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

K

L

q q = f(K, L)

Production set

K

L

qq = f(K, L)

All points “under” the production

function

Production Function and Isoquants

K

L

q q = f(K, L)

In the long run, all combinations of inputs are possible

Isoquants are horizontal cross sections of the

production function projected on the base

plane.

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.

K

L

q q = f(K, L)

In the short run, K is fixed and only L can vary The cross

section of the production function at a fixed K is the short run production function

More, fixed K

K

L

q q = f(K, L)

In the short run, K is fixed and only L can vary The cross

section of the production function at a fixed K is the short run production function

Three levels of K

L

q

q = f(K=K1, L)

q = f(K=K2, L)

q = f(K=K3, L)

In the short run, we assume, the quantity of at least one input used --but not all -- is fixed.

L constant

K

q

q = f(K,L=L1)

q = f(K,L=L2)

q = f(K,L=L3)

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.

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

L L

K K

qmarginal product of labor: MP f

Lq

marginal product of capital: MP fK

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.

0

0

L L

K K

qMP f

Lq

MP fK

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).

2L

LL2

2K

KK2

MP ff , , 0

L LMP f

f , , 0K K

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

2L

LL2

2K

KK2

MP ff 0

L LMP f

f 0K K

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.

Year

Per Capita Output

1800 1840 1880

Essay on the Principle of Population, 1st ed (1798)

Watts’s Steam Engine

Malthus Dies, 1834

Economic growth of IR first noticed in the

1830s

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.

Loutput q f(K,L)

AP =labor input L L

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

3 3

2 2 K Lq f K,L 12K L

100

2 3q f K,L 1,200L 10L

SR Production Function (K = 10)

0 20 40 60 80 100 1200

500000

1000000

1500000

2000000

2500000

3000000

q(K=10)

Series3

L

q2 3q 1,200L 10L

Marginal Product

• The marginal product function is

• When MPL = 0, total product is maximized at L = 80.

2L

2L

2

MP 2,400L 30L

MP 2,400L 30L 0

30L 2,400L30L 2,400L=80

SR Production Function (K = 10)

0 20 40 60 80 100 1200

500000

1000000

1500000

2000000

2500000

3000000

q(K=10)

Series3

L

q

Slope of function is MPL at that level of L

2 3q 1,200L 10L

Inflection Point

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

2L L

LLL

I

f MP 2,400L 30LdMP

f 2,400 60LdL

2,400 60L 0L =40

SR Production Function (K = 10)

0 20 40 60 80 100 1200

500000

1000000

1500000

2000000

2500000

3000000

q(K=10)

Series3

L

qAt inflection point, MPL is at its highest

LI

2 3

L

q 1,200L 10LMP 1,280,000 at L = 40

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

and solve 2

L

L

L

A

f K,LAP 1,200L 10L

LdAP

Maximized where = 0dL

dAP 1,200 20L

dL1,200 20L 020L 1,200L 60

SR Production Function (K = 10)

0 20 40 60 80 100 1200

500000

1000000

1500000

2000000

2500000

3000000

q(K=10)

Series3

L

q

Slope of ray from origin to curve at any L is = APL

LA

2 3

2L

q 1,200L 10L

AP 1200L 10L

Slope of this ray =36,000So APL =36,000 when L= 60

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

0 20 40 60 80 100 120

-80000

-60000

-40000

-20000

0

20000

40000

60000

MPL

APLLI LA

MPL and APL

0 10 20 30 40 50 60 70 80 90

-20000

-10000

0

10000

20000

30000

40000

50000

60000

MPL

APL

0 10 20 30 40 50 60 70 80 900

500000

1000000

1500000

2000000

2500000

3000000

q(K=10)

q(K=10)

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

L

K

• Each isoquant represents a different level of output, q0 = f(K0,L0), q1 = f(K1,L1)

q1 = 30

q0 = 20

dK0

dL

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.

LA

KA

KB

LB

A

B

L

K

q0 = 20

0q q

dKTRS

dL

TRS and Marginal Productivities• Take the total differential of the production

function:

• Along an isoquant dq = 0, soL K

ffdq dL dK

L Kdq MP dL MP dK

0

L K

L

q q K

MP dL MP dK

MPdKTRS defined as, TRS=

dL MP

Alternatively:Implicit Function Rule

L L

K K

Take an implicit function: f(K,L)-q=0 If the conditions of the implict function therorem hold, then there exists an equation K=g(L; q) and

ff MPdK dK dKL or or

fdL dL f dL MPK

0

L

q q K

MPdKAgain,TRS defined as, TRS=

dL MP

Diminishing TRS• Again, for demand (this time of inputs) to be

well behaved, we need production technology (akin to preferences) to be convex.

L

KL

K

MPdKdL MP

Which means, the slope rises, gets closer to zero as L increases.And means the TRS falls as L increases. dK

TRSdL

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.

0

2

2q q

d K0

dL

Alternatively (level curves)• As above, starting with q0 =f(K,L),

• So convexity if

L L

K K

f (K,L) f (K,L)dK TRS

dL f (K,L) f (K,L)

L K2

K L2

f K,L f (K,L)d df K,L f (K,L)d K dTRS

0 or 0dL dL dL dL

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 = ?

