Chapter 18: Technology - Econ 3023 Microeconomic Analysisd.umn.edu/~watanabe/econ3023fl12/docs/techho.pdf · Econ 3023 Microeconomic Analysis Chapter 18: Technology ... Exercise 2.5

Intro Technology Properties Trinity SR & LR Returns to Scale

Econ 3023 Microeconomic Analysis

Chapter 18: Technology

Instructor: Hiroki WatanabeFall 2012

Watanabe Econ 3023 18 Technology 1 / 74


1 Introduction

2 Technology

3 Properties of Technology

4 Trinity

5 Short Run & Long Run

6 Returns to Scale

7 Summary



1 Introduction

2 Technology


4 Trinity


6 Returns to Scale

7 Summary



Question 1.1 (Agenda for Producer Theory)1 How does Liz determine her hiring and investment

plan?2 Where does the supply curve come from?3 How does the market structure affect the Liz’s

decision making?



Project 1.2 (Overview of Producer Theory)18 Technology19 Profit Maximization Problem20 Cost Minimization Problem21 Cost Curves22 Firm Supply23 Aggregate Supply24 Monopoly25 Monopoly Behavior26 Oligopoly



Question 1.3 (Agenda for Today)1 Each firm has different capability / capacity to

produce an output. How do economists distinguishone from the other?

2 Trinity on the technology side3 What do economists mean by long term and short

term?



1 Introduction

2 TechnologyDescribing TechnologyTechnology with Multiple InputsPerfect SubstitutesCobb-Douglas Production FunctionFixed-Proportions Technology


4 Trinity


6 Returns to Scale

7 Summary



Describing Technology

Example 2.1 (Describing Technology)

Liz bakes two cheesecakes in an hour,four in two hours etc.If she works twice as long, she bakes twice as many.




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Production Function

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Technology doesn’t have to be linear.(usually not).




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A technology is a process by which inputs areconverted to an output.What does it take to produce this lecture?Several technologies produce the same product.Q: How do we compare and categorizetechnologies?




1, 2 denote the amount used of input 1 and 2.An input bundle = (1, 2) describes theamounts of input 1 and input 2. = (C, K) = (3,500) means 3 hours of chef’slabor and 500 units of kitchen equipment.y denotes the output level, e.g., a number ofcheesecakes produced.




Definition 2.2 (Production Function)A production function states the maximum amountof output y that can be produced with an input bundle:

y = ƒ (C, K , · · · ).




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A production plan is a combination of an inputbundle and an output level: (C, K , y).A production plan (C, K , y) is feasible if

y ≤ ƒ (C, K).

A production set is a collection of all feasibleproduction plans.




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Technology with Multiple Inputs

Chefs alone do not produce cheesecakes.How do we represent production technology withmultiple inputs?



Technology with Multiple Inputs

Definition 2.3 (Isoquant)

Isoquant at an input bundle = (C, K) is acollection of all input bundles that producey = ƒ (C, K).

If input bundles = (3,5) and ′ = (5,3) are on thesame isoquant, then they produce the samenumber of cheesecakes.



Perfect Substitutes

Definition 2.4 (Perfect Substitutes)Liz hires left-handed chefs L and right-handedchefs R to produce cheesecake y = ƒ (L, R).Each chef (lefty or righty) bakes 1 cheesecake.Her production function is

y = ƒ (L, R) = L + R.

Lefties and righties are perfect substitutes.



Perfect Substitutes

Exercise 2.5 (Isoquant of Perfect Substitutes)

Trace isoquant at (L, R) = (0,4) and (2,6) of aproduction function

y = ƒ (L, R) = L + R.



Perfect Substitutes

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Cobb-Douglas Production Function

Definition 2.6 (Cobb-Douglas ProductionFunction)

The most commonly used production function isCobb-Douglas production function:

ƒ (C, K) = CbK,

where > 0 and b > 0.E.g., ƒ (C, K) = CK .Isoquant at (C, K) = (1,10)?




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N.B. ƒ () = 1 and ƒ () = 2 contain more informationthan () = 1 and () = 2.Unlike utility functions, ƒ () = CK andg() = 2CK represent different technologies.



