microeconomics 1 - central european university · 2012-02-08 · course regulations technology...

Course regulations Technology

Microeconomics 1

Juan Manuel Puerta

September 27, 2009

First course in micro theory. Roughly consumer and producertheory and rudiments of general equilibrium.

Course prerequisites are calculus/algebra to the level of AlphaChiang or Simon and Blume.

1 Week 1-3: Producer Theory2 Week 4-6: Consumer Theory3 Week 7-8: Market Structure: Competition vs. Monopoly4 Week 9: General Equilibrium5 Week 10: Revision and Exam

This are approximate dates, you are responsible to periodicallycheck for changes (I will try to announce any major change)

Evaluation: Midterm and final (80%) and problem sets (20%).

“Big” and “Small” Varian

Breaks

The 3-tip method to doing well in Micro 1

Tip 1: Make sure your math level allows you to understandlectures. But don’t panic, there are a lot of good textbooks (seeprevious slides).

Tip 2: Work steadily througout the term. Do the homeworks andexercises at the end of the chapter. Remember: memory vs.solving

Tip 3: Talk, ask, participate... Don’t wait to the day of the examto tell me you haven’t understood anything! We have officehours to help you.

Slides vs. Blackboard action.

Remarks about grad school.

Questions?

Comments?

Short speeches?

Questions?

Comments?

Short speeches?

Questions?

Comments?

Short speeches?

Chapter 1: Technology

In the first part of the course we will be interested in studying thebehavior of firms.

The amount of output firms produce depends on thecharacteristics of the available technology.

In this first chapter we introduce important concepts regardingtechnology that will help us in order to study profit maximization(ch. 2-3) and cost minimization (ch. 4-5).

We will be interested in the combinations of inputs and outputsthat could be produced given the state of technology at a givenmoment. Think of outputs and inputs as defined over time(flows).

The aim of the chapter is to introduce some definitions abouttechnology and a few results that will be useful later.

Specification

Some definitions

n available goods that can be either inputs or outputs, yij and yo

jare the quantities of the good j used as an input and outputrespectively. The net ouput of good j is given by yj = yo

j − yij.

A production plan is a list of net outputs for the n goods. Aproduction plan is represented by a vectory = (y1, y2, ..., yj, ..., yn) in <n. If yj > (<)0 then the good is andinput(output).

The set of all technologically feasible production plans is theproduction possibility set, Y ⊂ <n

In the short run some inputs may be fixed, so some feasible plansmay not be “immediately feasible”. Let z denote the vector offixed inputs, the restricted production possibility set is the setof all feasible net output bundles consistent with z.

Specification

Consider a firm producing one output y using an input vector x.The input requirement set is defined as

V(y)={x in <n+ : (y,-x) is in Y}

This is simply the set of input requirements that produces at leasty.

We can also define the concept of isoquant using the inputrequirement set

Q(y)={x in <n+ : x is in V(y) and x is not in V(y′) for y′ > y}

In multi-output settings, there is a definition that would comehandy, this is the transformation functionA transformation function T : <n → < where T(y) = 0 if andonly if y is efficient, i.e. There is no y’ in Y such that y’ ≥ y andy’ 6= y

Monotonicity

Monotonic technologies

It may seem natural the possibility to “throw away” inputs.Assume we have 2 units of each of two inputs but we have atechnology with V(1)=(2,1),(1,2). It seems reasonable to use(2,1) to produce 1 unit of output and throw away the remainingunit of input 2. If this is possible, we say there is free disposal.The idea of free disposal is related to the monotonicityassumption of the input requirement set.

Monotonicity (Input requirement set). If x is in V(y) and x’ ≥ x,then x’ is in V(y).

Monotonicity (Production set) If y is in Y and y’ ≤ y, then y’ isin Y.

Note the particular sign convention. This is due to the fact thatinputs enter as negative values in Y. In our previous example,y=(1,-2,-1) and y’=(1,-2,-2).

Monotonicity

Monotonic technologies

It may seem natural the possibility to “throw away” inputs.Assume we have 2 units of each of two inputs but we have atechnology with V(1)=(2,1),(1,2). It seems reasonable to use(2,1) to produce 1 unit of output and throw away the remainingunit of input 2. If this is possible, we say there is free disposal.The idea of free disposal is related to the monotonicityassumption of the input requirement set.

Monotonicity (Input requirement set). If x is in V(y) and x’ ≥ x,then x’ is in V(y).

