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**Production Technology**

Chapter 7

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**Introduction Aim in this chapter Can substitute one factor for another**

Investigate purely technical relationship of combining inputs to produce outputs Presents a physical constraint on society’s ability to satisfy wants Classify factors going into production process Derive a production function that establishes a relationship between production factors and a firm’s output Discuss Law of Diminishing Marginal Returns and stages of production Develop concept of isoquants When two production factors are allowed to vary Can substitute one factor for another Measure of this ability is elasticity of substitution Effect of proportional changes in all inputs is called returns to scale Can classify production functions in terms of their elasticity of substitution and returns to scale attributes

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Factors of Production For economic modeling, factors of production are generally classified as Capital Durable manmade inputs Are themselves produced goods Labor Time or service individuals put into production Land All natural resources (for example, water, oil, and climate) Classification allows us to conceptualize simple cases first Then extend analysis to higher dimensions that are more general (realistic)

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**Factors of Production Time also enters into production process**

Economists generally divide time into three periods, based on ability to vary inputs Market period All inputs are fixed Short-run period Some inputs are fixed and some are variable Long-run period All inputs are variable In terms of actual time, market-period, short-run, and long-run intervals can vary considerably from one firm to another, Depends on nature of a particular firm Division of time into three periods is a simplification With intertemporal substitution among stages More general models incorporating numerous time stages are less restrictive in their assumptions Called dynamic models

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Production Functions Firms are interested in turning inputs into outputs with the objective of maximizing profit Formalized by a production function q = ƒ(K, L, M) Where q is output of a particular commodity K is capital L is labor M is land or natural resources For any possible combination of inputs, production function records maximum level of output that can be produced from that combination In market period all inputs are fixed, so level of output cannot be varied

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Production Functions Denote K°, L°, and M° as the fixed level of capital, labor, and land Production from these fixed inputs is fixed at q°, so q° = ƒ(K°, L°, M°) If capital and labor could be varied with only land fixed, then a short-run production function would be q = ƒ(K, L, M°) Now possible to vary output by changing either K or L Or both K and L In long run, all inputs could be varied, so only restriction on output is technology Production function represents set of technically efficient production processes Yields highest level of output for a given set of inputs

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Production Functions Generally, technical aspects of production do impose restrictions on profit Assumptions (axioms) concerning these aspects are required for developing economic models Two axioms generally underlie a production function Monotonicity Implies that if a firm can produce q with a certain level of inputs Should be able to produce at least q if there exists more of every input Assumes free disposal of inputs Implies that all marginal products of the variable inputs are positive at their profit-maximizing level Strict convexity Analogous to Strict Convexity Axiom in consumer theory

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**Variations in One Input (Short Run) Marginal Product**

Marginal product (MP) of variable input Change in output, Δq, resulting from a unit change of the variable input Holding all other inputs constant If capital is variable input, then marginal product of capital is Alternatively, if labor is variable input, then marginal product of labor is MP is analogous to concept of marginal utility except that MP is a cardinal number measured on the ratio scale Not an ordinal number

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**Variations in One Input (Short Run) Marginal Product**

Distances between any levels of MP are of a known size measured in physical quantities Bushels, crates, pounds, etc. Consider following cubic production function with labor as variable input q = 6L2 – ⅓L3 Marginal product of labor is MPL = 12L – L2 Graph of production function and MPL is provided in Figure 7.1

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**Figure 7.1 Stages of production and MPL and APL**

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**Variations in One Input (Short Run) Marginal Product**

At first, for low levels of labor, total product (TP) is increasing at an increasing rate Slope of TP or MPL is rising At point of inflection, slope is at its maximum MPL is also at a maximum To right of maximum MPL, TP is still increasing But at a decreasing rate MPL is positive, but falling At maximum TP, slope of TP curve is zero Corresponding to MPL = 0 When TP is falling, MPL is negative According to Monotonicity Axiom, given free disposal, a firm will not operate in negative range of MPL Generally assumed that MPL ≥ 0

