Tecnología y Estructura de Costos

Technologies u A technology is a process by which inputs are converted to an output. u E.g. labor, a computer, a projector, electricity, and software are being combined to produce this lecture.

Technologies u Usually several technologies will produce the same product -- a blackboard and chalk can be used instead of a computer and a projector. u Which technology is “best”? u How do we compare technologies?

Input Bundles u x i denotes the amount used of input i; i.e. the level of input i. u An input bundle is a vector of the input levels; (x 1, x 2, …, x n ).  E.g. (x 1, x 2, x 3 ) = (6, 0, 9  3).

Production Functions u y denotes the output level. u The technology’s production function states the maximum amount of output possible from an input bundle.

Production Functions y = f(x) is the production function. x’x Input Level Output Level y’ y’ = f(x’) is the maximal output level obtainable from x’ input units. One input, one output

Technology Sets u A production plan is an input bundle and an output level; (x 1, …, x n, y). u The collection of all feasible production plans is the technology set.

Technology Sets y = f(x) is the production function. x’x Input Level Output Level y’ y” y’ = f(x’) is the maximal output level obtainable from x’ input units. One input, one output y” = f(x’) is an output level that is feasible from x’ input units.

Technology Sets x’x Input Level Output Level y’ One input, one output y” The technology set

Technology Sets x’x Input Level Output Level y’ One input, one output y” The technology set Technically inefficient plans Technically efficient plans

Technologies with Multiple Inputs u What does a technology look like when there is more than one input? u The two input case: Input levels are x 1 and x 2. Output level is y. u Suppose the production function is

Technologies with Multiple Inputs u E.g. the maximal output level possible from the input bundle (x 1, x 2 ) = (1, 8) is u And the maximal output level possible from (x 1,x 2 ) = (8,8) is

Technologies with Multiple Inputs u The y output unit isoquant is the set of all input bundles that yield at most the same output level y.

Isoquants with Two Variable Inputs y  y  x1x1 x2x2

Technologies with Multiple Inputs u The complete collection of isoquants is the isoquant map. u The isoquant map is equivalent to the production function -- each is the other. u E.g.

Technologies with Multiple Inputs x1x1 x2x2 y

Cobb-Douglas Technologies u A Cobb-Douglas production function is of the form u E.g. with

x2x2 x1x1 All isoquants are hyperbolic, asymptoting to, but never touching any axis. Cobb-Douglas Technologies

x2x2 x1x1 All isoquants are hyperbolic, asymptoting to, but never touching any axis. Cobb-Douglas Technologies

x2x2 x1x1 All isoquants are hyperbolic, asymptoting to, but never touching any axis. Cobb-Douglas Technologies

x2x2 x1x1 All isoquants are hyperbolic, asymptoting to, but never touching any axis. Cobb-Douglas Technologies >

Fixed-Proportions Technologies u A fixed-proportions production function is of the form u E.g. with

Fixed-Proportions Technologies x2x2 x1x1 min{x 1,2x 2 } = min{x 1,2x 2 } = 8 min{x 1,2x 2 } = 4 x 1 = 2x 2

Perfect-Substitutes Technologies u A perfect-substitutes production function is of the form u E.g. with

Perfect-Substitution Technologies x1x1 x2x2 x 1 + 3x 2 = 18 x 1 + 3x 2 = 36 x 1 + 3x 2 = 48 All are linear and parallel

Marco de tiempo para las decisiones u Para estudiar la relación entre la decisión de producción de una empresa y sus costos, distinguimos dos marcos de tiempo para las decisiones: 1. El corto plazo 2. El largo plazo

Marco de tiempo para las decisiones El corto plazo y el largo plazo El corto plazo u Es un marco de tiempo en el que las cantidades de algunos recursos productivos (o factores) son fijas, y las cantidades de los otros factores de la producción pueden variarse. El largo plazo u Es un marco de tiempo en el que las cantidades de todos los recursos de la producción pueden variar.

Marco de tiempo para las decisiones El producto total u Es la cantidad total producida. El producto marginal u Es el cambio en la producción total causado al añadir una unidad de insumo variable (L) mientras todos los demás insumos permanecen constantes El producto promedio u Es el producto total dividido entre la cantidad de trabajo empleada (insumo variable).

Producto total, producto marginal y producto promedio ProductoProductoProducto Trabajo total marginal promedio (trabajadores (camisas (camisas por (camisas por día) por día) trabajador adicional) por trabajador) a 0 0 b 1 4 c 2 10 d 3 13 e 4 15 f 5 16

Producto total, producto marginal y producto promedio ProductoProductoProducto Trabajo total marginal promedio (trabajadores (camisas (camisas por (camisas por día) por día) trabajador adicional) por trabajador) a 0 0 b 1 4 c 2 10 d 3 13 e 4 15 f

Producto total, producto marginal y producto promedio ProductoProductoProducto Trabajo total marginal promedio (trabajadores (camisas (camisas por (camisas por día) por día) trabajador adicional) por trabajador) a 0 0 b c d e f

Alcanzable Curva del producto total Trabajo (trabajadores por día) PT No alcanzable Producto (camisas por día) a b c d e f

Curva del producto marginal u El producto marginal se mide también por la pendiente de la curva del producto total. Ley de los rendimientos decrecientes Principio que afirma que más alla de cierto punto el producto marginal disminuye a medida que se agregan unidades adicionales de un factor variable a un factor fijo.

Producto marginal Trabajo (trabajadores por día) PT Producción (camisas por día) Trabajo (trabajadores por día) Producto marginal (camisas por día por trabajador) PM c d

Marginal (Physical) Products u 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. u That is,

Marginal (Physical) Products E.g. if then the marginal product of input 1 is

Marginal (Physical) Products E.g. if then the marginal product of input 1 is

Marginal (Physical) Products E.g. if then the marginal product of input 1 is and the marginal product of input 2 is

Marginal (Physical) Products E.g. if then the marginal product of input 1 is and the marginal product of input 2 is

Marginal (Physical) Products u The marginal product of input i is diminishing if it becomes smaller as the level of input i increases. That is, if

Marginal (Physical) Products and E.g. ifthen

Marginal (Physical) Products and so E.g. ifthen

Marginal (Physical) Products and so and E.g. ifthen

Marginal (Physical) Products and so and Both marginal products are diminishing. E.g. ifthen

Returns-to-Scale u 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).

Returns-to-Scale If, for any input bundle (x 1,…,x n ), then the technology described by the production function f exhibits constant returns-to-scale. E.g. (k = 2) doubling all input levels doubles the output level.

Returns-to-Scale y = f(x) x’x Input Level Output Level y’ One input, one output 2x’ 2y’ Constant returns-to-scale

Returns-to-Scale If, for any input bundle (x 1,…,x n ), then the technology exhibits diminishing returns-to-scale. E.g. (k = 2) doubling all input levels less than doubles the output level.

Returns-to-Scale y = f(x) x’x Input Level Output Level f(x’) One input, one output 2x’ f(2x’) 2f(x’) Decreasing returns-to-scale

Returns-to-Scale If, for any input bundle (x 1,…,x n ), then the technology exhibits increasing returns-to-scale. E.g. (k = 2) doubling all input levels more than doubles the output level.

Returns-to-Scale y = f(x) x’x Input Level Output Level f(x’) One input, one output 2x’ f(2x’) 2f(x’) Increasing returns-to-scale