Intermediate Microeconomics (Producer Model) · PDF fileIntermediateMicroeconomics(ii)* * 6—1* * IntermediateMicroeconomics!(ii)! ProducerModel(* * 6Theproducermodelintheshort*term

Intermediate Microeconomics (ii) 6—1

Intermediate Microeconomics (ii) Producer Model 6 The producer model in the short term ....................................................................... 6—2 6.1 Comparing the consumer and the producer models .................................... 6—2 6.3 Creating the producer model .................................................................................. 6—3 6.3.1 Assumptions in the “short run” ..................................................................... 6—3 6.3.2 Production technologies .................................................................................. 6—3 6.3.3 The Marginal Product of Labor (MPL) ....................................................... 6—5 6.3.4 The law of diminishing marginal product ................................................ 6—7

6.4 Profits ................................................................................................................................ 6—8 6.4.1 Profit maximization ............................................................................................ 6—8 6.4.2 Isoprofits ................................................................................................................. 6—8 6.4.3 Marginal Revenue Product ........................................................................... 6—13 6.4.4 Cost curves .......................................................................................................... 6—17

7 The producer model in the long term ...................................................................... 7—27 7.1 Revision ......................................................................................................................... 7—27 7.2 Introduction to this week ...................................................................................... 7—27 7.3 Production ................................................................................................................... 7—28 7.3.1 Terms .................................................................................................................... 7—28 7.3.2 Production technologies (for this week) ............................................... 7—28 7.3.3 Technical Rate of Substitution, Returns to Scale and Marginal Products .............................................................................................................................. 7—29

7.4 Profit maximization for producers in the long-‐term model ................... 7—31 7.4.1 Example: Cobb-‐Douglas ................................................................................. 7—31 7.4.2 Step 1: deriving the costs curve (i) – isoquants .................................. 7—31 7.4.3 Step 2: deriving the cost curve (ii) -‐ isocosts ...................................... 7—32 7.4.4 Step 3: cost minimization ............................................................................. 7—32 7.4.5 Step 4: cost curves ........................................................................................... 7—35 7.4.6 Step 5: marginal and average cost curves ............................................. 7—36 7.4.7 Profit maximizing ............................................................................................. 7—37

7.5 Labor and Capital Demand Curves .................................................................... 7—38


6 The producer model in the short term In the first half of the semester, we built models to ascertain consumer preferences based on economic circumstances, and we modeled how choice and utility change with changing circumstances. Then, we extracted from these the demand function for goods, and the supply functions for labor and capital. This second chapter will be about the converse: modeling production preferences and creating a supply function for goods, and demand functions for labor and capital. Finally, we will equilibrate these to find optimal market production and consumption within the model. Note, since this is microeconomics, we assume that each firm, and each consumer, is small enough that they themselves are price takers: they are not large enough players to influence the price of goods. This week, we will build the producer model in the short term. The short-‐term constraint allows us to build a simplified model where capital (“k”) is fixed: only the labor input varies.

6.1 Comparing the consumer and the producer models Consumer Producer To be maximized: Utility (happiness)

u(x, l,k) Profit π (x) = px −wl − rk

Choice: Consumption Bundle x, l,k{ }

Production plan x, l,k{ }

Constraint: Budget p,w, r, l,L,e1,e2{ }

Production technology x = f (l,k)

Obtain: Demand for Goods xD (p, l)

Goods supply curve xS (p, l)

Supply of Labor l s (w,L)

Demand for Labor lD (w,L)

Supply of Capital KS (r,e1,e2 )

Demand for Capital KD (r,e1,e2 )


6.3 Creating the producer model

6.3.1 Assumptions in the “short run” A usual production model involves labor and at least one other input. In this simplified short run model, however, we assume that all non-‐labor inputs are fixed. Thus, x = f (l,k)becomes x = f (l,K ) which is equivalent to x = f (l) . Later, this restriction will be relaxed.

6.3.2 Production technologies Terminology:

• A production plan is the “choice” of production: a bundle of inputs and outputs (x, l)

• The output is a function of the input x = f (l) • The production choice set is the set of all feasible production plans given

a production technology • The production frontier is the boundary of the production choice set:

inputs are not wasted Example production technologies (note, any function can be used):

• A straight line production technology o x = f (l) = 4l

5 10 15 20 25 30 35 40

0

50

100

150

200

250

Labour

Prod

ucti

on o

f x

x=4l


• A cosine variable production technology

o x =200 1− cos π

100l

"

#$

%

&'

"

#$

%

&' | l ≤100

400 | l >100

)

