Production

Technology The physical laws of nature and limits of material availability and human understanding that govern what is possible in converting inputs into output.

Inputs, Factors of Production Land (incl. raw materials) Labor (including human capital) Capital (physical capital, like machinery and buildings)

Production Function A firm’s production function for a particular good (q) shows the maximum amount of the good that can be produced using alternative combinations of capital (K) and labor (L). q = f(K, L) Producing less than the maximum is always possible and all levels of output below the maximum are feasible and define the “production set.”

Production Function q q = f(K, L) K L

All points “under” the production function Production set q = f(K, L) q K All points “under” the production function L

Production Function and Isoquants q = f(K, L) q In the long run, all combinations of inputs are possible K Isoquants are horizontal cross sections of the production function projected on the base plane. L

Short Run, Long Run Long Run, quantities of ALL inputs used in production can be varied. Short Run, the quantity of at least one input used in production is fixed. ALL production takes place in a short run environment. You can think of the long run as the ability to move from one short run environment to another. Actual time it takes to make this move depends on many factors, technical, economic and regulatory.

The model Standard basic model to think of production as a function of K and L. L variable in the short run while K is fixed.

Short run, hold K fixed. L q = f(K, L) q In the short run, K is fixed and only L can vary K The cross section of the production function at a fixed K is the short run production function L

More, fixed K q = f(K, L) q In the short run, K is fixed and only L can vary K The cross section of the production function at a fixed K is the short run production function L

Three levels of K q = f(K=K3, L) q q = f(K=K2, L) In the short run, we assume, the quantity of at least one input used --but not all -- is fixed. q = f(K=K1, L) L

L constant q = f(K,L=L3) q L and K are just names for inputs. Either one could be fixed in the short run. Just intuitive that K is fixed and L variable in the SR. q = f(K,L=L2) q = f(K,L=L1) K

SR and then LR First we’ll think about the short run, and then turn to the long run.

Marginal Physical Product Marginal Product is the additional output that can be produced by employing one more unit of that input holding other inputs constant, so a short run concept

Marginal Productivity Assumptions We assume managers are not going to allow employees in the building if they bring total output down. However, over the range where profit is maximized, marginal products are positive.

Increasing and Diminishing Marginal Product (assumes something is fixed) Empirically, economists find that most production processes exhibit (as L increases from zero): Increasing Marginal Returns – each worker added causes output to increase by more than the previous worker (workers are not able to gain from specialization, K is fixed) And then… Decreasing Marginal Returns –workers added to production add less to output than the previous worker (workers crowd each other as they try to share a fixed amount of capital)

Marginal Productivity Assumptions Because of IMR and DMR, these are possible: Whether MP is always diminishing or whether it first increases and then diminishes depends on the context of the economic discussion. In economics classes, we think of increasing marginal returns and then diminishing marginal returns (need this for a U-shaped MC curve).

MP Assumptions As revenue or profit max means producing where MC is rising (MPL is falling), theoretically, we tend to ignore IMR and assume DMR

Malthus and Diminishing Marginal Productivity He argued that population growth meant declining marginal labor productivity His mistake was holding all else (except labor, i.e. population) constant. Ignored technological growth. Productivity was actually growing exponentially, but at such a slow rate that he did not see it. Per Capita Output Watts’s Steam Engine Economic growth of IR first noticed in the 1830s Essay on the Principle of Population, 1st ed (1798) Malthus Dies, 1834 Year 1800 1840 1880

Effect of Technology If we think of higher technology as being like having MORE capital, then you can think of the industrial revolution the result of fLK > 0 and a rapid expansion of K.

Average Physical Product Labor productivity is often measured by average productivity.

Specific Function Suppose the production function for tennis balls can be represented by To construct MPL and APL, we must assume a value for K let K = 10 The production function becomes

SR Production Function (K = 10) q L

Marginal Product The marginal product function is When MPL = 0, total product is maximized at L = 80.

SR Production Function (K = 10) q Slope of function is MPL at that level of L L

Inflection Point Output where MPL goes from increasing to decreasing (inflection point)

SR Production Function (K = 10) At inflection point, MPL is at its highest q LI L

Average Product To find average productivity, we hold K=10 and solve

SR Production Function (K = 10) Slope of ray from origin to curve at any L is = APL Slope of this ray =36,000 So APL =36,000 when L= 60 q LA L

MPL and APL In fact, when L = 60, both APL and MPL are equal to 36,000 Thus, when APL is at its maximum, APL and MPL are equal So long as a worker hired has a MPL higher than the overall APL, the APL will continue to rise. If the MPL = APL, But if a worker hired has a MPL below the overall APL, the APL will fall.

MPL and APL LI LA

MPL and APL Where the ray is also tangent, MPL = APL

Long Run All mixes of K and L are possible. Daily decisions about production always have some fixed inputs, so the long run is a planning time horizon.

Isoquant Map Each isoquant represents a different level of output, q0 = f(K0,L0), q1 = f(K1,L1) K q1 = 30 q0 = 20 L

Marginal Rate of Technical Substitution (TRS, RTS, MRTS) The slope of an isoquant shows the rate at which L can be substituted for K, or how much capital must be hired to replace one Laborer. K A KA B KB q0 = 20 L LA LB

TRS and Marginal Productivities Take the total differential of the production function: Along an isoquant dq = 0, so

Alternatively: Implicit Function Rule

Diminishing TRS Again, for demand (this time of inputs) to be well behaved, we need production technology (akin to preferences) to be convex. K Which means, the slope rises, gets closer to zero as L increases. And means the TRS falls as L increases. L

Diminishing TRS To show that isoquants are convex (that dK/dL increases – gets closer to zero) along all isoquants) That is, either: The level sets (isoquants) are strictly convex The production function is strictly quasi-concave

Convexity (level curves) dK/dL increases along all indifference curves We can use the explicit equation for an isoquant, K=K(L, q0) and find to demonstrate convexity. That is, while negative, the slope is getting larger as L increases (closer to zero). But we cannot always get a well defined equation for an isoquant. Binger and Hoffman, page 115

Alternatively (level curves) As above, starting with q0 =f(K,L), So convexity if

Convexity (level curves) And, that is *Note that fK3 > 0 What of: fL > 0, monotonacity fK > 0, monotonacity fLL < 0, diminishing marginal returns fKK < 0, diminishing marginal returns fLK = ? Binger and Hoffman, page 115

Strict Quasi-Convexity (production function) Also, convexity of technology will hold if the production function is strictly quasi-concave A function is strictly quasi-concave if its bordered Hessian is negative definite Binger and Hoffman, page 115

Negative Definite (production function) So the production function is strictly quasi-concave if 1. –fLfL < 0 2. 2fLfKfLK-fK2fLL -fL2fKK > 0 Related to the level curve result: Remembering that a convex level set comes from this We can see that strict convexity of the level set and strict quasi-concavity of the function are related, and each is sufficient to demonstrate that both are true.

TRS and Marginal Productivities Intuitively, it seems reasonable that fLK should be positive if workers have more capital, they will be more productive But some production functions have fKL < 0 over some input ranges assuming diminishing TRS means that MPL and MPK diminish quickly enough to compensate for any possible negative cross-productivity effects

TRS and MPL and MPK Back to our sample production function: For this production function

IMR and DMR vs. NMR Pull out a few terms If K = 10, then MPL = 0 at L=80

IMR vs. DMR Because If K = 10, then inflection point at L=40 fLL> 0 and fKK > 0 if K*L < 400 fLL< 0 and fKK < 0 if K*L > 400 If K = 10, then inflection point at L=40

Cross Effect Cross differentiation of either of the marginal productivity functions yields fLK > 0 if KL < 533 fLK < 0 if KL > 533 If K = 10 fLK> 0 when L < 53.3 fLK< 0 when L > 53.3

A Diminishing TRS? Strictly Quasi-Concave if Lots of parts that have different signs depending on K and L. + ? ? ? ? + ?

Returns to Scale How does output respond to increases in all inputs together? suppose that all inputs are doubled, would output double? Returns to scale have been of interest to economists since Adam Smith’s pin factory

Returns to Scale Two forces that occur as inputs are scaled upwards greater division of labor and specialization of function loss in efficiency (bureaucratic inertia) management may become more difficult fall of the Roman Empire? General Motors?

Returns to Scale Starting at very small scale and then expanding, firms tend to exhibit increasing returns to scale at small scale, which changes to constant returns over a range, and then when they get larger, face decreasing returns to scale. Obviously, the scale at each transition can vary. Vacuum Cleaner Repair Shops Steel Mills Doughnut Shops Automobile manufacture Empirical analysis reveals that established firms tend to operate at a CRS scale.

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 Constant Returns to Scale q = K.5L.5 What if we increase all inputs by a factor of t? (tK).5(tL).5 = ? t(K).5(L).5 = tq For t > 1, increase all inputs by a factor of t and output increases by a factor of t I.e. increase all inputs by x% and output increases by x%

Returns to Scale Decreasing Returns to Scale q = K.25L.25 What if we increase all inputs by a factor of t? (tK).25(tL).25 = ? t.5(K).25(L).25 = t.5q, which is < tq For t > 1, increase all inputs by a factor of t and output increases by a factor < t I.e. increase all inputs by x% and output increases by less than x%

Returns to Scale Increasing Returns to Scale q = K1L1 What if we increase all inputs by a factor of t? (tK)1(tL)1 = ? tq < t2(K)1(L)1 = t2q, which is > tq For t>1, increase all inputs by a factor of t and output increases by a factor > t I.e. increase all inputs by x% and output increases by more than x%

Returns to Scale Using the usual homogeneity notation, alternatively, it is notated, for t > 0. That is, production is homogeneous of degree k.

Returns to Scale, Example Solve for k q = K.4L.4 tkq = (tK).4(tL).4 = t.8(K).4(L).4 k ln(t) + ln(Q) = .8ln(t)+.4ln(K)+.4ln(L) k ln(t) = .8ln(t)+.4ln(K)+.4ln(L) - ln(Q) k ln(t) = .8ln(t)+.4ln(K)+.4ln(L)-.4ln(K)-.4ln(L) k ln(t) = .8ln(t) k ln(t) = .8ln(t)/ln(t) k = .8, production is Homogeneous of degree .8 k < 1 so DRS

Returns to Scale by Elasticity What is the % change in output for a t% increase in all inputs? Generally evaluated at t = 1 CRS: q,t =1 DRS: q,t < 1 IRS: q,t > 1

Returns to Scale by Elasticity What is the % change in output for a t% increase in all inputs? Evaluated at t = 1. In this example, RTS varies by K and L.

Constant Returns to Scale is Special Empirically, firms operate at a CRS scale. If a function is HD1, then the first partials will be HD0. If Then

Constant Returns to Scale is Special Obviously, if CRS, we can scale by any t > 0 But let’s pick a specific scale factor, 1/L: If Then Which tells us that if production is CRS, then it is also homothetic. Isoquants are radial expansions with the RTS the same along all linear expansion paths.

Constant Returns to Scale The marginal productivity of any input depends on the ratio of capital and labor not on the absolute levels of these inputs Therefore the TRS between K and L depends only on the ratio of K to L, not the scale of operation That is, increasing all inputs by x% does not affect the TRS The production function will be homothetic (TRS constant along ray from origin) Geometrically, this means all of the isoquants are radial expansions of one another

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

Economies of Scale (not Returns to Scale) In the real world, firms rarely scale up or down all inputs (e.g. management does not typically scale up with production). Economies of scale: %ΔLRAC/%ΔQ Economies of scale if < 0 Diseconomies of scale if > 0

Elasticity of Substitution The elasticity of substitution () measures the proportionate change in K/L relative to the proportionate change in the TRS along an isoquant And as was demonstrated earlier, elasticity is the effect of a change in one log on another. The value of  will always be positive because K/L and TRS move in the same direction

Elasticity of Substitution Both RTS and K/L will change as we move from point A to point B A B  is the ratio of these proportional changes K  measures the curvature of the isoquant TRSA TRSB (K/L)A q = q0 (K/L)B L

Elasticity of Substitution If  is low, the K/L will not change much relative to TRS the isoquant will be relatively flat If  is high, the K/L will change by a substantial amount as TRS changes the isoquant will be sharply curved More interesting when you remember that to minimize cost, TRS = pL/pK so TRS changes with input prices.

Elasticity of Substitution K q=g(K,L) It is possible for  to change along an isoquant or as the scale of production changes g >  f q=f(K,L) L

Elasticity of Substitution Solving for σ can be tricky, but, we can employ this calculus trick (especially useful for homothetic production functions): This allows us to turn this problem Into the (sometimes) easier

Elasticity of Substitution CRS is Special Again For CRS production functions only we have this option too Let q = f(K,L)

Common Production Functions Linear (inputs are perfect substitutes) Fixed Proportions (inputs are perfect compliments) Cobb-Douglas CES Generalized Leontief

The Linear Production Function (inputs are perfect substitutes) 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

Linear Production Function

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

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 q1 q2 q3 K  = 0 L

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

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 Statistically, this is how we estimate production functions via regression analysis.

CES Production Function Suppose that the production function is  > 1  increasing returns to scale  = 1  constant returns to scale  < 1  decreasing returns to scale

CES Production Function TRS Note, not a function of scale, γ

CES Production Function σ

CES Production Function For CES At limit as  → 1, σ → ∞, linear production function At limit as  → -, σ → ∞, fixed proportions production function When  = 0, Cobb-Douglas production function

A Generalized Leontief Production Function Suppose that the production function is TRS

A Generalized Leontief Production Function σ

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 inward

Technical Progress Suppose that the production function is q = A(t)f(K(t),L(t)) 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 Which simplifies to

Technical Progress Since And so

Technical Progress Dividing by q gives us

Technical Progress Expand by strategically adding in K/K and L/L

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 Note the elasticities Yielding Growth is a function of technical change and growth in the use of inputs.

Solow, US Growth 1909-1949 Solow estimated the following Plug these in Gq = 2.75% GL = 1.00% GK = 1.75% eq,L = .65 eq,K = .35 Plug these in And GA = 1.5% Conclusion, technology grew at a 1.5% rate from 1909-1949. 55% of GDP growth in the period.

Appendix Full derivations of TRS and convexity in production.

RTS and Marginal Productivities: Implicit Function Rule

Substitute

And get to…

And get to…

Convexity, Increasing dK/dL

Diminishing TRS TRS diminishing if this < 0 Which is the same thing.

Alternatively, the Bordered Hessian Strictly Quasi-Concave if and which looks a lot like the negative of this: