RtS: a 1 + a 2 = 4 3 > 0 Increasing RtS Ivan Etzo (UNICA) Lecture 1: Technolgy 49 / 61
50 Return to Scale and Marginal Product A marginal product is the rate-of-change of output as one input level increases, holding all other input levels fixed. Marginal product diminishes because the other input levels are fixed, so the increasing inputs units have each less and less of other inputs with which to work. Therefore, diminishing marginal product is a short run concept RtS imply that all inputs vary, thus it refers to the long run Ivan Etzo (UNICA) Lecture 1: Technolgy 50 / 61
51 The technical rate of substitution (TRS) Assume y = f (x 1, x 2 ) and that the firm is producing y = f (x 1, x 2 ) At what rate can a firm substitute one input for another without changing its output level? Using total differential of function f (x 1, x 2 ) impose dy = f x 1 dx 1 + f x 2 dx 2 dy = 0 f x 1 dx 1 + f x 2 dx 2 = 0 dx 2 dx 1 = f / x 1 f / x 2 Ivan Etzo (UNICA) Lecture 1: Technolgy 51 / 61
52 The technical rate of substitution (TRS) Ivan Etzo (UNICA) Lecture 1: Technolgy 52 / 61
53 The technical rate of substitution (TRS) Example: The Cobb-Douglas technology Consider f (x 1, x 2 ) = x α 1 x 1 α 2 TRS = f / x 1 = α f / x 2 1 α x 2 x 1 Ivan Etzo (UNICA) Lecture 1: Technolgy 53 / 61
54 The elasticity of substitution The elasticity of substitution measures the curvature of an isoquant. σ = (x 2 /x 1 ) x 2 /x 1 TRS TRS It measures the percentage change in the factor ratio divided by the percentage change in TRS. Using logarithmic derivative ɛ = dlny dlnx Ivan Etzo (UNICA) Lecture 1: Technolgy 54 / 61
55 The elasticity of substitution Example: The Cobb-Douglas technology Consider f (x 1, x 2 ) = x1 αx 2 1 α we know that TRS = α 1 α x 2 x 1 x 2 = 1 α x 1 α TRS taking logs and deriving with respect to TRS ln x 2 = ln 1 α + ln TRS x 1 α σ dln (x 2/x 1 ) dln TRS = 1 Ivan Etzo (UNICA) Lecture 1: Technolgy 55 / 61
56 The CES production function The Constant Elasticity of Substitution has the following form y = [a 1 x ρ 1 + a 2x ρ 2 ] 1 ρ the CES exhibits constant returns to scale It contains other production functions as a special cases 1 If ρ = 1 Perfect-substitutes production function 2 If ρ = 0 Cobb-Douglas production function 3 If = Fixed-proportion production function Ivan Etzo (UNICA) Lecture 1: Technolgy 56 / 61
57 The CES production function 1 ρ = 1 y = a 1 x 1 + a 2 x 2 2 ρ = 0 Consider the TRS of the CES produciton function TRS = a 1 a 2 ( ) ρ 1 x1 when ρ 0 then the TRS tends to the following limit TRS = a ( ) 1 x1 a 2 x 2 that is, the TRS of the Cobb-Douglas production function (i.e. the isoquant is the same). 3 ρ = then the TRS of the CES TRS=0 if x 2 > x 1 ; TRS = if x 1 > x 2 i.e. Just as the TRS of the Fixed-proportion production function. x 2 Ivan Etzo (UNICA) Lecture 1: Technolgy 57 / 61
58 The CES production function Proof: the CES has a constant elasticity of substitution We need to compute the elasticity of substitution σ From the TRS of the CES (where, for simplicity, a 1 = a 2 = 1) taking logs TRS = ( ) ρ 1 x1 x 2 x 2 x 1 = TRS 1 1 ρ ln x 2 = 1 ln TRS x 1 1 ρ and applying the definition of σ using the log derivative. σ = dln(x 2/x 1 ) dln TRS = 1 1 ρ Ivan Etzo (UNICA) Lecture 1: Technolgy 58 / 61
59 Well-Behaved Technologies A well-behaved technology is monotonic convex. Ivan Etzo (UNICA) Lecture 1: Technolgy 59 / 61
60 Well-Behaved Technologies Monotonicity Monotonicity: More of any input generates more output. Ivan Etzo (UNICA) Lecture 1: Technolgy 60 / 61
61 Well-Behaved Technologies Convexity Convexity: If the input bundles x and x both provide y units of output then the mixture tx + (1 t)x provides at least y units of output, for any 0 < t < 1. Convexity implies that the TRS decreases as x 1 increases. Ivan Etzo (UNICA) Lecture 1: Technolgy 61 / 61
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