july25

Outline for Friday, July 25

� Remember

� Homework due Monday

� Midterm review Monday 4:30 Rackley 105

� Midterm Tuesday� Midterm Tuesday

� Technology

� Review Thursday

� Graphs: TP, MP, AP

� Returns to scale

� Productivity and technical change

Production

� Labor, L, and capital, K, combine to produce output, q

� K cannot be changed in the short run, which ends when K can be changed

Cobb-Douglas Technology

BAKLq =

K

L

Fixed-Proportions Technologies

K

aL = bK

},min{ bKaLq =

L

aL = bK

Perfect-Substitution Technologies

KbKaLy +=

L

Production

Consumers Firms

PreferencesHow much happiness from a

combination of goods

TechnologyHow much output from a

combination of inputs

Production

Consumers Firms

PreferencesHow much happiness from a

combination of goods

TechnologyHow much output from a

combination of inputs

Utility functionA numerical representation of

technology – rescalable

Production functionA numerical representation of

technology – not rescalable

Production

Consumers Firms

Preferences Technology

Utility function ~ rescalable Production function ~ not rescalable

Indifference curves IsoquantsIndifference curvesWhat combos give exactly U

IsoquantsWhat combos make exactly q

Production

Consumers Firms



Indifference curves Isoquants

MRSMarginal Rate of Technical

Substitution (MRTS)

Production

Consumers Firms




MRS MRTS

U(X, Y0) and U(X0, Y) F(L, K0) and F(L0, K)

Production

Consumers Firms




MRS MRTS


MUX and MUY MPL and MPK

Production

Consumers Firms




MRS MRTS



MRS = - MUX/MUY MRTS = -MPL/MPK

Production

Consumers Firms




MRS MRTS



MRS = - MUX/MUY MRTS = -MPL/MPK

Diminishing MUX Diminishing MPL ~ beyond a point

Production - differences

Consumers Firms


Diminishing MUX Diminishing MPL ~ beyond a point

Eventually diminishing MPL

� At low levels of labor, there may be increasing marginal returns to specialization



� At some point, these returns will start falling ~ “diminishing marginal returns set in”“diminishing marginal returns set in”



� At some point, these returns will start falling ~ “diminishing marginal returns set in”“diminishing marginal returns set in”

� It’s still positive

� The same holds for MPK:

� If L is held constant,

� Marginal returns to capital will eventually fall

Figure 6.1 Production Relationships with Variable Labor

B

C

110

90

(a)

© 2007 Pearson Addison-Wesley. All rights reserved.

6–19

A

11640

L , Workers per day

56

Graphing TP, MP, AP

� Total product of labor is f(L, K0) – the output

Graphing TP, MP, AP


� Marginal product of labor is MPL – the slope of f(L, K0)

Graphing TP, MP, AP



� Average product of labor is f(L, K0)/L – the slope of a line from the origin to fline from the origin to f

Graphing TP, MP, AP




� When MP is above AP…AP is increasing

� Each additional unit costs more than the average so far

�The average goes up

Graphing TP, MP, AP







� When MP is below AP…AP is decreasing

� Where do MP and AP cross?

Graphing TP, MP, AP







� When MP is below AP…AP is decreasing

� MP crosses AP at its peak

Graphing TP, MP, AP


AP

b

20

15

(b)

6–26

Marginal product, MPL

Average product, APL

c

11640

L , Workers per day

15


Graphing TP, MP, AP


� It’s also where the AP ray is tangent to TP

B

A

C

11640

L, Workers per day

110

90

56

(a)

a

(b)

6–27

TP


Marginal product, MPL

Average product, APL

b

a

c

11640

L, Workers per day

20

15

Graphing TP, MP, AP

� Problem 8. q = 1000 min{L, 3K}

� Draw isoquant map (e.g., L = 2, K = 1 � q = 2000)

� Draw the TP, AP, MP of labor curves for a K = K0

Returns to scale

� What is the marginal effect of increasing both inputs?

Returns to scale


� How does f(tL0, tK0) vary with t?

Returns to scale

q = f(tL0, tK0)

Output Level

Increasingreturns-to-scale

t

Input Level

Decreasingreturns-to-scale

returns-to-scale

Returns to scale



� Look at f(tL0, tK0)/ f(L0, K0)

Returns to scale



� Look at f(tL0, tK0)/ f(L0, K0)

� > t Increasing returns to scale� > t Increasing returns to scale

� < t Decreasing returns to scale

� = t Constant returns to scale

Returns to scale



� Example: q = LAKB

Returns to scale




� Increasing returns to scale (IRS) if A+B > 1� Increasing returns to scale (IRS) if A+B > 1

� CRS if A+B = 1

� DRS if A+B < 1

Returns to scale





� CRS if A+B = 1

� DRS if A+B < 1

� Problem 29. q = L0.27K0.16M0.61

Returns to scale





� CRS if A+B = 1

� DRS if A+B < 1

� Problem 29. q = L0.27K0.16M0.61

� Problem 26.

� (a) q = L+K

� (c) q = L+K+LAKB

Returns to scale

� Problem 22. q = L3/4K1/4

� Fix K=K0. What is the AP of labor?

� Keep K0. What is the MP of labor?

� Are there IRS, CRS, or DRS?� Are there IRS, CRS, or DRS?

Scaling matters

� Unlike the utility, the quantity of output is meaningful

� f(L, K) and 10f(L, K) represent different technology

� Different firms have different productivity

Scaling matters

� Unlike the utility, the quantity of output is meaningful

� f(L, K) and 10f(L, K) represent different technology


� We can compare a firm to the most productive firm in the � We can compare a firm to the most productive firm in the industry

� How much could the most productive firm have made with our firm’s inputs?

Scaling matters


� We can compare a firm to the most productive firm in the industry

� How much could the most productive firm have made with our firm’s inputs?

� If our firm uses f0 and inputs L0 and K0, while the most productive firm uses f1, then the relative productivity is expressed as a percentage

100),(

),(

001

000×

KLf

KLf

Changing technology

� Technology may change over time

� Firms invest in R&D another model

� Technology advances exogenously

Changing technology




� How can technology change?� How can technology change?

� Overall productivity can increase

� The productivity of one input could increase

Changing technology






� f � 2f


� MPL � 2MPL

Changing technology






� f � 2f

� This is a neutral change


� MPL � 2MPL

� This is a nonneutral change

Outline for Friday, July 25

� Remember

� Homework due Monday

� Midterm review Monday 4:30 Rackley 105

� Midterm Tuesday� Midterm Tuesday

� Technology

� Review Thursday

� Graphs: TP, MP, AP

� Returns to scale

� Productivity and technical change

� Next time: input costs

july25

Technology

midterm review

ap returns

scale productivity

technical change

cobbdouglas technologyka

short run