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Average Product According to U.S. Department of Labor, output per hour of labor for nonfarm business increased at an annual rate of 2.1% from 1991 to 2000 Measure of productivity is measured in physical quantities Called average product (AP) of an input Defined, for labor, as APL = q/L In general, average product (AP) is output (TP) divided by input In Figure 7.1, APL at first increases, reaches a maximum, then declines Productivity of labor, as measured by APL, changes as additional workers are employed Results from short-run condition that all other inputs remain fixed At first, with a relatively small number of workers for a large amount of other inputs Adding an additional worker increases productivity of all workers APL increases However, a point is reached where labor is no longer relatively limited compared with fixed inputs An additional worker will result in APL declining

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Average Product Graphically, we can determine APL from TP curve by considering a line (cord) through origin Slope of a cord through origin is TP divided by labor Since APL is defined as TP divided by labor Slope of a cord through origin is APL at a level of labor where cord intersects TP As number of workers increases, at first cord shifts upward and slope of the cord increases Resulting in increased APL Can continue to shift cord upward and it will continue to intersect TP curve until it finally is tangent to TP curve At this point, APL is at its maximum

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**Law of Diminishing Marginal Returns and Stages of Production**

A firm’s costs will depend on Prices it pays for inputs Technology of combining inputs into output In short run firm can change its output by adding variable inputs to fixed inputs Output may at first increase at an increasing rate However, given a constant amount of fixed inputs, output will at some point increase at a decreasing rate Occurs because at first variable input is limited compared with fixed input As additional workers are added, productivity remains very high Output, or TP, increases at an increasing rate However, as more of variable input is added, it is no longer as limited Eventually, TP will still be increasing, but at a decreasing rate MPL will still be positive, but declining Called Law of Diminishing Marginal Returns (or just diminishing returns)

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**Law of Diminishing Marginal Returns and Stages of Production**

As indicated in Figure 7.1, diminishing marginal returns starts at point A MPL is at a maximum To the left of point A there are increasing returns and at point A constant returns exist Between points A and B, where MPL is declining, diminishing marginal returns exist To the right of point B, marginal productivity is both diminishing and negative (MPL < 0), which violates Monotonicity Axiom TP curve will at some point increase only at a decreasing rate (concave) due to Law of Diminishing Marginal Returns Some production functions may not exhibit increasing returns at first In fact no firm with a profit-maximizing objective will operate in area of increasing returns or negative returns Production functions generally will only be concave With diminishing marginal returns throughout production process Depicted in Figure 7.2

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**Figure 7.2 Production function with diminishing marginal returns throughout**

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**Law of Diminishing Marginal Returns and Stages of Production**

In Figure 7.2 MPL and APL decline throughout Cobb-Douglas production function can also only exhibit diminishing marginal returns throughout production process Can characterize production where all marginal products are positive Useful for representing firms’ technology constraints Given that profit-maximizing firms will only operate in area of diminishing marginal returns where all marginal products are positive Illustrated in Figure 7.3 for a variable level of labor

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**Figure 7.3 Cobb- Douglas production function with labor as the only variable input**

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**Relationship of Marginal Product to Average Product**

In area of diminishing marginal returns, marginal product can intersect with average product As indicated in Figure 7.1, this intersection of MPL and APL occurs where APL is at a maximum If addition to total, marginal unit, is greater (less) [equal to] than overall average Average will rise (fall) [neither rise nor fall] Taking derivative of average results in relationship between marginal product and average product Marginal product is average product plus an adjustment factor (APL/L)L If slope of APL is zero (rising) [falling] Adjustment factor is zero (> 0) [< 0] MPL = APL (MPL > APL) [MPL < APL]

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Output Elasticity Another important relation between an average and marginal product is output elasticity Measures how responsive output is to a change in an input For example, output elasticity of labor, denoted L, is defined as proportionate rate of change in q with respect to L Given production function q = ƒ(K, L) Output elasticity of labor is L = (ln q)/ (ln L) = (q/L)(L/q) = MPL/APL When MPL > APL, L > 1; when 0 < MPL < APL, 0 < L < 1; and when MPL < 0, L < 1 Illustrated in Figure 7.1

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**Table 7.1 Estimated output elasticities for milk**

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Stages of Production Firm must determine profit-maximizing amount of an available input it should employ Use technology of production to determine at what stage of production to add a variable input, say, labor Exact profit-maximizing level of labor within this stage depends on Cost of labor Price received for the firm’s output Specifically, we divide short-run production function into three stages of production