L2 22

K LK L K K LL L KK2 3

K

L2 2

K LK L K K LL L KK3

K

f (K,L)d

f (K,L) 2ff ff ff fd K0

dL dL f

f (K,L)d

f (K,L) 2ff ff ff fdTRS0

dL dL f

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

L K

L LL LK

K KL KK

0 ffH ff f

ff f

00

0 0L K

LL LL LK

L LLK KL KK

fff

H and H ff fff

ff f

Negative Definite (production function)• So the production function is strictly quasi-concave if

– 1. –fLfL < 0

– 2. 2fLfKfLK-fK2fLL -fL

2fKK > 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.

2 2L K LK K LL L

2 3K

KK2 2( )d K

0d

ff fL f

ff ff

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

3 3

2 2 K Lq f K,L 12K L

100

3 2 2 32 2

L K

3K L 3K LMP 24K L MP 24KL

100 100

IMR and DMR vs. NMR• Pull out a few terms

• If K = 10, then MPL = 0 at L=80

3 22

L

3 22

L

L

L

L

3K LMP 24K L

1003K L

MP 0, 24K L100

f 0 if (2,400 3KL) 0f 0 if KL 800f 0 if KL 800

2 32

K

2 32

K

K

K

K

3K LMP 24KL

1003K L

MP 0, 24KL100

f 0 if (2,400 3KL) 0f 0 if KL 800f 0 if KL 800

IMR vs. DMR• Because

• 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

32

LL

32

LL

6K Lf 24K -

1006K L

f 0, 24K100

32

KK

32

KK

6KLf 24L

1006KL

f 0 24L100

,

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

2 2

LK KL

2 2

LK

9K Lff 48KL

1009K L

f 0, 48KL100

A Diminishing TRS?

• Strictly Quasi-Concave if

• Lots of parts that have different signs depending on K and L.

2 2K LL K L KL L KKff 2ff ff f 0

++ ? ? ? ??

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

Effect on Output Returns to Scale

f(tK1,tL1) = tf(K1,L1) Constant

f(tK1,tL1) < tf(K1,L1) Decreasing

f(tK1,tL1) > tf(K1,L1) Increasing

• 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 Scaleq = 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 Scaleq = 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

Effect on Output Returns to Scale

f(tK1,tL1) = tkf(K1,L1)

k = 1, Constant

k < 1, Decreasing

k > 1, Increasing

• 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

q,tf(tK,tL) t

e , and evaluated at t=1tf (tK,tL)

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.

.4 .2

q,t

.4 .2

q,t .4 .2

.4 .2

q,t .4 .4 .2.6

.4 .2

q,t .4 .2 1.4

.4 .2

q,t .4 .2

q K L L

f (tK, tL) te

t f (tK, tL)

tK tL tL te

t tK tL tL

.6 K L L te

t t K L tL

.6t K L tLe

t K L t L

.6 K L Lat t 1, e

K L 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

K

L

f(tK,tL) tf (K,L)

f(K,L) f(tK,tL)MP

K Kf(K,L) f(tK,tL)

MPL L

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.

K L

1 1 Kf K, L f ,1

L L L

K Kf ,1 f ,1

L LMP , MPK L

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

L

K

• Along a ray from the origin (constant K/L), the TRS will be the same on all isoquants

q = 3

q = 2q = 1

The isoquants are equallyspaced as output expands

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

L

K

K K% d TRSL LK% TRS dTRS

L

K Kdln dlnL LfdlnTRS dln f

Elasticity of Substitution

L

K

• Both RTS and K/L will change as we move from point A to point B

A

B q = q0

TRSA

TRSB

(K/L)A

(K/L)B

is the ratio of theseproportional changes

measures thecurvature of theisoquant

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

• It is possible for to change along an isoquant or as the scale of production changes

K

L

q=f(K,L)

q=g(K,L)

g > f

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

L

K

Kdln Lfdln f

1y

xxy

L

K

1fdln fKdln L

Elasticity of SubstitutionCRS is Special Again

• For CRS production functions only we have this option too

• Let q = f(K,L)

K L

K,L

ffq f

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 isq = 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

L

K

q aK bLf b

TRS f a

K Kln ln

1 1L Lln TRS 0b b

ln lna a

Kln

L

The Linear Production Function

L

K

q1q2 q3

Capital and labor are perfect substitutes

TRS is constant as K/L changes

slope = -b/a =

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

L

K

q1

q2

q3

No substitution between labor and capital is possible

= 0

K/L is fixed at b/a

q3/b

q3/a

Cobb-Douglas Production Function

• Suppose that the production function isq = 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 Functiona b

