30E00300 Productivity and Efficiency Analysis Abolfazl Keshvari, Ph.D.

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1 30E00300 Productivity and Efficiency Analysis 2016 Abolfazl Keshvari, Ph.D.

2 Mathematics and statistics We need to know some basics of math and stat What is a function, and its first order derivatives Vectors, matrices, and their operations Basic concepts of regression analysis and linear programming It is good to use some software: Excel, and Excel Solver (or equivalent) Matlab, R, Stata, or similar software 2

3 Textbook An Introduction to Efficiency and Productivity Analysis Authors: Timothy J. Coelli, D.S. Prasada Rao, Christopher J. O Donnell, George E. Battese Get the book or the e-book from the library: 3

4 Performance evaluation Think about any operation Like a factory, workshop, school, hospital, bank, Assume there is only one input and one output Ratio of output to input: Output Input More is better! More shows a better productivity. This is Productivity ratio. 4

5 Performance evaluation Productivty changes over time Firms become more productive, or less productive If a firm is more productive than the last year, it is more efficient now Productivity and efficiency are not the same, but people may use them interchangeably! 5

6 Firm, unit, and DMU Here these are the same for us DMU: Decision making unit They refer to any business: Micro level: factories, bank branches, schools, Macro level: economy of the countries, banks, educational systems, Private or public sectors Note DM: Decision maker. The person who makes a decision. 6

7 Production function (frontier) Output Production frontier Firm Input 7

8 Technical efficiency Firm i is shown by: x i, y i Productivity ratio: y i x i Production function shows the maximum output that can be obtained from the input Firms on the production function are technically efficient Firms below the production functions are technically inefficient y y i O C B A x i Firm i Production frontier x Technical efficiency 8

9 Production possibility set Firm i is shown by: x i, y i Productivity ratio: y i x i If the firm is not technically efficient, it is below the production function y y i C B A Production frontier Firm i All points below the production function are technically feasible O x i x This is production possibility set (PPS) Or the other name feasible production set 9

10 Problems? How to estimate the production function? Methods: regression analysis, data envelopment analysis (DEA), stochastic frontier analysis (SFA) How to evaluate the efficiency of the firm? Distance functions Efficiency measurements: technical, allocative, cost, Productivity measurements: total factor productivity (TFP), partial productivity 10

11 Example In reality, efficiency scores are almost around 0.5 to 1 (just an observation, not a rule!) We can use percentages for efficiency scores, 50%, 90% In general, we expect all the efficiency scores be less than or equal to 1 Our examples here are samples of Cobb- Douglas production function Usually, firms use several inputs to produce several outputs It is typical to have a model with two inputs (labor and capital) and one outputs 11

12 Inputs and outputs Examples of inputs: labour, raw material, machinery, energy, and uncontrolable inputs such as environment Examples of outputs: products, services, and by-products (desirable or undesirable, such as pollution) We start by cases with one output. Inputs: x = x 1 x 2 x N, output: y 12

13 Production function N inputs, 1 output Production function is shown as y = f x Properties: 1. Nonnegativity f x 0 2. Weak essentiality f Nondecreasing in inputs (monotonicity) x 0 x 1 f x 0 f x 1 4. Concave in inputs f x is a concave function 13

14 Example Which properties are violated? Note that we assume these properties in this course. There are very popular in researches and applications. There are production functions that violate some of these properties, and hold some other properties. 14

15 Transition from 2 indices to 3 indices Index (indices): inputs and outputs Familiar concept: weather maps, topographical maps Contour lines An output isoquant is a contour line of the production function that shows the various combinations of two inputs that can produce the same level of output. Shows the possibilities for substituting inputs without changing the level of output 15

16 Some useful definitions 1 Marginal product: The extra output generated by an extra input The slope of the production function MP n = f x x n, x n is one of the inputs n = 1,, N Marginal rate of technical substitution: Represents the rate at which inputs may be substituted while the output level remains constant MRTS nm = MP m MP n, 1 n, m N 16

17 Example Production function: q = f L, K K 6 L is labout, K is capital, q is the product 5 4 B MP L = f L,K L, MRTS L,K = MP K MP L A 1 MRTS K,L = MP L MP K = f L,K L f L,K K = K L L MRTS K,L A, B = 2 1 = 2 17