Fixed-Proportions Technology

Definition 2.7 (Fixed-Proportions Technology)

1 lb of filling (1) and .5 lb of crust (2) make 1cheesecake (y = 1).# of cheesecakes made out of 1 lb of filling and392847 lbs of crust is still y = 1.This type of technology is calledfixed-proportions technologies. a

aproducer theory’s analogue of perfect complements inconsumer’s theory.



Fixed-Proportions Technology

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1 Introduction

2 Technology

3 Properties of TechnologyMonotonic TechnologyConvex TechnologyMarginal Product

4 Trinity


6 Returns to Scale

7 SummaryWatanabe Econ 3023 18 Technology 29 / 74


Definition 3.1 (Measuring Productivity)

Three ways to measure the productivity:1 Monotonicity

(Does Kenneth increase or decrease output level?)2 Returns to scale

(What if Kenneth works twice as long?)3 Marginal productivity

(What if Kenneth works for one more hour?)



Monotonic Technology

Technology is monotonic if more of any inputgenerate more output.Some technology may not be monotonic.1,000,000 pages are probably not as productive as10 pages at NBC.




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Convex Technology

Definition 3.2 (Convex Technology)

If a balanced combination of C and K produces morecheesecakes than the extreme combination of C andK , the technology is convex.

ƒ (5,5) = 25 > ƒ (1,9) = 9.



Convex Technology

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Marginal Product








Marginal Product

Definition 3.4 (Marginal Product)

Marginal product MPC(C, K) of input C at (C, K)measures an increase in output level y when Liz hiresone more chef holding the input level K constantat K . a

aMPC(C, K ) =

Δy

ΔC=Δƒ (C, K )

ΔC

=ƒ (C + ΔC, K )− ƒ (C, K )

ΔC.



Marginal Product

For Cobb-Douglas utility function

ƒ (C, K) =pCpK , MPC(C, K) =

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Let’s say C = 4. How many additional cheesecakescan you make, if you increase C by one unit?

MPC(C = 4, K = 4) = 12 .

MPC(C = 4, K = 16) = 1.MPC(C, K) depends on K as well as C.



Marginal Product

Definition 3.5 (Diminishing Marginal Product)

Input C exhibits diminishing marginal product ifthe additional increase in y from a unit increase in C(= MPC()) becomes smaller as C increases.

MPC(1C, K) > MPC(2C, K), (1C< 2

C).

When there is only one chef (C = 1), adding onemore chef will result in MPC(C = 1, K) = 50 ( =added chef’s contribution).When there are 30 chefs (C = 30), adding onemore chef will result in MPC(C = 30, K) = 1.



1 Introduction

2 Technology


4 TrinityTrinityMarginal Rate of Technical Substitution


6 Returns to Scale



Trinity

Definition 4.1 (Trinity on the Production Side)

The following are equivalent:1 The slope of isoquant.2 Marginal rate of technical substitution.3 The ratio between MPC() and MPK().a

aYou can forget this one.



Marginal Rate of Technical Substitution

What does the slope of isoquant mean?




Definition 4.2 (Marginal Rate of TechnicalSubstitution)Marginal rate of technical substitution(MRTS(C, K)) measures the unit of K Liz can forgo tooffset the increase in output level y due to a unitincrease in C.

A producer version of MRS().




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Fact 4.3 (Convex Technology & MRTS)

MRTS drops in magnitude with C along the isoquant iftechonology is convex.




1 The slope of isoquant.2 Marginal rate of technical substitution.3 The ratio between MPC() and MPK().1

1You can forget this one.Watanabe Econ 3023 18 Technology 47 / 74


1 Introduction

2 Technology


4 Trinity

5 Short Run & Long RunDefinitionsRelation between Short Run & Long RunShort-Run Production Function & Marginal Product

6 Returns to Scale



Definitions

Definition 5.1 (Short Run & Long Run)1 A short run is a circumstance in which a firm is

restricted in its choice of at least one input level.2 A long run is the circumstance in which a firm is

unrestricted in its choice of input levels.

