Chapter 9 Production Functions

Chapter Overview Marginal and Average Products Isoquants 1. The Production Function Marginal and Average Products Isoquants The Marginal Rate of Technical Substitution 2. Returns to Scale Technical Progress Some Special Functional Forms

Key Concepts Productive resources, such as labor and capital equipment, that firms use to manufacture goods and services are called inputs or factors of production. The amount of goods and services produces by the firm is the firm’s output. Production transforms a set of inputs into a set of outputs Technology determines the quantity of output that is feasible to attain for a given set of inputs.

Key Concepts Production Function: Q = output K = Capital L = Labor The production function tells us the maximum possible output that can be attained by the firm for any given quantity of inputs. Production Function: Q = output K = Capital L = Labor

The Production Function & Technical Efficiency Technically efficient: Sets of points in the production function that maximizes output given input (labor) Technically inefficient: Sets of points that produces less output than possible for a given set of input (labor)

The Production Function & Technical Efficiency

Short run versus Long run Short run - a period of time so brief that at least one factor of production cannot be varied practically. Fixed input - a factor of production that cannot be varied practically in the short run. Variable input - a factor of production whose quantity can be changed readily by the firm during the relevant time period. Long run - a lengthy enough period of time that all inputs can be varied.

Short-Run Production In the short run, the firm’s production function is q = f(L, K) where q is output, L is workers, and K is the fixed number of units of capital.

Total Product, Marginal Product, and Average Product of Labor with Fixed Capital

Total Product of Labor Total product of labor- the amount of output (or total product) that can be produced by a given amount of labor.

Marginal Product of Labor Marginal product of labor (MPL ) - the change in total output, Dq, resulting from using an extra unit of labor, DL, holding other factors constant: MPL = Q/L (holding constant all other inputs) MPK = Q/K

Marginal Physical Product Marginal physical product is the additional output that can be produced by employing one more unit of that input holding other inputs constant

Diminishing Marginal Productivity Marginal physical product depends on how much of that input is used In general, we assume diminishing marginal productivity

Average Product of Labor Average product of labor (APL ) - the ratio of output, q, to the number of workers, L, used to produce that output: APL = Q/L APK = Q/K

Production Relationships with Variable Labor 110 Output, q, Units per day 90 B 56 A Diminishing Marginal Returns sets in! 4 6 11 L, Workers per day (b) L a MP 20 , L AP b 15 Average product, APL Marginal product, MPL c 4 6 11 L, Workers per day

Total Product

Law of Diminishing Marginal Returns If a firm keeps increasing an input, holding all other inputs and technology constant, the corresponding increases in output will become smaller eventually. That is, if only one input is increased, the marginal product of that input will diminish eventually.

Example: Short run Production Function Suppose the production function for a firm can be represented by q = f(k,l) = 600k 2l2 - k 3l3 To construct MPl and APl, we must assume a value for k let k = 10 The production function becomes q = 60,000l2 - 1000l3

Long-Run Production In the long run both labor and capital are variable inputs. It is possible to substitute one input for the other while holding output constant.

Isoquants Isoquant - a curve that shows the efficient combinations of labor and capital that can produce a single (iso) level of output (quantity). Equation for an isoquant: q = f (L, K).

Isoquant Map Each isoquant represents a different level of output output rises as we move northeast q = 30 q = 20 k per period l per period

Properties of Isoquants The farther an isoquant is from the origin, the greater the level of output. Isoquants do not cross. Isoquants slope downward

Substituting Inputs Marginal rate of technical substitution (MRTS) - the number of extra units of one input needed to replace one unit of another input that enables a firm to keep the amount of output it produces constant. Slope of Isoquant!

How the Marginal Rate of Technical Substitution Varies Along an Isoquant 16 , Units of capital D K = –6 b 10 D L = 1 K –3 1 c 7 –2 d 1 5 e –1 4 1 q = 10 1 2 3 4 5 6 7 8 9 10 L , W o r k Ers

Production Function With Two Variable Inputs: Moving from short run to long run Q = f(L, K)

Production Function With Two Variable Inputs

Production With Two Variable Inputs Isoquants show combinations of two inputs that can produce the same level of output. Firms will only use combinations of two inputs that are in the economic region of production.