18 Cobb-Douglas production function A well-known production function Two-input case: q = AL α K β MP L = AαL α 1 K β MRTS KL = AαLα 1 K β AβL α K β 1 = α β K L 18

19 Some useful definitions 2 Output elasticity: measures the responsiveness of output to a change in levels of one of the inputs used in production, ceteris paribus. E n = f x x n x f x Returns to scale: measures the output response when all inputs are varied simultaneously. f λx = λf x Constant returns to scale (CRS) f λx > λf x Increasing returns to scale IRS f λx < λf x Decreasing returns to scale DRS 19

20 Example: Cobb-Douglas E L = q L L q = AαLα 1 K β q = AL α K β L AL α K β = α So α = 0.2 means that a 1% increase in labor will approximately increase output by 0.2%. If α + β = 1 the CD production function exhibits CRS 20

21 Cost function Inputs: x = x 1, x 2,, x N presents the quantities of inputs. Example: L is labor in man-hour. Where are the costs of inputs? w = w 1, w 2,, w N presents the prices of inputs How to mix inputs so that the cost of production is minimized? Cost function: c w, q = w 1 x w N x N = w x 21

22 Cost minimization The cost minimization problem is to move the cost isoquant so that it is tangent to the output isoquant. This optimal point shows the cost efficient combination of inputs. Mathematical formulation K c 1 > c 2 > c f L, K = L 22

23 Long-run and short-run It is not possible to change some of the inputs in a short period If we assume all inputs are changeable we have a long-run model If some inputs are not changeable in the time horizon of the evaluation, we have a short-run model Remedy: fix the unchangeable inputs to their current values 23

24 Cost minimization problem Interpretation. min w 1 x 1 + w 2 x 2 s. t. y = f x 1, x 2 x 1 and x 2 are variables. w 1, w 2 and y are known. The solution is in the form of x 1 = x 1 w 1, w 2, y and x 2 = x 2 w 1, w 2, y Conditional input demands 24

25 Cost function and input demands Cost minimization problem: min c w, q s. t. q = f x The solution is the values of inputs (x) that generate q output with the minimum cost (w), So, the value of inputs are related to the output and the inputs costs Let show them by x n w, q n = 1,, N We call them conditional input demand functions They tell us how much of the inputs the firm would use at prices w 1, w 2,, w N if it wanted to produce the output q at least price 25

26 Necessary condition for cost minimization min w 1 x 1 + w 2 x 2 s. t. y = f x 1, x 2 The necessary condition for cost minimization is that w 1 w 2 = MP 1 MP 2 Or MP 2 w 2 = MP 1 w 1 26

27 Shephard s lemma In our context, this lemma connects the cost function to conditional input demand functions: x n w, q = c w, q w n An economist may find it easier to estimate the cost function than the production function of the firm. Shephard s lemma tells us what functional forms are acceptable candidates for cost function and how to recover the production function. 27

28 Example Let s assume the cost function is estimated as c w 1, w 2, y = Aw 1 a w 2 1 a y b Using Shephard s lemma: x 1 w, y = c w 1 = Aay b w 2 w 1 1 a,x 2 w, y = c w 2 = A 1 a y b w 2 w 1 a a 1 a y = λx b b 1 x 2 λ = Aa a 1 a 1 a 1 b 28

29 Revenue function If we are interested in the maximum revenue from a given level of resources, we use revenue function. Note that there may be more than one output: y = f x where both x and y are vectors Let p be the outputs prices Revenue function is r p, x = p y Revenue maximization problem is max r p, x s. t. y = f x. 29

30 Profit function Cost function: input prices are known, output level is fixed Revenue function: output prices are known, input level is fixed Profit function: input and output prices are known. Find the inputs and outputs to maximize the profit π p, w = max p q w x s. t. q = f x 30

31 Profit function 31

32 Hotelling s lemma What is Shephard s lemma? Hotelling s lemma: x n p, w = q m p, w = π p,w w n π p,w p m Input demand function Output supply function 32

33 Recap We are interested in the technology used by the firm An standard way to represent the technology is to use a production function It is also important to understand the behavioral relationships that arise from firms optimizing decisions These decisions are represented by input demand and output supply functions 33

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