Monotonicity (Production set) If y is in Y and y’ ≤ y, then y’ isin Y.

Note the particular sign convention. This is due to the fact thatinputs enter as negative values in Y. In our previous example,y=(1,-2,-1) and y’=(1,-2,-2).

Monotonicity

Monotonicity in action. The left chart represents that original inputrequirement set. The right one includes all the possible combinationsthat imply wasting inputs, i.e. the combinations if monotonicity holds.

Convexity

Imagine we want to produce 100 units of output. By a replicationargument, (200,100) and (100,200) should be in V(100). Arethere any other ways of producing 100 units of output?

Imagine we use 50% of the times the first technology (producing50 units of output) and 50% the second technology (producingthe other 50).

0.5(200, 100) + 0.5(100, 200) = (150, 150)In general, t(200, 100) + (1− t)(100, 200) should also be inV(100) for t = 0.01, 0.02, ..., 1

Convexity. If x and x’ are in V(y), then tx+(1-t)x’ is in V(y) forall 0 ≤ t ≤ 1

Convexity

Some remarks on convexity

1 Convexity was motivated through a replication argument. Seemsappropriate for large output relative to scale.

2 Convexity could be motivated in production plans that areimplemented over relatively long times (you can switchproduction plans in the middle)

3 Convexity for production sets. A production set is convex if yand y’ are both in Y, then ty+(1-t)y’ is also in Y.

Note that this is more restrictive assumption. In particular it rulesout fixed costs. Why?

Convexity

Some useful results

Theorem 1. Convex production sets imply convex inputrequirement sets Proof:

1 Take y=(y,-x) and y’=(y,-x’) both in Y. This implies x and x’ arein V(y)

2 Convexity of Y implies that ty + (1− t)y’ is also in Y. But then3 t(y,−x) + (1− t)(y,−x’) = (ty + (1− t)y,−tx− (1− t)x’) =

(y,−tx− (1− t)x’)4 But then by definition of V(y), (tx+(1-t)x’) is in V(y).5 So x and x’ in V(y) imply (tx+(1-t)x’) is in V(y). In other words,

V(y) is convex.�

Theorem 2. Convex input requirement set⇐⇒ quasiconcaveproduction function

Convexity

Does the converse statement hold true, i.e. V(y) convex =⇒ Yconvex?

It turns out this claim is not true (ex. 1.1)

How would you go about proving this is not the case?

Counterexample? Contradiction? (You can establish A=⇒B ifyou prove that -B=⇒-A or similarly we the contradiction afterassuming A and -B). So imagine a production function that likey = f (x) = x2. This production function does not imply a convexproduction set (can you see that?) Y={(y,-x) in <2: y ≤ x2}

But V(y)={x in <+: x2 ≥y} which is a convex set.

Convexity

Regular Technologies

1 Weak regularity condition regarding the input requirement setV(y)

2 REGULAR. V(y) is a closed, non-empty set for all y ≥ 03 Non-emptiness: To avoid trivial situations, we assume that y can

be produced.4 Closeness: Informally, the input requirement set should include

its own boundary.5 Think of it also as an assumption of “smoothness”.6 Imagine there is a sequence xi such that each produces at least y7 If that sequence converges to an input bundle x0, then this bundle

should produce at least y

Parametric representations of technology

Many possible ways of produce the same output.Use a function to summarize these possible input set“Smoothing an isoquant”

Parametric technological representations should be thought of asapproximations.Not only useful for pedagogical purposes, it will allow us tobring calculus tools in order to investigate the production choicesof firms.

Assume: Technology could be summarized by a smoothproduction function and we are producing at y∗ = f (x∗1, x

∗2).

Idea: If we increase the amount of input 1, how much can wedecrease input 2 given that we want to produce the same amount

This is called the Marginal Rate of Substitution, TRS/MRTSbetween factor 1 and 2

Two Dimensional Case: Assume x2(x1) is the implicit functionthat tells you how much of x2 it takes to produce y given that weare also using x1 of input 1. x2(x1) should fulfill the followingidentity:

f (x1, x2(x1)) ≡ y (1)

∗2).

f (x1, x2(x1)) ≡ y (1)

∗2).

f (x1, x2(x1)) ≡ y (1)

∗2).

f (x1, x2(x1)) ≡ y (1)

Differentiating the identity we get,

∂f (x∗1, x∗2)

∂x1+∂f (x∗1, x

∂x2(x∗1)∂x1

= 0 (2)

and rearranging the equation,

∂x2(x∗1)∂x1

= −∂f (x∗1 ,x

∗2 )

∂f (x∗1 ,x∗2 )

Alternatively, you can obtain it using the concept of TotalDifferential.The term ∂f (x∗1 ,x

∗2 )

∂xiis the Marginal Product of Factor i.

∂f (x∗1, x∗2)

∂x1+∂f (x∗1, x

∂x2(x∗1)∂x1

= 0 (2)

∂x2(x∗1)∂x1

= −∂f (x∗1 ,x

∗2 )

∂f (x∗1 ,x∗2 )

∗2 )

∂f (x∗1, x∗2)

∂x1+∂f (x∗1, x

∂x2(x∗1)∂x1

= 0 (2)

∂x2(x∗1)∂x1

= −∂f (x∗1 ,x

∗2 )

∂f (x∗1 ,x∗2 )

∗2 )

∂f (x∗1, x∗2)

∂x1+∂f (x∗1, x

∂x2(x∗1)∂x1

= 0 (2)

∂x2(x∗1)∂x1

= −∂f (x∗1 ,x

∗2 )

∂f (x∗1 ,x∗2 )

∗2 )

What would be associated change in output given a small changein each of the inputs dx = (dx1, dx2).

dy =∂f∂x1

dx1 +∂f∂x2

dx2 (4)

solving for ∂x2(x∗1 )∂x1

you get equation 3.

Simple graphical interpretation: TRS is the slope of the isoquant

The Technical Rate of Substitution

What would be associated change in output given a small changein each of the inputs.

dy =∂f∂x1

dx1 +∂f∂x2

dx2 (5)

solving for ∂x2(x∗1 )∂x1

you get equation 3.

Example: TRS for a CES function f (x1, x2) = [a1xρ1 + a2xρ2]1ρ

where a1 > 0, a2 > 0 and ρ < 1

TRS = ∂x2(x∗1 )∂x1

= −∂f∂x1∂f∂x2

= − (1/ρ)[a1xρ1 +a2xρ2 ]1ρ−1

ρa1xρ−11

(1/ρ)[a1xρ1 +a2xρ2 ]1ρ−1

ρa2xρ−12

TRS = −a1a2

( x1x2

)ρ−1

where a1 > 0, a2 > 0 and ρ < 1

TRS = ∂x2(x∗1 )∂x1

= −∂f∂x1∂f∂x2

= − (1/ρ)[a1xρ1 +a2xρ2 ]1ρ−1

ρa1xρ−11

(1/ρ)[a1xρ1 +a2xρ2 ]1ρ−1

ρa2xρ−12

TRS = −a1a2

( x1x2

)ρ−1

where a1 > 0, a2 > 0 and ρ < 1

TRS = ∂x2(x∗1 )∂x1

= −∂f∂x1∂f∂x2

= − (1/ρ)[a1xρ1 +a2xρ2 ]1ρ−1

ρa1xρ−11

(1/ρ)[a1xρ1 +a2xρ2 ]1ρ−1

ρa2xρ−12

TRS = −a1a2

( x1x2

)ρ−1

Elasticity of substitution

Remembering elasticities...

Elasticities are a measure of the responsiveness of one variablegiven a 1% change in another variable of interest.

E.g. Price elasticity of demand: Percentage change in thequantity demanded given a 1% increase in price.

Mathematically, the elasticity of y with respect to x can bewritten like εy,x = 4y/y

TRS measures the slope of the isoquant, the elasticity ofsubstitution measures the curvature of the isoquant

In practice the change (4) is thought to be very small

So a useful working definition of elasticity is εy,x = dy/ydx/x = d(ln y)

d(ln x)

Elasticity of substitution

The Elasticity of Substitution

Measures how much the factor ratio changes (4(x2/x1)) given achange in the slope of the isoquant (TRS)

Mathematically, σ = d(x2/x1)/(x2/x1)d(TRS)/TRS = TRS

(x2/x1)d(x2/x1)d(TRS)

or using the logarithmic derivative shown above, σ = d ln(x2/x1)d ln |TRS|

where the absolute value (|) is needed as TRS is negative.Ex. Cobb-Douglas: f (x1, x2) = xa

1x1−a2

TRS = −∂f/∂x1

∂f/∂x2= −

axa−11 x1−a

xa1(1− a)x−a

= − a1− a

x1⇒ x2

x1= −1− a

Then taking logs, ln x2x1

= ln 1−aa + ln |TRS|,

Finally, differentiation yields σ = d ln(x2/x1)d ln |TRS| = 1

Returns to Scale

Idea: We are producing y using input vector x.What would happen to the output if we scale the inputs by afactor t ≥ 0?

