Technologies Chapter Eighteen Technolog A technolog is a process b which inputs are converted to an output. E.g. labor, a computer, a projector, electricit, and software are being combined to produce this lecture. Technologies Input Bundles Usuall several technologies will produce the same product -- a blackboard and chalk can be used instead of a computer and a projector. Which technolog is best? How do we compare technologies? 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 n ). E.g. (,, x ) = (6, 0, 9 ). Production Functions denotes the output level. The technolog s production function states the maximum amount of output possible from an input bundle. = f( x,, x n ) Output Level Production Functions One input, one output = f(x) is the production function. = f(x ) is the maximal output level obtainable from x input units. x x Input Level
Technolog Sets A production plan is an input bundle and an output level; (,, x n, ). A production plan is feasible if f( x,, x n ) The collection of all feasible production plans is the technolog set. Output Level Technolog Sets One input, one output = f(x) is the production function. = f(x ) is the maximal output level obtainable from x input units. = f(x ) is an output level that is feasible from x input units. x x Input Level Technolog Sets The technolog set is T = {( x,, xn, ) f( x,, xn) and x 0,, xn 0}. Output Level Technolog Sets One input, one output The technolog set x x Input Level Output Level Technolog Sets One input, one output Technicall inefficient plans Technicall efficient plans The technolog set x x Input Level What does a technolog look like when there is more than one input? The two input case: Input levels are and. Output level is. Suppose the production function is = f( x, x) = x / x /.
E.g. the maximal output level possible from the input bundle (, ) = (, 8) is / / / / = x = 8 = = 4. And the maximal output level possible from (, ) = (8,8) is / / / / = x = 8 8 = = 8. Output, (8,) (8,8) Isoquants with Two Variable Inputs The output unit isoquant is the set of all input bundles that ield at most the same output level. 8 4 Isoquants with Two Variable Inputs Isoquants with Two Variable Inputs Isoquants can be graphed b adding an output level axis and displaing each isoquant at the height of the isoquant s output level. Output, 8 4
Isoquants with Two Variable Inputs Isoquants with Two Variable Inputs More isoquants tell us more about the technolog. 8 6 4 Isoquants with Two Variable Inputs Output, 8 6 4 The complete collection of isoquants is the isoquant map. The isoquant map is equivalent to the production function -- each is the other. / / E.g. = f ( x, x ) = x x 4
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Cobb-Douglas Technologies A Cobb-Douglas production function is of the form = A xn a n. E.g. / / = with n=, A =, a = anda =. Cobb-Douglas Technologies All isoquants are hperbolic, asmptoting to, but never touching an axis. = x x Cobb-Douglas Technologies All isoquants are hperbolic, asmptoting to, but never touching an axis. = x x Cobb-Douglas Technologies All isoquants are hperbolic, asmptoting to, but never touching an axis. = x x a a = " = " = 7
Cobb-Douglas Technologies All isoquants are hperbolic, asmptoting to, but never touching an axis. = x x " > = " = Fixed-Proportions Technologies A fixed-proportions production function is of the form = min{ ax, ax,, anxn}. E.g. = min{ x, x} with n =, a = and a =. Fixed-Proportions Technologies 7 4 4 8 4 = min{ x, x} = min{, } = 4 min{, } = 8 min{, } = 4 Perfect-Substitutes Technologies A perfect-substitutes production function is of the form = ax + ax + + anxn. E.g. = x + x with n =, a = and a =. Perfect-Substitution Technologies = x + x + = 8 + = 6 8 6 9 + = 48 All are linear and parallel 8 4 Marginal (Phsical) Products = f( x,, x n ) 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, MPi = xi 8
Marginal (Phsical) Products E.g. if / = f( x, x) = x / x then the marginal product of input is Marginal (Phsical) Products E.g. if / = f( x, x) = x / x then the marginal product of input is MP x / = = x / x Marginal (Phsical) Products E.g. if / = f( x, x) = x / x then the marginal product of input is MP x / = = x / x and the marginal product of input is Marginal (Phsical) Products E.g. if / = f( x, x) = x / x then the marginal product of input is MP x / = = x / x and the marginal product of input is MP x / = = x / x. Marginal (Phsical) Products Tpicall the marginal product of one input depends upon the amount used of other inputs. E.g. if MP x / = x / then, MP x / / 4 = 8 = x / if = 8, and if = 7 then MP x / / = 7 = x /. Marginal (Phsical) Products The marginal product of input i is diminishing if it becomes smaller as the level of input i increases. That is, if MP i = = < 0. xi xi xi x i 9
Marginal (Phsical) Products / / E.g. if = then MP = x / x / MP = x / and x / Marginal (Phsical) Products / / E.g. if = then MP = x / x / MP = x / and x / so MP = x 5 / x / < 0 x 9 Marginal (Phsical) Products / / E.g. if = then MP = x / x / MP = x / and x / so MP = x 5 / x / < 0 x 9 and MP = x / x 4 / x < 0. 9 Marginal (Phsical) Products / / E.g. if = then MP = x / x / MP = x / and x / so MP = x 5 / x / < 0 x 9 and MP = x / x 4 / x < 0. 9 Both marginal products are diminishing. 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). If, for an input bundle (,,x n ), fkx (, kx,, kxn) = kfx (, x,, xn) then the technolog described b the production function f exhibits constant returns-to-scale. E.g. (k = ) doubling all input levels doubles the output level. 0
Output Level One input, one output = f(x) Constant returns-to-scale If, for an input bundle (,,x n ), f( kx, kx,, kxn) < kf( x, x,, xn) then the technolog exhibits diminishing returns-to-scale. E.g. (k = ) doubling all input levels less than doubles the output level. x x x Input Level Output Level f(x ) f(x ) f(x ) One input, one output = f(x) Decreasing returns-to-scale If, for an input bundle (,,x n ), f( kx, kx,, kxn) > kf( x, x,, xn) then the technolog exhibits increasing returns-to-scale. E.g. (k = ) doubling all input levels more than doubles the output level. x x x Input Level One input, one output Output Level Increasing returns-to-scale f(x ) = f(x) A single technolog can locall exhibit different returns-to-scale. f(x ) f(x ) x x x Input Level
Output Level One input, one output Increasing returns-to-scale = f(x) Decreasing returns-to-scale Examples of The perfect-substitutes production function is = ax + ax + + anxn. Expand all input levels proportionatel b k. The output level becomes a( kx) + a( kx) + + an( kxn) Input Level x Examples of The perfect-substitutes production function is = ax + ax + + anxn. Expand all input levels proportionatel b k. The output level becomes a( kx) + a( kx) + + an( kxn) = k( ax + ax + + anxn) Examples of The perfect-substitutes production function is = ax + ax + + anxn. Expand all input levels proportionatel b k. The output level becomes a( kx) + a( kx) + + an( kxn) = k( ax + ax + + anxn) = k. The perfect-substitutes production function exhibits constant returns-to-scale. Examples of The perfect-complements production function is = min{ ax, ax,, anxn}. Expand all input levels proportionatel b k. The output level becomes min{ a( kx), a( kx),, an( kxn)} Examples of The perfect-complements production function is = min{ ax, ax,, anxn}. Expand all input levels proportionatel b k. The output level becomes min{ a( kx), a( kx),, an( kxn)} = k(min{ ax, ax,, anxn})
Examples of The perfect-complements production function is = min{ ax, ax,, anxn}. Expand all input levels proportionatel b k. The output level becomes min{ a( kx), a( kx),, an( kxn)} = k(min{ ax, ax,, anxn}) = k. The perfect-complements production function exhibits constant returns-to-scale. Examples of The Cobb-Douglas production function is = xn a n. Expand all input levels proportionatel b k. The output level becomes a ( kx ) a ( kx ) ( kxn ) a n Examples of The Cobb-Douglas production function is = xn a n. Expand all input levels proportionatel b k. The output level becomes a ( kx ) a ( kx ) ( kxn ) a n = k k k n x n Examples of The Cobb-Douglas production function is = xn a n. Expand all input levels proportionatel b k. The output level becomes a ( kx ) a ( kx ) ( kxn ) a n = k k k n x n a k + a+ + an = xn a n Examples of The Cobb-Douglas production function is = xn a n. Expand all input levels proportionatel b k. The output level becomes a ( kx ) a ( kx ) ( kxn ) a n = k k k n x n a k + a+ + a = n x xn a k + + = n. n Examples of The Cobb-Douglas production function is = xn a n. a ( kx ) a ( kx ) ( kxn ) a n k a + + a n =. The Cobb-Douglas technolog s returnsto-scale is constant if a + + a n =
Examples of The Cobb-Douglas production function is = xn a n. a ( kx ) a ( kx ) ( kxn ) a n k a + + a n =. The Cobb-Douglas technolog s returnsto-scale is constant if a + + a n = increasing if a + + a n > Examples of The Cobb-Douglas production function is = xn a n. a ( kx ) a ( kx ) ( kxn ) a n k a + + a n =. The Cobb-Douglas technolog s returnsto-scale is constant if a + + a n = increasing if a + + a n > decreasing if a + + a n <. Q: Can a technolog exhibit increasing returns-to-scale even though all of its marginal products are diminishing? Q: Can a technolog exhibit increasing returns-to-scale even if all of its marginal products are diminishing? A: Yes. / / E.g. =. = x / x / = 4 a + a = > so this technolog exhibits increasing returns-to-scale. = x / x / = 4 a + a = > so this technolog exhibits increasing returns-to-scale. But MP = x / x / diminishes as x increases 4
= x / x / = 4 a + a = > so this technolog exhibits increasing returns-to-scale. But MP = x / x / diminishes as x increases and MP x / = x / diminishes as x increases. So a technolog can exhibit increasing returns-to-scale even if all of its marginal products are diminishing. Wh? A marginal product is the rate-ofchange 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 input s units have each less and less of other inputs with which to work. When all input levels are increased proportionatel, there need be no diminution of marginal products since each input will alwas have the same amount of other inputs with which to work. Input productivities need not fall and so returns-to-scale can be constant or increasing. Technical Rate-of-Substitution Technical Rate-of-Substitution At what rate can a firm substitute one input for another without changing its output level? 00 5
Technical Rate-of-Substitution The slope is the rate at which input must be given up as input s level is increased so as not to change the output level. The slope of an isoquant is its technical rate-of-substitution. Technical Rate-of-Substitution How is a technical rate-of-substitution computed? 00 Technical Rate-of-Substitution How is a technical rate-of-substitution computed? The production function is = f( x, x). A small change (d, d ) in the input bundle causes a change to the output level of d = x dx + x d. Technical Rate-of-Substitution d = x dx + x d. But d = 0 since there is to be no change to the output level, so the changes d and d to the input levels must satisf 0 = + x dx x dx. Technical Rate-of-Substitution 0 = + x dx x dx rearranges to x dx = x d so dx x = /. dx / x Technical Rate-of-Substitution dx x = / dx / x is the rate at which input must be given up as input increases so as to keep the output level constant. It is the slope of the isoquant. 6
Technical Rate-of-Substitution; A Cobb-Douglas Example = f( x, x) = x a x b so a b ax = x bx a x b and =. x x The technical rate-of-substitution is a b dx / x ax x ax = = dx x a b bx x bx =. / Technical Rate-of-Substitution; A Cobb-Douglas Example / / = x ; a = andb = ax x x TRS = ( / ) = = bx ( / ) x x Technical Rate-of-Substitution; A Cobb-Douglas Example 8 / / = x ; a = andb = ax x x TRS = ( / ) = = bx ( / ) x x x TRS = 8 = = x 4 Technical Rate-of-Substitution; A Cobb-Douglas Example 6 / / = x ; a = andb = ax x x TRS = ( / ) = = bx ( / ) x x x TRS = 6 = = x 4 4 Well-Behaved Technologies A well-behaved technolog is monotonic, and convex. Well-Behaved Technologies - Monotonicit Monotonicit: More of an input generates more output. monotonic not monotonic x x 7
Well-Behaved Technologies - Convexit Well-Behaved Technologies - Convexit Convexit: If the input bundles x and x both provide units of output then the mixture tx + (-t)x provides at least units of output, for an 0 < t <. " " 00 Well-Behaved Technologies - Convexit Well-Behaved Technologies - Convexit " ( tx t x tx t x ) + ( ) ", + ( ) " 00 " " ( tx t x tx t x ) + ( ) ", + ( ) " 0 00 " Well-Behaved Technologies - Convexit Convexit implies that the TRS increases (becomes less negative) as increases. Well-Behaved Technologies higher output " " 50 00 00 8
The long-run is the circumstance in which a firm is unrestricted in its choice of all input levels. There are man possible short-runs. A short-run is a circumstance in which a firm is restricted in some wa in its choice of at least one input level. Examples of restrictions that place a firm into a short-run: temporaril being unable to install, or remove, machiner being required b law to meet affirmative action quotas having to meet domestic content regulations. A useful wa to think of the long-run is that the firm can choose as it pleases in which short-run circumstance to be. What do short-run restrictions impl for a firm s technolog? Suppose the short-run restriction is fixing the level of input. Input is thus a fixed input in the short-run. Input remains variable. 9
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x / / = is the long-run production function (both and are variable). Four short-run production functions. The short-run production function when x is / / / = x = x. The short-run production function when 0 is / / / = x 0 = 5x.
/ / = 0 / / = 5 / / = / / = Four short-run production functions.