Production With Two Variable Inputs Isoquants

Economic Region of Production The firm should not use certain combinations of outputs in the long run no matter how cheap they are. These input combinations are represented by the portion of the isoquant curve that has a positive slope. A positively sloped isoquant means that merely to maintain the same level of production, the firm must use more of both the inputs if it increases its use of one of the inputs. Ridge Lines: The lines connecting the points where the marginal product of an input is equal to zero (one line for each input) in the isoquant map and forming the boundary for the economic region of production.

Production With Two Variable Inputs Economic Region of Production

The Economic Region of Production The economic region of production is the range in an isoquant diagram where both inputs have a positive marginal product. It lies inside the ridge lines. A profit maximizing firm will never try to produce using input combinations outside the ridge lines.

MARGINAL RATE OF TECHNICAL SUBSTITUTION Isoquants are downward sloping and convex to the origin. The slope of the isoquant (dK / dL) defines the degree of substitutability of the factors of production. The slope of the isoquant decreases (in absolute terms) as we move downwards along the isoquant, showing the increasingdifficulty in substituting in substituting K for L.

Production Functions—Two Special Cases Two extreme cases of production functions show the possible range of input substitution in the production process: 1) the case of perfect substitutes and 2) the fixed proportions production function The fixed-proportions production function describes situations in which methods of production are limited.

Production With Two Variable Inputs Perfect Substitutes Perfect Complements When the isoquants are straight lines, the MRTS is constant. The rate at which K and L can be substituted for each other is the same no matter what level of inputs is being used. When the isoquants are L-shaped, only one combination of labor and capital can be used to produce a given output. Adding more labor alone does not increase output, nor does adding more capital alone.

Substitutability of Inputs

Elasticity of Substitution A measure of how easy is it for a firm to substitute labor for capital. It is the percentage change in the capital-labor ratio for every one percent change in the MRTSL,K along an isoquant.

Elasticity of Substitution The elasticity of substitution () measures the proportionate change in K/L relative to the proportionate change in the MRTSx,y along an isoquant The value of  will always be positive because K/L and MRTS move in the same direction.

Elasticity of Substitution Both MRTS and K/L will change as we move from point A to point B A B  is the ratio of these proportional changes K per period  measures the curvature of the isoquant RTSA RTSB (k/l)A (k/l)B q = q0 L per period

Elasticity of Substitution If  is high, the MRTS will not change much relative to K/L the isoquant will be relatively flat If  is low, the MRTS will change by a substantial amount as K/L changes the isoquant will be sharply curved It is possible for  to change along an isoquant or as the scale of production changes

Elasticity of Substitution K "The shape of the isoquant indicates the degree of substitutability of the inputs…"  = 0  = 1  = 5  =  L

Elasticity of Substitution If we define the elasticity of substitution between two inputs to be the proportionate change in the ratio of the two inputs to the proportionate change in MRTS, we need to hold output and the levels of other inputs constant

Returns to Scale How much will output increase when ALL inputs increase by a particular amount? suppose that all inputs are doubled, would output double?

Returns to Scale If the production function is given by q = f(k,l) and all inputs are multiplied by the same positive constant (t >1), then

Returns to Scale RTS = [%Q]/[% (all inputs)] How much will output increase when ALL inputs increase by a particular amount? RTS = [%Q]/[% (all inputs)] If a 1% increase in all inputs results in a greater than 1% increase in output, then the production function exhibits increasing returns to scale. If a 1% increase in all inputs results in exactly a 1% increase in output, then the production function exhibits constant returns to scale. If a 1% increase in all inputs results in a less than 1% increase in output, then the production function exhibits decreasing returns to scale.