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Stages of Production Stage I includes area of increasing returns and extends up to point where average product reaches a maximum Illustrated in Figure 7.1 Includes a portion of marginal product curve that is declining Marginal product is greater than average product, so average product is rising As long as average product is rising, firm will add variable inputs Fixed inputs are present in uneconomically large proportion relative to variable input Variable input is limited relative to fixed inputs Rational profit-maximizing producer would never operate in Stage I of production Firm would not produce in short run Would produce by using fewer units of fixed inputs in long run Fixed inputs become variable Reduction of fixed inputs would result in entire set of product curves shifting leftward Results in Stage I ending at a lower level of output Illustrated in Figure 7.4

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**Figure 7.4 Shifts in stages of production with a reduction in the level of fixed inputs**

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Stages of Production Rational producer will also not operate in Stage III of production Range of negative marginal product for variable input In Stage III, TP is actually declining as more of the variable input is added Figures 7.1, 7.2, and 7.4 illustrate Stage III Additional units of the variable input Stage III actually cause a decline in total output Even if units of variable input were free, a rational producer would not employ them beyond the point of zero marginal product In Stage III, variable input is combined with fixed input in uneconomically large proportions Indeed, point of zero MP, for variable input, is called intensive margin Point of maximum AP of variable input is called extensive margin A firm will operate between extensive and intensive margins Stage II of production Both AP and MP of variable input are positive but declining Output elasticity is between 0 and 1 In contrast, output elasticity for variable input is < 0 in Stage III and > 1 in Stage I

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Two Variable Inputs Assumed a different combination of, say two, inputs will produce same level of output For example, in manufacturing microwave ovens, greater use of plastics may be substituted for a reduction in metal use Indifference curves represent a consumer’s preferences for different combinations of two goods with utility remaining constant In production theory isoquants represent different input combinations that may be used to produce a specified level of output Iso means equal and quant stands for quantity An isoquant is a locus of points representing same level of output or equal quantity For movements along an isoquant Level of output remains constant Input ratio changes continuously Isoquants are the same concept as indifference mapping Equal utility along same indifference curve replaced by equal output level along same isoquant Figure 7.5 represents a possible production function for two inputs

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**Figure 7.5 Isoquant map for two variable inputs, capital, K, and labor, L**

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**Marginal Rate of Technical Substitution (MRTS)**

In Figure 7.5, isoquants are drawn with a negative slope Based on assumption that substituting one input for another can result in output not changing A measure for this substitution is marginal rate of technical substitution (MRTS) Defined as negative of slope of an isoquant Measures how easy it is to substitute one input for another holding output constant Similar to concept of MRS in consumer theory MRTS measures reduction in one input per unit increase in the other that is just sufficient to maintain a constant level of output

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**Convex and Negatively Sloping Isoquants**

Can establish underlying assumptions of negatively sloped and convex-to-the-origin isoquant by developing relationship between MRTS and MPs MRTS (K for L) = MPL ÷ MPK Take total derivative of production function, q = ƒ(K, L) dq = MPLdL + MPKdK Along an isoquant dq = 0, output is constant Thus MPLdL = -MPKdK Solving for the negative of the slope of the isoquant yields Along an isoquant, gain in output from increasing L slightly is exactly balanced by loss in output from a suitable decrease in K For isoquants to be negatively sloped, both MPL and MPK must be positive Ridgelines trace out boundary in isoquant map where marginal products are positive See Figure 7.6 Ridgelines are isoclines (equal slopes) where MRTS is either zero or undefined for different levels of output

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**Figure 7.6 Ridgelines in the isoquant map**

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**Convex and Negatively Sloping Isoquants**

MRTS results in isoquants drawn strictly convex to origin Result is analogous to relationship between MRS and strictly convex indifference curves For high ratios of K to L MRTS is large Indicating that a great deal of capital can be given up if one more unit of labor becomes available Assumption of strictly convex isoquants is related to Law of Diminishing Marginal Returns Given MRTS(K for L) = MPL/MPK Movement from A to B in Figure 7.6 results in an increase in labor Corresponding decrease in MPL Decrease in capital with a corresponding increase in MPK A firm will always operate in Stage II of production Characterized by diminishing marginal returns Stage II of production, for both the variable inputs, is represented by strictly convex isoquants In Figure 7.6, a rational producer will only operate somewhere between points D and C