CONSTANT RETURNS TO SCALE. A technology exhibitsconstant returns to scale if any of the following are satisfied:(1) y in Y implies ty is in Y, for all t ≥ 0;(2) x in V(y) implies tx is in V(ty) for all t ≥ 0;(3) f(tx)=tf(x) for all t ≥ 0; i.e., the production function f(x ishomogeneous of degree 1

Intuition: if we use (say) 5 units of K and 5 of L to produce 15units of the final good, it is in reasonable to assume that using(L,K)=(10,10) we will produce 30. This is the replicationargument

Returns to Scale

However, there are some cases in which this reasoning wouldn’t work

(1) The replication story is about scaling up but it might notmake a lot of sense if we need to scale down production (e ≤ 1)(2) The replication story is about t = 1, 2, 3, ..., what aboutnon-integer values?

(3) What if doubling the inputs leads to a more than doubling inproduction? Ex. oil-pipe example.(4) Maybe we cannot replicate an input (ex. farm than needsmore land but it is unavailable)

CRS and the “Law of Diminishing Marginal Return”

Returns to Scale

Increasing and Decreasing Returns to Scale

INCREASING(DECREASING) RETURNS TO SCALE. Atechnology exhibits increasing(decreasing) returns to scale iff(tx)>(<)tf(x) for all t>1

Decreasing returns appear naturally when we cannot replicateinputs, therefore we should expect them see decreasing returns inrestricted production sets.

Imagine a decreasing return technology f (x). “Mythical” input zintroduced to define a broader production functionF(z, x) = zf (x

z ). The “broader” technology displays CRS whilethe original technology has DRS. Exercise: Prove it!

Returns to Scale

Returns to scale may vary depending on the values of x. That’swhy in many cases it is useful to have a local measure of returnsto scale.

Elasticity of scale measures the percentage increase in outputgiven a 1% increase in all inputs

Mathematically, ε(x) = df (tx)/f (tx)dt/t |t=1 = df (tx)

f (tx) |t=1

We say that the function exhibits localincreasing/constant/decreasing returns to scale is ε(x)>/=/<1Example: Scale elasticity of the CES technology (ex. 1.5),f (x) = [xρ1 + xρ2]

Step 1: f (tx) = [(tx1)ρ + (tx2)ρ]1/ρ = [tρxρ1 + tρxρ2]1/ρ =

[tρ(xρ1 + xρ2)]1/ρ = t[xρ1 + xρ2]

1/ρ = tf (x)Step 2: ε(x) = df (tx)

f (tx) |t=1 = f (x) ttf (x) |t=1 = 1

In addition to showing that CES has scale elasticity equal to 1,we have established something else above...

Homogeneous/Homothetic

Definition: A function is homogeneous of degree k iff (tx) = tkf (x) for all t>0

We will often deal with functions that are homogeneous ofdegree 0 or 1. CES?

CRS⇐⇒ production function homogeneous of degree 1.

Definition: A function g : < → < is said to be a positivemonotonic transformation if g is a strictly increasing function;i.e. a function for which x>y implies that g(x) > g(y)Definition: A homothetic function is a monotonictransformation of a function that is homogeneous of degree 1.f (x) if homothetic if and only if it can be written asf (x) = g(h(x)) where h(.) is homogeneous of degree 1 and g(.)is a monothonic transformation.

Intuition: Think of a monotonic transformation as if you wouldmeasure output in different units.

Intuition: A homothetic technology is one in which there is away of measuring output in such a way that it would look as if ithad constant returns to scale.

Geometric interpretation

Isoquants look like “blown-up” versions of a single isoquant inboth homogeneous (of degree 1) and homothetic technologies

The main difference: In homogeneous technologies, “samedistance between isoquants” (say, the 2y isoquant is related tooutput levels 2x). In homothetic technologies, the distancebetween isoquants may differ.

microeconomics 1 - central european university · 2012-02-08 · course regulations technology...

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