Returns to Scale K 2K Q = Q1 K Q = Q0 L L 2L

Returns to Scale

Returns to Scale It is possible for a production function to exhibit constant returns to scale for some levels of input usage and increasing or decreasing returns for other levels the degree of returns to scale is generally defined within a fairly narrow range of variation in input usage

Constant Returns to Scale Constant returns-to-scale production functions are homogeneous of degree one in inputs f(tk,tl) = t1f(k,l) = tq The marginal productivity functions are homogeneous of degree zero if a function is homogeneous of degree k, its derivatives are homogeneous of degree k-1

Constant Returns to Scale Along a ray from the origin (constant K/L), the MRTS will be the same on all isoquants k per period q = 3 q = 2 q = 1 The isoquants are equally spaced as output expands l per period

f(tx1,tx2,…,txn) = tkf(x1,x2,…,xn)=tkq Returns to Scale Returns to scale can be generalized to a production function with n inputs q = f(x1,x2,…,xn) If all inputs are multiplied by a positive constant t, we have f(tx1,tx2,…,txn) = tkf(x1,x2,…,xn)=tkq If k = 1, we have constant returns to scale If k < 1, we have decreasing returns to scale If k > 1, we have increasing returns to scale

The Linear Production Function Suppose that the production function is q = f(k,l) = ak + bl This production function exhibits constant returns to scale f(tk,tl) = atk + btl = t(ak + bl) = tf(k,l) All isoquants are straight lines

The Linear Production Function Capital and labor are perfect substitutes k per period MRTS is constant as K/L changes slope = - b/a q1 q2 q3  =  l per period

Fixed Proportions Suppose that the production function is q = min (ak,bl) a,b > 0 Capital and labor must always be used in a fixed ratio the firm will always operate along a ray where K/L is constant Because K/L is constant,  = 0

Fixed Proportions No substitution between labor and capital is possible K/L is fixed at b/a q3/b q3/a k per period q1 q2 q3  = 0 l per period

Cobb-Douglas Production Function Suppose that the production function is q = f(k,l) = Akalb A,a,b > 0 This production function can exhibit any returns to scale f(tk,tl) = A(tk)a(tl)b = Ata+b kalb = ta+bf(k,l) if a + b = 1  constant returns to scale if a + b > 1  increasing returns to scale if a + b < 1  decreasing returns to scale

Cobb-Douglas Production Function The Cobb-Douglas production function is linear in logarithms ln q = ln A + a ln k + b ln l a is the elasticity of output with respect to k b is the elasticity of output with respect to l

CES Production Function Suppose that the production function is q = f(k,l) = [k + l] /   1,   0,  > 0  > 1  increasing returns to scale  < 1  decreasing returns to scale For this production function  = 1/(1-)  = 1  linear production function  = -  fixed proportions production function  = 0  Cobb-Douglas production function

A Generalized Leontief Production Function: A special case Suppose that the production function is q = f(k,l) = k + l + 2(kl)0.5 Marginal productivities are fk = 1 + (k/l)-0.5 fl = 1 + (k/l)0.5 Thus,

A Generalized Leontief Production Function: A special case This function has a CES form with  = 0.5 and  = 1 The elasticity of substitution is

Technological Progress Definition: Technological progress (or invention) Shifts the production function by allowing the firm to achieve more output from a given combination of inputs or the same output with fewer inputs.

Neutral Technological Progress Technological progress that decreases the amounts of labor and capital needed to produce a given output. Affects MRTSK,L

Technological Progress Labor saving technological progress results in a fall in the MRTSL,K along any ray from the origin Capital saving technological progress results in a rise in the MRTSL,K along any ray from the origin.

Labor Saving / Capital deepening Technological Progress Technological progress that causes the marginal product of capital to increase relative to the marginal product of labor

Capital Saving / Labour deepening Technological Progress Technological progress that causes the marginal product of labor to increase relative to the marginal product of capital

Technical Progress Methods of production change over time Following the development of superior production techniques, the same level of output can be produced with fewer inputs the isoquant shifts in

Technical Progress Suppose that the production function is q = A(t)f(k,l) where A(t) represents all influences that go into determining q other than k and l changes in A over time represent technical progress A is shown as a function of time (t) dA/dt > 0

Technical Progress Differentiating the production function with respect to time we get Dividing by q gives us

Technical Progress For any variable x, [(dx/dt)/x] is the proportional growth rate in x denote this by Gx Then, we can write the equation in terms of growth rates

Technical Progress Since