Review of Production Theory
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1 Review of Production Theory based on OnFront, Reference Guide Coelli, et al., Introduction to Efficiency and Productivity Springer, 25, Ch 3. April 9, 28
2 Technology: Set Representations Technology describes the relationship between inputs x and outputs y (denoted by q in Coelli et al), where x n n = 1,..., N n indexes different types of inputs (1) x = (x 1,..., x N ) is the vector of inputs y m, m = 1,..., M m indexes different types of outputs y = (y 1,..., y M ) is the vector of outputs
3 Equivalent Technology Sets: Input-Output Set: GR or S in Coelli et al. Set of all feasible input and output bundles. Output Set: P(x) Set of all outputs producible from x. Input Set L(y) (L(q) in Coelli et al): The set of all inputs that can produce output vector y
4 Input-Output Set Set of all feasible input and output bundles. GR = {(x, y) : x can produce y} (2) Output (y) Production frontier GR Input (x)
5 Output Set Output Set P(x): the set of all outputs producible from given inputs, x P(x) = {(y) : (x, y) belongs to GR} (3) Output 2 (y2) P(x) Output 1 (y1)
6 Input set Input Set L(y) (L(q) in Coelli et al): The set of all inputs that can produce output vector y L(y) = {(x) : (x, y) belongs to GR} (4) Input 2 (x2) L(y) Input 1 (x1)
7 Returns to Scale: Input-Output Set Constant Returns to Scale (CRS) doubling inputs doubles outputs Decreasing Returns to Scale (DRS) doubling inputs yields less than double outputs Increasing Returns to Scale (IRS) doubling inputs yields more than double outputs Variable Returns to Scale (VRS) allows for IRS, CRS and DRS
8 Returns to Scale y y y CRS DRS IRS GR GR GR x x x y VRS GR x
9 Function Representations of Technology Production Function: where y is a single output f (x) = max{y : y P(x)} (5) Output (y) Production frontier Y=f(x) GR X Input (x)
10 Distance Functions: multiple inputs and outputs Output Distance Function D o (x, y) = 1 if y is technically efficient, i.e., on the boundary of P(x) If y is single-valued, then D o (x, y) = y/f (x). Output 2 (y2) (y1*,y2*) (y1,y2) P(x) Output 1 (y1)
11 Farrell output based technical efficiency 1/D o (x, y) =Farrell output measure of technical efficiency, F o (x, y) F o (x, y) = maximal output/observed output Output 2 (y2) b F_o(x,y)=b/a a (y1,y2) P(x) Output 1 (y1)
12 Input Distance Function D i (y, x) = 1 if x is technically efficient, i.e., on the boundary of L(y) 1/D i (y, x) =Farrell input measure of technical efficiency Input 2 (x2) (x1,x2) (x1*,x2*) L(y) Input 1 (x1)
13 Farrell input based technical efficiency F i (y, x) = efficient input/observed input Input 2 (x2) c Fi(y,x)=d/c d L(y) Input 1 (x1)
14 Other Functions requiring information on prices Cost Function C(y, w) = min x {wx : x L(y)}, (6) where w is a vector of input prices. Input 2 (x2) Obs cost = w1*x1 + w2*x2 Min cost = w1*x1* + w2*x2* isoquant Input 1 (x1)
15 Cost Efficiency Cost efficiency = minimal cost/observed cost Input 2 (x2) Cost eff = f/e e (Cost at f=cost at g) f g isoquant Input 1 (x1)
16 Revenue Function R(x, p) = max{py : y P(x)}, (7) y where p is a vector of output prices. Output 2 (y2) Max rev = p1*y1*+p2*y2* Obs rev = p1*y1+p2*y2 Output 1 (y1)
17 Profit Function Π = max{py wx : (x, y) GR} (8) x,y Output (y) Max profit = p*y*-w*x* Obs profit=p*y w*x Input (x)
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