a b 1L

a 1 bK

q AK L

f bAK L bKTRS

f aAK L aL

K Kln ln

1 1L L 1bK 1b K b Kln ln ln ln lnaL a L a L

Kln

L

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

1 0 0q K L , , ,

CES Production Function

• TRS

• Note, not a function of scale, γ

1 1L

1 1K

1 1L

1 1K

q K L

f K L L

f K L K

f L 1 1 KTRS

f K LL LK K

CES Production Function• σ

1

1

K Kln ln

L LTRS K

lnL

1 1 1K KK 1 ln 1ln L LL

K KK

ln L LL

11

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

q K L , 1, 0, 0

11

A Generalized Leontief Production Function

• Suppose that the production function is

• TRS0.5 0.5 0.5

0.50.5L

0.5 0.5 0.5 0.5K

0.5

.5

q K L 2 KL

K K L1f KL LTRS

L Kf LL1 KK

KTRS

L

A Generalized Leontief Production Function

• σ

0.5 0.5

Kln

1 1 1LK .5K K .5 lnln ln LL L

KlnK Lln

L2

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

q A tf K t ,L t

dq dA(t) df dK df dLf(K(t),L(t)) A(t)

dt dt dK dt dL dt

dq dA df dK df dLf(K,L) A

dt dt dK dt dL dt

Technical Progress

• Since

• And so

K Ldq dA q q dK dL

ffdt dt A f(K,L) dt dt

q qq A f K,L , then A , and f K,L

f K,L A

dq dA df dK df dLf(K,L) A

dt dt dK dt dL dt

Technical Progress

• Dividing by q gives us

K L

K L

dq dA q q dK dLff

dt dt A f(K,L) dt dt

dqffdA 1 dK dLdt

q dt A f(K,L) dtf (K,L) dt

Technical Progress

• Expand by strategically adding in K/K and L/L

K Ldq / dt dA / dt dK / dt dL / dt

ffq A f(K,L) f(K,LK L

K L)

K L

dqffdA 1 dK dLdt

q dt A f(K,L) dtf (K,L) dt

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

q A K K L LK L

G G f G f Gf(K,L) f(K,L)

K Ldq / dt dA / dt K dK / dt L dL / dt

ffq A f(K,L) K f(K,L) L

Technical Progress• Note the elasticities

• Yielding

• Growth is a function of technical change and growth in the use of inputs.

q,K q,L

q A q,K K q,L L

f(K,L) K f(K,L) Le , e

K f(K,L) L f(K,L)

G G e G e G

q A K K L LK L

G G f G f Gf(K,L) f(K,L)

Solow, US Growth 1909-1949• Solow estimated the following

• 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.

q A q,K K q,L L

A q q,K K q,L L

G G e G e G

G G e G e G

Appendix

• Full derivations of TRS and convexity in production.

RTS and Marginal Productivities:Implicit Function Rule

Generalize the implicit function: f(K,L)-q=0 to F(K,L,q) 0F F F

dK dL dq 0K L q

If the conditions of the implict function therorem hold, then there exists an equation K=g(L,q) and

g gdK dL dq

L q

Substitute

Subsitute into

F F FdL dq 0 to get

K L q

F F FdL dq 0

K L

g

qF F F F

dL dq 0K L K q

F g F F g FdL dq 0

K

gdL dq

L q

g gdL dq

L qg g

d

L L K q

L d

q

K

K

L

d

q

d

q

And get to…F g F F g F

dL dq 0K L L K q q

Since the dL and dq are independent, the bracketedexpressions must = 0 for the equation to h

F g F0

K L

old. That is:

F g F and 0

qWe

L K q

are interested in:

Fg L 0 and K=g(L,q) and F(K,L,q) f(K,L)-q=0

F

F g FK

K

0L

L

L

And get to…

L

K

Fg LSolve this to get 0

FLK

g dKAnd since K=g(L,q) with q held constant, and

L dLF f F f

F(K,L,q) f(K,L)-q=0, so and L L K K

f(K,L)dK L

f(K,L)dLKMPdK

dL

F g F0

K

MP

L L

Convexity, Increasing dK/dL

LLL LK K L KL KK2

K L2 2

K K

L L LLL LK K L LK KK

K K K2K

LLL K LK

K

f L,K dL dK dL dKd ff ff fff L,K dL dL dL dL fd K dK

, Note: dL dL f dL f

f L,K ffd ff ff fff L,K ffdL U

f L,Kd ff fff L,K

dL

2L

L KL L KKK

2K

K

K

L

2 2K LL K LK L K KL L K KK L

3K

L

2 2K LK L K LL K KK L

3K

fff f

ff

fMultiply by:

f

f L,Kd

f L,K ff ff ff ff ffdL f

f L,Kd

f L,K 2ff ff ff f0

dL f

Diminishing TRS

• TRS diminishing if this < 0

• Which is the same thing.

L

2 2K K LL K L KL L KK

3K

f L,Kd

f L,Kd TRS ff 2ff ff f0

dL dL f

Alternatively, the Bordered Hessian

L K

L LL LK

K KL K

2 2LK L K L KK K LL

K

0 f f

det f f f 2f f f f 0

f f

f

f

f f

Strictly Quasi-Concave if

L 2L

L LL

0 fdet f 0

f f

and

2 2K L KL L KK

3K

K LL2f f f f f fdRTS0 if 0

dL f

f

which looks a lot like the negative of this: