Technology Molly W. Dahl Georgetown University Econ 101 – Spring 2009 1 Technologies A technology is a process by which inputs are converted to an output. E.g. labor, a computer, a projector, electricity, software, chalk, a blackboard are all being used to produce this lecture. 2 Production Functions xi denotes the amount of input i y denotes the output level. The technology’s production function states the maximum amount of output possible from an input bundle. y f ( x1 ,, xn ) 3 Production Functions One input, one output Output Level y’ y = f(x) is the production function. y’ = f(x’) is the maximum output obtainable from x’ input units. x’ Input Level x 4 Technology Sets One input, one output Output Level y’ y” y = f(x) is the production function. y’ = f(x’) is the maximum output obtainable from x’ input units. y” = f(x’) is an output level that is feasible from x’ input units. x’ x Input Level 5 Technology Sets One input, one output Output Level y’ The technology set y” x’ Input Level x 6 Technology Sets One input, one output Output Level Technically efficient plans y’ y” The technology Technically set inefficient plans x’ Input Level x 7 Technologies with Multiple Inputs What does a technology look like when there is more than one input? Suppose the production function is 1/3 1/3 y f ( x1 , x 2 ) 2x1 x 2 . 8 Technologies with Multiple Inputs The isoquant is the set of all input bundles that yield the same maximum output level y. More isoquants tell us more about the technology. 9 Isoquants with Two Variable Inputs x2 x2 y y y y x1 x1 10 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 1/ 3 1/ 3 f ( x1 , x2 ) 2 x1 x2 11 Technologies with Multiple Inputs x2 x2 y x1 x1 12 Cobb-Douglas Technologies A Cobb-Douglas production function is of the form a1 a 2 an y A x1 x 2 xn . E.g. with 1/3 1/3 y x1 x 2 1 1 n 2, A 1, a1 and a 2 . 3 3 13 Cobb-Douglas Technologies x2 All isoquants are hyperbolic, asymptoting to, but never touching any axis. y" > y' a1 a 2 y x1 x 2 x1 14 Fixed-Proportions Technologies A fixed-proportions production function is of the form y min{a1 x1 , a 2x 2 ,, an xn }. E.g. with y min{x1 , 2x 2 } n 2, a1 1 and a 2 2. 15 Fixed-Proportions Technologies y min{x1 , 2x 2 } x2 x1 = 2x2 7 4 2 4 8 min{x1,2x2} = 14 min{x1,2x2} = 8 min{x1,2x2} = 4 14 x1 16 Perfect-Substitutes Technologies A perfect-substitutes production function is of the form y a1 x1 a 2x 2 an xn . E.g. with y x1 3x 2 n 2, a1 1 and a 2 3. 17 Perfect-Substitution Technologies y x1 3x 2 x2 x1 + 3x2 = 18 x1 + 3x2 = 36 x1 + 3x2 = 48 8 6 3 All are linear and parallel 9 18 24 x1 18 Well-Behaved Technologies A well-behaved technology is monotonic, and convex. 19 Well-Behaved Technologies Monotonicity Monotonicity: More of any input generates more output. y y monotonic not monotonic x x 20 Well-Behaved Technologies Monotonicity higher output x2 y y y x1 21 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. 22 Well-Behaved Technologies Convexity x2 x'2 tx'1 (1 t )x"1 , tx'2 (1 t )x"2 x"2 y x'1 x"1 x1 23 Well-Behaved Technologies Convexity x2 x'2 tx'1 (1 t )x"1 , tx'2 (1 t )x"2 y y x"2 x'1 x"1 x1 24 Marginal (Physical) Products y f ( x1 ,, xn ) 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. That is, y MPi xi 25 Marginal (Physical) Products E.g. if 1/3 2/ 3 y f ( x1 , x 2 ) x1 x 2 then the marginal product of input 1 is 26 Marginal (Physical) Products E.g. if 1/3 2/ 3 y f ( x1 , x 2 ) x1 x 2 then the marginal product of input 1 is y 1 2/ 3 2/ 3 MP1 x1 x 2 x1 3 27 Marginal (Physical) Products E.g. if 1/3 2/ 3 y f ( x1 , x 2 ) x1 x 2 then the marginal product of input 1 is y 1 2/ 3 2/ 3 MP1 x1 x 2 x1 3 and the marginal product of input 2 is y 2 1/3 1/3 MP2 x1 x 2 . x2 3 28 Marginal (Physical) Products The marginal product of input i is diminishing if it becomes smaller as the level of input i increases. That is, if MPi y 2y 0. 2 xi xi xi xi 29 Marginal (Physical) Products 1/3 2/ 3 E.g. if y x1 x 2 then 1 2/ 3 2/ 3 2 1/3 1/3 MP1 x1 x 2 and MP2 x1 x 2 3 3 30 Marginal (Physical) Products 1/3 2/ 3 E.g. if y x1 x 2 then 1 2/ 3 2/ 3 2 1/3 1/3 MP1 x1 x 2 and MP2 x1 x 2 so 3 3 MP1 2 5 / 3 2/ 3 x1 x 2 0 x1 9 31 Marginal (Physical) Products 1/3 2/ 3 E.g. if y x1 x 2 then 1 2/ 3 2/ 3 2 1/3 1/3 MP1 x1 x 2 and MP2 x1 x 2 so and 3 3 MP1 2 5 / 3 2/ 3 x1 x 2 0 x1 9 MP2 2 1/ 3 4 / 3 x1 x 2 0. x2 9 Both marginal products are diminishing. 32 Technical Rate-of-Substitution At what rate can a firm substitute one input for another without changing its output level? 33 Technical Rate-of-Substitution The slope is the rate at which input 2 must be given up as input 1’s level is increased so as not to change the output level. The slope of an isoquant is its technical rate-of-substitution. x2 x'2 y x'1 x1 34 Technical Rate-of-Substitution How is a technical rate-of-substitution computed? 35 Technical Rate-of-Substitution How is a technical rate-of-substitution computed? The production function is y f ( x1 , x 2 ). A small change (dx1, dx2) in the input bundle causes a change to the output level of y y dy dx1 dx 2 . x1 x2 36 Technical Rate-of-Substitution y y dy dx1 dx 2 . x1 x2 But dy = 0 since there is to be no change to the output level, so the changes dx1 and dx2 to the input levels must satisfy y y 0 dx1 dx 2 . x1 x2 37 Technical Rate-of-Substitution y y 0 dx1 dx 2 x1 x2 rearranges to y y dx 2 dx1 x2 x1 so dx 2 y / x1 . dx1 y / x2 38 Technical Rate-of-Substitution dx 2 y / x1 dx1 y / x2 is the rate at which input 2 must be given up as input 1 increases so as to keep the output level constant. It is the slope of the isoquant. 39 TRS: A Cobb-Douglas Example a b y f ( x1 , x 2 ) x1 x 2 so y x1 a1 b ax1 x 2 and y a b 1 bx1 x 2 . x2 The technical rate-of-substitution is a1 b dx 2 y / x1 ax1 x 2 ax 2 . 1 dx1 y / x2 bx1 bx1axb 2 40 The Long-Run and the ShortRun In the long-run a firm is unrestricted in its choice of all input levels. There are many possible short-runs. In the short-run a firm is restricted in some way in its choice of at least one input level. 41 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 equal proportion e.g. all input levels doubled, or halved 42 Constant Returns-to-Scale If, for any input bundle (x1,…,xn), f (kx1 , kx 2 ,, kxn ) kf ( x1 , x 2 ,, xn ) then the technology exhibits constant returns-to-scale (CRS). E.g. (k = 2) If doubling all input levels doubles the output level, the technology exhibits CRS. 43 Constant Returns-to-Scale One input, one output Output Level y = f(x) 2y’ Constant returns-to-scale y’ x’ 2x’ Input Level x 44 Decreasing Returns-to-Scale If, for any input bundle (x1,…,xn), f (kx1 , kx 2 ,, kxn ) kf ( x1 , x 2 ,, xn ) then the technology exhibits decreasing returns-to-scale (DRS). E.g. (k = 2) If doubling all input levels less than doubles the output level, the technology exhibits DRS. 45 Decreasing Returns-to-Scale One input, one output Output Level 2f(x’) y = f(x) f(2x’) Decreasing returns-to-scale f(x’) x’ 2x’ Input Level x 46 Increasing Returns-to-Scale If, for any input bundle (x1,…,xn), f (kx1 , kx 2 ,, kxn ) kf ( x1 , x 2 ,, xn ) then the technology exhibits increasing returns-to-scale (IRS). E.g. (k = 2) If doubling all input levels more than doubles the output level, the technology exhibits IRS. 47 Increasing Returns-to-Scale One input, one output Output Level Increasing returns-to-scale y = f(x) f(2x’) 2f(x’) f(x’) x’ 2x’ Input Level x 48 Examples of Returns-to-Scale The Cobb-Douglas production function is 2 x an . y x1a1 xa n 2 Expand all input levels proportionately by k. The output level becomes (kx1 ) a1 (kx 2 ) a2 (kxn ) an 49 Examples of Returns-to-Scale The Cobb-Douglas production function is 2 x an . y x1a1 xa n 2 Expand all input levels proportionately by k. The output level becomes (kx1 ) a1 (kx 2 ) a2 (kxn ) an a1 a 2 an a1 a 2 an k k k x x x 50 Examples of Returns-to-Scale The Cobb-Douglas production function is 2 x an . y x1a1 xa n 2 Expand all input levels proportionately by k. The output level becomes (kx1 ) a1 (kx 2 ) a 2 (kxn ) an k a1k a 2 k an x a1 x a 2 x an 2 x an k a1 a 2 an x1a1 x a n 2 51 Examples of Returns-to-Scale The Cobb-Douglas production function is 2 x an . y x1a1 xa n 2 Expand all input levels proportionately by k. The output level becomes (kx1 ) a1 (kx 2 ) a 2 (kxn ) an k a1k a 2 k an x a1 x a 2 x an 2 x an k a1 a 2 an x1a1 x a n 2 k a1 an y. 52 Examples of Returns-to-Scale The Cobb-Douglas production function is 2 x an . y x1a1 xa n 2 (kx1 )a1 (kx 2 )a 2 (kxn )an ka1 an y. The Cobb-Douglas technology’s returnsto-scale is constant if a1+ … + an = 1 increasing if a1+ … + an > 1 decreasing if a1+ … + an < 1. 53 Examples of Returns-to-Scale The perfect-substitutes production function is y a1 x1 a 2x 2 an xn . Expand all input levels proportionately by k. The output level becomes a1 (kx1 ) a 2 (kx 2 ) an (kxn ) 54 Examples of Returns-to-Scale The perfect-substitutes production function is y a1 x1 a 2x 2 an xn . Expand all input levels proportionately by k. The output level becomes a1 (kx1 ) a 2 (kx 2 ) an (kxn ) k( a1x1 a 2x 2 anxn ) 55 Examples of Returns-to-Scale The perfect-substitutes production function is y a1 x1 a 2x 2 an xn . Expand all input levels proportionately by k. The output level becomes a1 (kx1 ) a 2 (kx 2 ) an (kxn ) k( a1x1 a 2x 2 anxn ) ky. The perfect-substitutes production function exhibits constant returns-to-scale. 56 Examples of Returns-to-Scale The perfect-complements production function is y min{a1 x1 , a 2x 2 , , an xn }. Expand all input levels proportionately by k. The output level becomes min{a1 (kx1 ), a 2 (kx 2 ), , an (kxn )} 57 Examples of Returns-to-Scale The perfect-complements production function is y min{a1 x1 , a 2x 2 , , an xn }. Expand all input levels proportionately by k. The output level becomes min{a1 (kx1 ), a 2 (kx 2 ), , an (kxn )} k(min{a1x1 , a 2x 2 , , anxn }) 58 Examples of Returns-to-Scale The perfect-complements production function is y min{a1 x1 , a 2x 2 , , an xn }. Expand all input levels proportionately by k. The output level becomes min{ a1 (kx1 ), a 2 (kx 2 ), , an (kxn )} k(min{ a1x1 , a 2x 2 , , anxn }) ky. The perfect-complements production function exhibits constant returns-to-scale. 59