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**Stages of Production in the Isoquant Map**

Can illustrate stages of production in isoquant map by fixing one of the inputs A situation where capital is fixed at some level is indicated by horizontal line at A in Figure 7.7 In short run, firm must operate somewhere on this line At Stage I, labor input is small relative to fixed level of capital Marginal product of capital and MRTS are negative Isoquants have positive slopes At point B, MRTS is undefined, MPK is zero, and APL equals MPL This is demarcation between Stages I and II of production In Stage II of production, all isoquants are strictly convex and have negative slopes At point C, marginal product of labor is zero Corresponds to line of demarcation between Stages II and III

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**Figure 7.7 Stages of production in the isoquant map**

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**Classifying Production Functions**

Production functions represent tangible (measurable) productive processes Economists pay more attention to actual form of these functions than to form of utility functions Resulted in classification of production functions in terms of returns to scale and substitution possibilities Empirical estimates of actual production functions For some production processes it may be extremely difficult if not impossible to substitute one input for another

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Returns to Scale Measure how output responds to increases or decreases in all inputs together Long-run concept since all inputs can vary For example, if all inputs are doubled, returns to scale determine whether output will double, less than double, or more than double In many cases, it is difficult to change some inputs at will and increase inputs proportionally Firms do attempt to control as much of environmental conditions as feasible Examples in agriculture include greenhouses or pesticides Assuming it is possible to proportionally change all inputs, a production function can exhibit constant, decreasing, or increasing returns to scale across different output ranges However, it is generally assumed, for simplicity, production functions only exhibit either constant, decreasing, or increasing returns to scale

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**Returns to Scale Specifically, given production function**

q= ƒ(K, L) A explicit definition of constant returns to scale is ƒ(K, L) = ƒ (K, L) = q, for any > 0 If all inputs are multiplied by some positive constant , output is multiplied by that constant also If production function is homogeneous Constant returns to scale production function is homogeneous of degree 1 or linear homogeneous in all inputs Isoquants are radial blowups and equally spaced as output expands (Figure 7.8)

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**Figure 7.8 Returns to scale**

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Returns to Scale Decreasing returns to scale exists if output is increased proportionally less than all inputs ƒ(K, L) < ƒ(K, L) = q Increasing returns to scale exists if output increases more than proportional increase in inputs ƒ(K, L) > ƒ(K, L) = q

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**Determinants of Returns to Scale**

Adam Smith established that returns to scale is result of two forces Division of labor An increase in all inputs increases division of labor and results in increased efficiency Production might more than double Managerial difficulties Result in decreased efficiency Production might not double Early 20th century concept of assembly-line mass production is based on division of labor Each worker has a specialized task to perform for each product being assembled Worker becomes very skilled at this task Increases productivity Example: Henry Ford experienced increasing returns to scale in automobile manufacturing

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**Determinants of Returns to Scale**

One cause of managerial difficulties in mass production is required stockpiling of parts and supplies Inventory control must be maintained, where an accounting of parts is required Results in a significant amount of inputs allocated to storage and accounting of inventories Results in decreasing returns to scale Just-in-time delivery systems are helping to mitigate these factors One problem with just-in-time production Increased vulnerability of firms to supply disruptions Without a stockpile of parts, such disruptions could shut down production fairly quickly

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**Determinants of Returns to Scale**

Postindustrial manufacturing is shifting away from mass production of a standardized product and evolving toward mass customization Called agile manufacturing Results in increasing returns to scale

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**Determinants of Returns to Scale**

As a firm increases in size by increasing all inputs, another possible cause of decreasing returns to scale is Allocation of inputs for environmental and local service projects As a firm employs more inputs and increases output, it becomes increasingly more exposed to public concerns associated with its production practices To enhance and maintain goodwill within its community, firm will allocate additional inputs for environmental and local service projects Contributes to decreasing returns to scale

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**Returns to Scale and Stages of Production**

Determine relationship between returns to scale and stages of production by assuming a linear homogeneous production function (homogeneous of degree 1) Implies a constant returns to scale production function Applying Euler’s Theorem to production function q = ƒ(K, L) we obtain q = L(MPL) + K(MPK) Dividing by L gives APL = MPL + (K/L)MPK Solving for MPK yields MPK = (L/K)(APL – MPL)