Chef’s working hours (C) can be made longer orshorter on a daily basis.Kitchen size (K) can’t.In the long run, Liz can change both of them.In the long-run environment the firm can choose asit pleases in which short-run circumstance to be.



Relation between Short Run & Long Run

What do short-run restrictions imply for atechnology?Suppose the short-run restriction is fixing the levelof K .K is thus a fixed input in the short-run. C remainsvariable.




Perfect substitutes:1 Long run: y = ƒ (L, R) = L + R.2 Short run: y = ƒ (L, R) = L + R.

Cobb-Douglas:1 Long run: y = ƒ (C, K ) = CK .2 Short run: y = ƒ (C, K ) = CK .




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Short-Run Production Function & Marginal Product

The slope of short-run production function denotesthe marginal product of chefs.Recall: The marginal product of C at (C, K)measures the increase in output level y when Lizincreases C by one unit holding the input levelK fixed.MP varies depending on K even when C is thesame.



1 Introduction

2 Technology


4 Trinity


6 Returns to ScaleDefinitionsReturns to ScaleExamples



Definitions








Definitions

MPC(C, K) measures the change in y when weincrease C by one unit while holding Kconstant.What if we multiply all the inputs C, K altogether?

Definition 6.2 (Returns to Scale)Returns to scale describes how the output level ychanges as all input levels C, K change in directproportion.



Returns to Scale

Definition 6.3 (Constant Returns to Scale (CRS))If for any input level ,

ƒ (2) = 2ƒ ()

then exhibits constant returns to scale.

Doubling the input level doubles the output level.Ingredients on cooking recipes.A benchmark technology.



Returns to Scale

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Returns to Scale

Definition 6.4 (Decreasing Returns to Scale(DRS))If for any input level ,

ƒ (2) < 2ƒ ()

hen ƒ (C, K) exhibits decreasing (diminishing)returns to scale.

Doubling the input level less than doubles theoutput level.Chefs in a crowded kitchen.



Returns to Scale

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Linear TechnologyDRS

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Returns to Scale

Definition 6.5 (Increasing Returns to Scale (IRS))

If for any input bundle ,

ƒ (2) > 2ƒ ()

hen ƒ (C, K) exhibits increasing returns to scale.

Doubling all input levels more than doubles theoutput level.Chefs in a spacious kitchen (sharing work etc).



Returns to Scale

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Returns to Scale

A technology can be locally IRS and DRS.



Returns to Scale

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Examples

Example 6.6 (Returns to Scale)

What are the return-to-scales for the followingproduction functions?

1 y = ƒ (C) = C + 5.2 y = ƒ (L, R) = L + R.3 y = ƒ (C, K) = CK .4 y = ƒ (C, K) =

pCpK .



Examples

Discussion 6.7 (Returns to Scale of UtilityFunction)Does it make sense to discuss returns to scale of autility function?



1 Introduction

2 Technology


4 Trinity


6 Returns to Scale

7 Summary



consumer theory producer theory

utility function () production function ƒ ()indifference curve isoquant

MRS MRTSperfect complements fixed proportions



Description of technology.MonotonicityConvexity.Trinity.Time frames.Returns to scale.


Cobb-Douglas productionfunction, 23, 51constant returns to scale,61convex, 34, 46decreasing (diminishing)returns to scale, 63diminishing marginalproduct, 39feasible production plan,16fixed-proportionstechnology, 27, 72increasing returns to scale,65indifference curve, 72input bundle, 13isoquant, 19, 46, 72local DRS, 67local IRS, 67long run, 49, 51

marginal product, 37, 57marginal productivity, 30,36, 59marginal rate of technicalsubstitution, 43, 46, 72monotonicity, 30, 36, 59MRTS, see marginal rate oftechnical substitutionoutput level, 13perfect complements, 72perfect substitutes, 20, 51production function, 14production plan, 16production set, 16returns to scale, 30, 36,59, 60short run, 49, 51technology, 12utility function, 70, 72, see input bundle

Documents

Chapter 18: Technology - Econ 3023 Microeconomic Analysisd.umn.edu/~watanabe/econ3023fl12/docs/techho.pdf · Econ 3023 Microeconomic Analysis Chapter 18: Technology ... Exercise 2.5