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**Returns to Scale and Stages of Production**

Assuming constant returns to scale, we define stages of production as Stage I MPL > APL > 0, MPK < 0 Stage II APL > MPL > 0, APK > MPK > 0 Stage III MPL < 0, MPK > APK > 0 Stages I and III are symmetric for a constant returns to scale production function Given Monotonicity Axiom, only relevant region for production is Stage II

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**Elasticity of Substitution**

A firm may compensate for a decrease in use of one input by an increase in use of another Heinrich von Thunen collected evidence from his farm in Germany that suggested ability of one input to compensate for another was significant Postulated principle of substitutability Possible to produce a constant output level with a variety of input combinations Principle of substitutability is not an economic law There are production functions for which inputs are not substitutable However, for those functions where inputs are substitutable Degree that inputs can be substituted for one another is an important technical relationship for producers Production functions may also be classified in terms of elasticity of substitution Measures how easy it is to substitute one input for another Determines shape of a single isoquant

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**Elasticity of Substitution**

In Figure 7.9 consider a movement from A to B Results in capital/labor ratio (K/L) decreasing Profit-maximizing firm is interested in determining a measure of ease in which it can substitute K for L If MRTS does not change at all for changes in K/L, the two inputs are perfect substitutes If MRTS changes rapidly for small changes in K/L, substitution is difficult If there is an infinite change in the MRTS for small changes in K/L (called fixed proportions), substitution is not possible A scale-free measure of this responsiveness is elasticity of substitution

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**Figure 7.9 Capital/labor ratio° and MRTS, K°/L° > K'/L'**

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**Elasticity of Substitution**

Defined as percentage change in K/L divided by percentage change in MRTS Along a strictly convex isoquant, K/L and MRTS move in same direction Elasticity of substitution is positive In Figure 7.9, a movement from A to B results in both K/L and MRTS declining Relative magnitude of this change is measured by elasticity of substitution If it is high, MRTS will not change much relative to K/L and the isoquant will be less curved (less strictly convex) A low elasticity of substitution gives rather sharply curved isoquants Possible for the elasticity of substitution to vary for movements along an isoquant and as the scale of production changes However, frequently elasticity of substitution is assumed constant

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**Elasticity of Substitution: Perfect-Substitute**

= , a perfect-substitute technology Analogous to perfect substitutes in consumer theory A production function representing this technology exhibits constant returns to scale ƒ(K, L) = aK + bL = (aK + bL) = ƒ(K, L) All isoquants for this production function are parallel straight lines with slopes = -b/a See Figure 7.10

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**Figure 7.10 Elasticity of substitution for perfect-substitute technologies**

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**Elasticity of Substitution: Leontief**

= 0, a fixed-proportions (or Leontief ) technology Analogous to perfect complements in consumer theory Characterized by zero substitution A production technology that exhibits fixed proportions is This production function also exhibits constant returns to scale

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**Elasticity of Substitution: Leontief**

Figure 7.11 illustrates a fixed proportions function Capital and labor must always be used in a fixed ratio Marginal products are constant and zero Violates Monotonicity Axiom and Law of Diminishing Marginal Returns Isoquants for this technology are right angles Are not smooth curves, but are kinked At kink, MRTS is not unique—can take on an infinite number of positive values K/L is a constant, d(K/L) = 0, which results in = 0

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**Figure 7.11 Elasticity of substitution for fixed-proportions technologies**

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**Elasticity of Substitution; Cobb-Douglas**

= 1, Cobb-Douglas technology Isoquants are strictly convex Assumes diminishing MRTS (Figure 7.12) An example of a Cobb-Douglas production function is q = ƒ(K, L) = aKbLd a, b, and d are all positive constants Useful in many applications because it is linear in logs

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**Figure 7.12 Isoquants for a Cobb-Douglas production function**

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**Elasticity of Substitution; Cobb-Douglas**

= some positive constant Constant elasticity of substitution (CES) production function can be specified q = [K- + (1 - )L-]-1/ > 0, 0 ≤ ≤1, ≥ -1 is efficiency parameter is a distribution parameter is substitution parameter Elasticity of substitution is = 1/(1 + ) Useful in empirical studies

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Chapter 9 Production Functions

Chapter 9 Production Functions

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