*+

,+

-

.+

/+

• An exponential production technology

o x = 40l0.5

0 20 40 60 80 100 120

450

0

50

100

150

200

250

300

350

400

Labour

Prod

ucti

on o

f x

variable cosine

0 20 40 60 80 100

0

50

100

150

200

250

300

350

400

450

Labour

Prod

ucti

on o

f x

x=40l^.5


6.3.3 The Marginal Product of Labor (MPL) The Marginal Product of Labor MPL is the increase in productivity caused by a one-‐unit increase in labor. Graphically, it is the slope of the production frontier. Mathematically, it is the derivative of the function of labor with respect to labor (the derivative of the production function with respect to labor):

MPL =δ f (l)δl

6.3.3.1 MPL of a linear production technology Where the production technology is x = 4l :

f (l) = x = 4l

MPL =δ f (l)l

=δxδl

= 4

Graphically, it is the slope. Here, the slope does not change with output: it is thus constant:

Labour

Prod

ucti

on o

f x

x=4l

Production Frontier MPL=slope of Production frontier

MPL=dx/dl=4


6.3.3.2 MPL of a cosin variable production function

Where the production function is x =200 1− cos π

100l

"

#$

%

&'

"

#$

%

&' | l ≤100

400 | l >100

)

*+

,+

-

.+

/+

x =200 1− cos π

100l

"

#$

%

&'

"

#$

%

&' | l ≤100

400 | l >100

)

*+

,+

-

.+

/+

MPL =δ f (l)δl

=δxδl

=200 1

100sin π100

l"

#$

%

&' | l ≤100

0 | l ≥100

)

*+

,+

-

.+

/+

Graphically: the slope increases until the inflexion point at l = 50 , where it decreases until l =100 , where the production frontier is flat, thus, the slope is 0.

Labour

Prod

ucti

on o

f x

variable cosine

MPL=dx/dl


6.3.3.3 MPL of an exponential production function Where the production function is x = 40l0.5 :

x = 40l0.5

MPL = δ f (l)δl

=δxδl

= 40•0.5l−(0.5) = 20l−(0.5) = 20l

Graphically, the slope is positive, and decreasing:

6.3.4 The law of diminishing marginal product Realistically, as the amount of labor hours increases, the benefit to production eventually begins decreasing. This is captured by the variable cosine function and the exponent function. Further, the variable cosine function is the most realistic because productivity increases until a point of maximum efficiency (the inflexion point) and then decreases to zero.

Labour

Prod

ucti

on o

f x x=40l^.5

MPL=20l^-0.5


6.4 Profits Profits are the revenue from production (amount produced multiplied by the price of the good) with the economic costs (cost of inputs) subtracted. Mathematically, in the short-‐term model, it is expressed as follows: π = px −wl Or profits are equal to revenue (price multiplied by quantity produced) minus cost of labor (wage multiplied by labor hours). Note, in the short-‐term model, wage is the only cost considered.

6.4.1 Profit maximization Whereas under the consumer model we attempted to maximize the consumer’s utility subject to their budget. Here, we attempt to maximize profit π = px −wlsubject to the production technology: x = f (l) . Mathematically, this profit maximization problem is expressed as follows:

maxx,l

π = px −wl

subject.to | x = f (l)

There are many ways to solve this problem, but we will focus on the following three methods: isoprofits, marginal revenue product and cost curves.

6.4.2 Isoprofits Isoprofits act like utility functions in the consumer model: they represent combinations of labor and output where the resulting profit is the same. Further, like utility functions, they create an isoprofit map, and the further the isoprofit line that you are on, the more profit for the producer. P(π, p,w) = x, l( )∈ R+

2 |π = px −wl#$ %& Or, the isoprofits map is dependent on profits, price and wages, and is the map of all real combinations of x and l so that the profits remain unchanged. After we plot the isoprofit map, we attempt to go on the “furthest” isoprofit line by seeing where it is tangential to the production frontier (dependent on the production technology).


6.4.2.1 Example part 1 – the isoprofit map: π = px −wl;π = (5)x − (10)l Here, let us construct an example isoprofit, where π = 500 . Note, to work out the “x”, rearrange the equation so that “x” is the subject:

π = 5x −10l

x = π5+105l

l x π = 5x −10l 0 100 500 25 150 500 50 200 500 75 250 500 100 300 500 Note: here, the slope of the isoprofit line is w p =10 5= 2 . Thus, we can graph it as follows:

Note: this is just one isoprofit line. Like indifference curves, they come in a map, represented by the same graph but with profit changing between them (like we changed utility levels in the consumer model). Thus, a isoprofit map with this profit function would be as follows (again, noting that each possible profit level has an associated isoprofit line):

labor

outp

ut (

x)

isoprofit: x=pi/p+(w/p)(l)

labor

outp

ut (

x)

isoprofit: x=pi/p+(w/p)(l)


6.4.2.2 Example part 2(a) – maximizing profit with a linear production technology and an isoprofit map

We maximize profits by seeing where the production frontier is tangential to the isoprofit line. Given a linear production technology: x = 4l ; and the above profit function: π = px −wl |π = 5x −10l we solve the profit maximization problem as below. Since the isoprofit and production frontier are linear, we compare the slope of the production frontier (MPL ) and the slope of the isoprofit line (w p ). Since MPL > w p | (4 > 2) , there is no point of tangency and the producer should continue adding units of labor (each added unit puts the producer on a higher isoprofit line). This is only the case (labor increase always leading to increasing profit) in a simplified linear production technology, since it does not capture the concept of diminishing marginal returns. Graphically, this can be shown as follows:

Since increases in labor do not stop shifting the producer onto higher isoprofit lines, the producer should add an infinite amount of labor to maximize profit.

labor

outp

ut (

x)

isoprofits

Prod

uctio

n fro

ntier


6.4.2.3 Example part 2(b) – maximizing profit with a cosine variable production technology and an isoprofit map

We maximize profits by seeing where the production frontier is tangential to the isoprofit line.

Given a cosine variable production technology: x =200 1− cos π

100l

"

#$

%

&'

"

#$

%

&' | l ≤100

400 | l >100

)

*+

,+

-

.+

/+;

and the above profit function: π = px −wl |π = 5x −10l we solve the profit maximization problem as below. First, let us do this graphically, for the sake of simplicity. The tangent point is the point of optimal labor use for maximizing profit.

Mathematically, profit is maximized where MPL = w p . Since we see that the point of intersection is before the point “l>100” (the production frontier is still curved at the point of intersection), we find the formula for the MPL in the first section and equate this with w / p .

x =200 1− cos π

100l

"

#$

%

&'

"

#$

%

&' | l ≤100

400 | l >100

)

*+

,+

-

.+

/+

MPL =δ f (l)δl

=δxδl

MPL = 200π100

sin π100

l"

#$

%

&'

MPL = 2π sinπ100

l"

#$

%

&'

w / p =10 / 5= 2

I can’t solve that, but it is at approximately (90,390).

labor

outp

ut (

x)

Profit maximisation


6.4.2.4 Example part 2(c) – maximizing profit with an exponent production technology and an isoprofit map

We maximize profits by seeing where the production frontier is tangential to the isoprofit line. Given a cosine variable production technology: x = 40l0.5 ; and the above profit function: π = px −wl |π = 5x −10l we solve the profit maximization problem as below. Mathematically, to find the tangent point between the production frontier and the isoprofit line, we equate MPL = w / p .

x = 40l0.5

MPL =δ f (l)δl

=δxδl

MPL = 20l−(0.5)

wp=10 / 5

wp= 2

MPL =wp

20l−12 = 2

l−12 =

220

l−12

"

#$

%

&'

−21=110"

#$

%

&'−21

l = 101

"

#$

%

&'2

=102 =100

Now, we substitute the “l” we found back into the production technology to find the output at that level of labor input:

x = 40l0.5

x = 40(100)0.5

x = 400

Thus, profit maximization occurs at l=100, x=400. Graphically, we find the point of tangency.

labor

outp

ut (

x)

profit max


6.4.3 Marginal Revenue Product The marginal revenue product (MRP), as the name implies, denotes the increase in revenue for each additional unit of labor, i.e. MRP = p•MPL . We solve the profit maximization problem here by seeing which is the last unit of labor where the marginal revenue product is greater than the marginal cost, in this model, purely the wage; mathematically, we keep adding labor where MRP > w . Mathematically, we can find the point where this stops being the case by equating the MRP with wage:

MRP = wMRP = p•MPL

!

"#

$

%&

MPL • p = w

MPL =wp

Graphically, we plot the MRP and the wage rate on a (labor,$) plane.


6.4.3.1 Example 1 – profit maximization using MRP where there is a linear production technology

We maximize profits by seeing where the MRP is greater than wages.

Given a linear production technology: x = 4l ; and π = px −wl |π = 5x −10l∴w =10

we

solve the profit maximization problem as below. Mathematically: Find MPL Find the MRP

x = 4l

MPL =δ f (l)δl

=δxδl

MPL = 4

MPL = 4p = 5!

"#

$

%&

MRP =MPL • p = 4•5MRP = 20

Equate with wage

MRP = w20 ≠10

Since MRP is greater than wage, the producer can keep increasing labor whilst at the same time increasing profits. Graphically:

Here, the MRP is always greater that wage, thus, profit is always increasing (and is represented by the green shaded area.

labor

$

MRP

Wage

Profit = MRP-w


6.4.3.2 Example 2 – profit maximization using MRP where there is cosine variable We maximize profits by seeing where the MRP is greater than wages.

Given a cosine variable production technology: x =200 1− cos π

100l

"

#$

%

&'

"

#$

%

&' | l ≤100

400 | l >100

)

*+

,+

-

.+

/+;

and π = px −wl |π = 5x −10l∴w =10

we solve the profit maximization problem as below.

Mathematically, this sucks; so let’s do this graphically:

As shown in the graph, the first few units of labor do not produce enough product to justify the wage of the laborers. However, after a point, and until “Profit max.” the MRP is greater than wages and the hiring of extra laborers is justified. After “Profit max.” the laborers cost more than they make for the firm.

labor

$

Wage

Profit = MRP-w

Profit max.

MRP=MPL*P


6.4.3.3 Example 2 – profit maximization using MRP where there is cosine variable We maximize profits by seeing where the MRP is greater than wages. Given a cosine variable production technology: x = 40l0.5 ; and π = px −wl |π = 5x −10l∴w =10

we solve the profit maximization problem as below.

Mathematically, we solve this as follows: Step 1: find MRP Step 2: Profit maxed at MRP=w:

MPL =δ f (l)δl

=δxδl

x = 40l0.5

!

"

###

$

%

&&&

MPL = 20l−(0.5)

MRP =MPL • pπ = px −wl |π = 5x −10l∴ p = 5

!

"

###

$

%

&&&

MRP = 20l−(0.5) •(5)=100l−(0.5)

MRP = w

100l−12 =10

l−12

"

#$

%

&'

−21=110"

#$

%

&'−21

l = 110"

#$

%

&'−2

=102 =100

Step 3: find optimal “x”

x = 40l0.5

l =100

!

"#

$

%&

x = 40(100)0.5 = 400

Therefore optimal consumption is at x = 400 | l =100 Graphically:

Since MRP>W (until labor=100), it is profitable for a producer to continue to hire labor until that point (profit increases with each unit of labor, at a decreasing rate, until l=100)

Labour

$

MRP=100l^-0.5

10

100


6.4.4 Cost curves Using cost curves, we maximize profits using two steps: first, we find the combination of inputs that produce any quantity of output at the lowest possible price (note, this is simple in the single-‐input, i.e. short term, model); second, we find the amount of output that maximizes profit (here, the difference between the cost curve we derived in the first step and the moneys earned from selling the output).

6.4.4.1 Step 1: deriving cost curves (total cost curve) First, we find the lowest amount of labor for each level of output: l(x) = f −1(x) (Note: f −1 is the inverse function of f (x) ) In the short-‐term model, this is just given by the production frontier (since labor is the only input, any less labor means we wouldn’t be able to produce a sufficient amount of output, any more and labor would be wasted) Second, we derive the cost curve, a formula tying minimum cost of input with requisite amount of output. C(x,w) = w• l(x)

6.4.4.1.1 Example 1: deriving cost curves (total cost) for a linear production technology

Cost curves describe the relationship between output and the cost of inputs. Given a linear production technology: x = 4l ; and π = px −wl |π = 5x −10l we find the associated cost curves as below.

x(l) = 4l[ ]∴l(x) = l4

C(x,w) = w• l(x)[ ] = 10( )• 14l

C(x,w) = 2.5l

Here, we took the inverse of f (l) = x(l) to find f −1(l) = l(x) : we had the function of x with respect to l, now we have the function of l with respect to x.

5 10 15 20 25 30 35 40

0

20

40

60

80

100

120

140

160

output (x)

l=x/4

C(x,w)=w*l(x)=2.5x

$


6.4.4.1.2 Example 2: deriving cost (Total cost) curves for a cosine variable production function

Cost curves describe the relationship between output and the cost of inputs.

Given a linear production technology: x =200 1− cos π

100l

"

#$

%

&'

"

#$

%

&' | l ≤100

400 | l >100

)

*+

,+

-

.+

/+; and

π = px −wl |π = 5x −10l we find the associated cost curves as below.

x(l) = 200 1− cos π100

l"

#$

%

&'

"

#$

%

&'

l(x) = x−1(l)

(

)

***

+

,

---

x = 200 1− cos π100

l"

#$

%

&'

"

#$

%

&'

x200

=1− cos π100

l"

#$

%

&'

1− x200

= cos π100

l"

#$

%

&'

cos−1 200− x200

"

#$

%

&'=

π100

l

l =100cos−1 200− x

200"

#$

%

&'

π

l(x) = ±100cos−1 200− x

200"

#$

%

&'

πC(x,w) = p• l(x)π = px −wl |π = 5x −10l(

)*

+

,-

∴C(x,w) = ±500cos−1 200− x

200π

This is complicated mathematically, but we remember that we can find the inverse of a function by flipping it around the line y=x, so graphically, it is much less complex.

l(x)

y=x

x(l)

C(x,w)


6.4.4.1.3 Example 3: deriving cost curves (total cost) for an exponent production function

Cost curves describe the relationship between output and the cost of inputs. Given a linear production technology: x = 40l0.5 ; and π = px −wl |π = 5x −10l we find the associated cost curves as below.

l−1(x) = x(l)

l(x) = 40l12

"

#

$$

%

&

''

x = 40l12

x40(

)*

+

,-2

= l12

(

)*

+

,-

2

l = x2

402=

x2

1600

C(x,w) = p• x(l)

= 5( )• x2

1600!

"#

$

%&

=x2

320

Graphically:

x(l)

x=y

C(x,w)

l(x)


6.4.4.2 Step 2: deriving cost curves (marginal and average costs) The marginal and average cost curves are required to solve the profit maximization problem, and are derived from the total cost curve as derived above. The marginal cost curve represents the additional cost for each unit increase in output (i.e. the change in total cost for a given change in output). The average cost is merely the total cost divided by the amount of output. Step 1: find total cost curve

x(l) = l−1(x)C(x,w) = x(l)•w

Step 2: find the marginal cost curve

MC(x,w) = δC(x,w)δx

Step 3: find the average cost curve

AC(x,w) = C(x,w)x

The marginal cost curve will always intercept the average cost curve at its minimum. Profits are made past this point of intersection, and this defines the product supply function.


6.4.4.2.1 Example 1: deriving cost curves (marginal and average cost) for a linear production technology

Mathematically, given a production technology of x = 4l and a profit function of π = px −wl |π = 5x −10l : Step 1: find the total cost function

x(l) = l−1(x) = x4

C(x,w) = w• x(l)

= 10( ) x4"

#$%

&'

= 2.5x

Step 2: find the marginal cost function

MC x,w( ) = δC(x,w)

δxC x,w( ) = 2.5x

!

"

###

$

%

&&&

MC x,w( ) = 2.5

Step 3: find the average cost function

AC x,w( ) =C x,w( )

xC x,w( ) = 2.5x

!

"

###

$

%

&&&

AC x,w( ) = 2.5xx

= 2.5

Graphing this:

output (x)

Tota

l Cos

t=C(

x,w)=

2.5x

$

AC/MC=2.5

$


6.4.4.2.2 Example 2: deriving cost curves (marginal and average cost) for a cosine variable production technology

Given:

• Production technology of x =200 1− cos π

100l

"

#$

%

&'

"

#$

%

&' | l ≤100

400 | l >100

)

*+

,+

-

.+

/+

• Profit function of π = px −wl |π = 5x −10l : Graphically:

C(x,w)

AC

MC

x

$


6.4.4.2.3 Example 3: deriving cost curves (marginal and average cost) for an exponential production technology

Given:

• Production technology of x = 40l0.5 • Profit function of π = px −wl |π = 5x −10l :

Mathematically Step 1: find the total cost curve

x l( ) = l−1 x( )l x( ) = 40l1/2

x l( ) = l40"

#$

%

&'2

=l2

1600C x,w( ) = w• x(l)

=10• l2

1600=l2

160

TC =C(x,w) = l2

160

Step 2: find the marginal cost curve

MC =δC x,w( )

δx

C x,w( ) = x2

160=1160

• x2

MC = 2160

• x = x80

Step 3: find the average cost curve

AC =c x,w( )x

=x2

160•1x

=x160

Graphically

Total Cost = C(x,w)

MC=dTC/dx

AC=TC/x


6.4.4.3 Step 3: profit maximization using cost curves The curves we derived above – the total cost, marginal cost and average cost curves – allow us to analyze the cost of producing each unit of output. However, this does not tell us how much the producer should produce given different economic circumstances. To work out optimum output, we have to derive the output supply curve “ x p,w( ) ” – output given cost of inputs and market given prices. How much to produce, given the cost curves derived above? Economic reasoning suggests that we should produce more and more output until the cost of producing an additional unit (MC) is equal to the amount of revenue generated by the sale of that unit of output: i.e. MC=P. If we produce less, than there are profit-‐making units that we will miss out on. If we produce more, we lose money on the additional units (since we pay more for the inputs than we receive, i.e. MC>P).

6.4.4.3.1 Breakeven analysis creating the x(p,w) curve Where P=MC=AC, we have the breakeven price. Below that, there is not enough revenue to cover the average costs. Thus, x p,w( ) , giving x* , is the MC curve above where MC=AC. If MC is always above AC, then x p,w( ) is the whole of the MC curve. Formally, it can be expressed as follows:

x p,w( ) =MC x,w( ) |MC < AC0 |MC > AC

!"#

$#

%&#

'#

Why does the slide say MC−1 ? Is it because we are inverting the function to make “x” the subject?


6.4.4.3.2 Graphing the production function

6.4.4.3.2.1 Linear

In the linear model, MC is the same as the AC curve, so the whole of the MC curve gives the production function. Given P1, P>MC for all values of “x”, therefore there should be unlimited production, since any increase in production increases profit. Given P2, P<MC for all “x”, therefore there should be no production, since each unit of output incurs a loss.

6.4.4.3.2.2 Cosine variable function

To the right of the “x(be)” point, the MC curve gives the production function. The producer cannot profit by producing less than x(be) units of output, or where there is a price lower than P(be). At P2, the producer should produce any output since P2<P(be); i.e. the revenue from each unit of output is less than the costs for the same unit of output. At P1, the producer should produce x1

* units of output. Any more, and the cost of producing one additional unit of output (MC) is greater than the revenue gained.

output (x)

$

AC/MC/TC

$

P1

P2

x

$

P(be)

P1

P2

x(be) x*(1)

ACMC

Production function


6.4.4.3.2.3 Exponential function

Here, the average cost is always below the marginal cost, thus the whole of the marginal cost curve is the production function. At P1, we should produce x*1 units of output to maximize profit.

x

$

P(be)

P1

P2

x(be) x*(1)

ACMC

Production function


7 The producer model in the long term Last week, we solved the profit maximization problem for producers that had a single input (labor) and no (or, rather, unconsidered) fixed costs. This week we begin to relax these assumptions by unfixing capital ( x = f l,k( )→ x = f l,k( ) ) and allowing firms to dodge fixed costs by entering and exiting the market.

7.1 Revision We solved the profit maximization problem π

max= px −wl subject to x = f l( ) in

three ways. • Using isoprofits, we saw what the furthest isoprofit line that could be

reached using the given production technology. • Using MRP (Marginal Revenue Product), we worked out the increase in

revenue for each additional unit of output, and equated it with the price in order to find the last unit of output that was still profitable.

• Using the cost curves, we derived the production function (where the Marginal Cost Curve is above the Average Cost curve), which, given any price, gives an optimal output.

This week, we will focus on the cost curves method, so we will revise it here. There were two steps here:

1. Find the lowest cost of producing any volume of output a. In the f l,k( ) model this was simple, we just used the production

frontier 2. Find the output supply curve – x(p,w, r) as the MC curve where MC>AC

a. x p,w, r( ) =MC−1 p,w( ) | p >min AC w, x( )( )0 | p <min AC w, x( )( )

"#$

%$

&'$

($

7.2 Introduction to this week This week, we move the producer model to the long run, and relax the assumption that capital is fixed: x = f l,k( )→ x = f l,k( ) . In this model, capital represents all non-‐labor investment, and the associated cost is “r” – the interest rate on the purchased capital. Here, we, too, use the cost curve method of working out profit maximization, as we learned last week. So, what happens to the two-‐step process?

1. Find the lowest cost of producing any volume of output a. This is a bit more difficult here than last week, since we have to

find the cheapest was of combining capital and labor for any given amount of output

2. Find the output supply curve – identical to last week (the MC curve where MC>AC)


7.3 Production

7.3.1 Terms • A production plan is a combination of inputs (l, k) and outputs (x): (x, k, l) • Outputs are a function of inputs: x = f l,k( ) • The production choice set is the set of all feasible choices of inputs and

outputs • The production frontier is the limit of this choice set – here lie production

plans where inputs are fully utilized (there is no waste)

7.3.2 Production technologies (for this week) Last week we talked about linear, cosine variable and exponential production functions, but these were only appropriate where there was a single input. This week, we shift to production technologies that account for both capital and labor. We use the same production functions as we did with utility functions in the consumer model, since, as in consumers where consumption was substitutable in different ways, so too are inputs in the production model.

• Cobb-‐Douglas o x = f l,k( ) = lαkβ

• Perfect complements o x =min αl,βk{ }

• Perfect substitutes o x =αl +βk

Trying to plot the production frontier where there are two inputs would create a three-‐dimensional graph, which is impractical for problem solving. Here is an example of a Cobb-‐Douglas production function:

Thus, it is beneficial to work out the profit maximization problem mathematically.


7.3.3 Technical Rate of Substitution, Returns to Scale and Marginal Products

7.3.3.1 Technical Rate of Substitution This is similar to the old consumer model idea of the marginal rate of substitution. There, it represented the amount of x1you would be willing to give up to get one additional unit of x2 -‐ mathematically, it represented the substitutability of goods such that utility remains unchanged. Now, since we have shifted to the producer model, we look to it in a different way. How much of one input (e.g. “l”) must be substituted for another (e.g. “k”) such that output (“x”) remains unchanged. Mathematically, it is expressed as follows:

TRS l,k( ) = −δx l,k( ) lδx l,k( ) / k

= −MPLMPK

à “l” and “k” not derivatives?

Is this the same as − ΔlΔk

???

What does an output mean? For example, a TRS(l,k) of -‐3 suggests that if you reduce capital by 3 units, and add one unit of labor, then output will remain unchanged. Not the other way around? But this alone is not enough, as illustrated by the following two examples: Example 1: given a production technology of g l,k( ) = l0.5k0.5 :


g l,k( ) = l0.5k0.5

"

#

$$$

%

&

'''

δx l,k( )l

= 0.5l−0.5k0.5 |δx l,k( )

k= 0.5l0.5k−0.5


= −0.5l−0.5k0.5

0.5l0.5k−0.5

= −k0.5k0.5

l0.5l0.5= −

k0.5+0.5

l0.5+0.5= −

kl

Example 2: given a production technology of h l,k( ) = lk


h l,k( ) = lk

"

#

$$$

%

&

'''

TRS l,k( ) = − k • l0

l •k0= −

kl

The two production technologies have identical TRS’s, but their scale is different. TRS does not factor in scale, for that, we go to “returns to scale”.


7.3.3.2 Returns to scale Let us start with scaling all of the inputs by a constant: “t”.

• g l,k( ) = l0.5k0.5 → (tl)0.5(tk)0.5

o = t0.5l0.5t0.5k0.5 = t0.5+0.5l0.5k0.5 = t l0.5( ) k0.5( )= t •g l,k( )

• h l,k( ) = lk→ tl • tk

o = t2 lk( )= t2 •h l,k( )

Here, the production functions are differentiated by scale. Now, let us create some terminology to describe these different returns to scale:

• Iff f tl, tk( ) < t • f l,k( ) , we have decreasing returns to scale • Iff f tl, tk( ) = t • f l,k( ) , we have constant returns to scale • Iff f tl, tk( ) > t • f l,k( ) , we have increasing returns to scale

With the consumer model, the “level” of utility did not matter: if it was maximized, than that was the optimal choice for the consumer. However, in the producer model, returns to scale are crucial! Where optimum production is 100, or 1000 can mean the difference between being profitable or not (and closing the business).

7.3.3.3 Marginal Product of inputs Since we have different inputs, the MPL and MPk may be different, and are not necessarily congruent with the returns to scale. Let us take the above as an example:

• g l,k( ) = l0.5k0.5

o

MPL =δ l,k( )l

= 0.5k0.5l−0.5

=k0.5

2l0.5

MPk =δ l,k( )k

= 0.5l0.5k−0.5

=l0.5

2k0.5

• h l,k( ) = lk→ tl • tk

o MPL =δ l,k( )l

= k MPk =

δ l,k( )k

= l

What does this mean though? • g l,k( ) -‐ as either input increases, the marginal product decreases, there

are decreasing marginal products • h l,k( ) -‐ there are constant returns

But, we can also do it the other way, by increasing one input and seeing the effect on output: if f l, tk( ) > tf l,k( ) , we have an increasing marginal product of Capital.


7.4 Profit maximization for producers in the long-term model Like we mentioned before, our goal is to maximize the profits of the producer subject to their production technology. Mathematically, our problem may be expressed as π

max= px −wl subject to x = f l,k( ) .

To do this, there is a two-‐step process:

• First, find out the cheapest possible combination of inputs for each level of output;

o C w, r, x( ) = wl w, r, x( )+ rk w, r, x( ) • Second, find the amount of output to produce given a market price by

equating price with marginal cost.

7.4.1 Example: Cobb-‐Douglas

Let’s learn this using the example production technology f l,k( ) = 20• l25 •k

25 given

the market prices w=20, r=10, p=5.

Decreasing returns to scale: f (tl, tk) = 20• t

25 • l

25 • t

25 •k

25

= t45 20l

25k

25 = t

45 f l, l( ) < tf l, l( )

Decreasing MPL : f tl,k( ) = 20t25l25k

25 = t

25 • f l,k( ) < tf l,k( )

Decreasing MPK : f l, tk( ) = 20t25l25k

25 = t

25 • f l,k( ) < tf l,k( )

7.4.2 Step 1: deriving the costs curve (i) – isoquants An isoquant is a line on a (l, k) plane that describes all of the combinations of l & k that give the same level of output. The slope of this line is the TRS (as described above), the amount of one good that would be given up to replace with another such that output remains constant. They are equivalent to indifference curves.

TRS l,k( ) = −δx l,k( ) δlδx l,k( ) /δk

f l,k( ) = 20• l25 •k

25

δx l,k( )l

= 20• 25l−35 •k

25 |δx l,k( )

k= 20• 2

5k−35 • l

25


= −20 • 2

5• l

−35 •k

25

20 • 25•k

−35 • l

25

= −k25 •k

35

l25 • l

35

= −kl


7.4.3 Step 2: deriving the cost curve (ii) -‐ isocosts Isocosts are lines that describe combinations of l & k that cost the same – this is equivalent to a budget line. To draw this, we rewrite the cost equation with k as

the subject: k = Cr−wrl . The slope is –w/r.

7.4.4 Step 3: cost minimization So, our goals here are to find the lowest cost combination of labor and capital to produce any given value of x. As we could expect, we look for the lowest possible isocost curve that is tangent to the requisite isoquant, and we do this by equating their slope functions. The slope of the isoquant is the TRS, and the slope of the isocost is the ratio of wages to interest rates, so we have to find:

TRS(l,k) = wr

This can be illustrated as follows:

Mathematically, we express the problem as follows: The cost minimization problem is expressed as: min

l,kCost = wl + rk

subject to x l,k( ) = x We solve the problem by ensuring that:

TRS l,k( ) = wr

wherex l,k( ) = x

Cap

ital

Isoquant

Isocost

cost minimisation point (TRS=w/r)

Labor


But this doesn’t solve the whole problem, in order to do so; we have to follow the following steps:

1. Set TRS l,k( ) = wr and solve to find “l” or “k”

2. Substitute this into the production function 3. Solve for the other input 4. Substitute this into the equation for the original input

7.4.4.1 Example

Let’s take thee example production technology f l,k( ) = 20• l25 •k

25 given the

market prices w=20, r=10, p=5. Find the tangent point of the isoquant and isocost curves Slope of isoquant = TRS Slope of isocost = w/r

f l,k( ) = 20• l25 •k

25

TRS l,k( ) =δ f l,k( ) /δlδ f l,k( ) /δk

=20• 2

5• l

−35 •k

25

20• 25•k

−35 • l

25

=kl

wr=20( )10( )

= 2

Equate the two

wr=kl

k = l •wr


Substitute this into the production function

f l,k( ) = 20• l25 •k

25

k = l •wr

!

"

###

$

%

&&&

x = 20• l25 • l •w

r'

()

*

+,

25

Solve for “l”

x = 20• l25 • l w

r!

"#

$

%&

25

x20!

"#

$

%&

52= l • l •w

r

l2 = x20!

"#

$

%&

52•rw

l = x20!

"#

$

%&

54•rw!

"#

$

%&

12

Solve for “k” by plugging this into the production function

l = x20!

"#

$

%&

54•rw!

"#

$

%&

12

k = wrl

'

(

))))

*

+

,,,,

k = wr

x20!

"#

$

%&

54•rw!

"#

$

%&

12

!

"

##

$

%

&&=wr•

x20!

"#

$

%&

54•rw!

"#

$

%&

12

k = wr

!

"#

$

%&

12•

x20!

"#

$

%&

54


7.4.5 Step 4: cost curves If we do this kind of problem solving for each possible “x”, we can derive the cost curve, the minimum cost for each possible value of “x”.

This gives us the cost curve, but now we want to put it in dollar terms.

C w, r, x( ) = rk +wl

k = wr

!

"#

$

%&

12•

x20!

"#

$

%&

54

l = x20!

"#

$

%&

54•rw!

"#

$

%&

12

C w, r, x( ) = r wr

!

"#

$

%&

12•

x20!

"#

$

%&

54+w x

20!

"#

$

%&

54•rw!

"#

$

%&

12=

x20!

"#

$

%&

54r wr

!

"#

$

%&

12+w r

w!

"#

$

%&

12

!

"

##

$

%

&&

=x20!

"#

$

%&

54

rw( )12 + rw( )

12

!

"#

$

%&

C w, r, x( ) = x20!

"#

$

%&

542 rw( )

12

Cap

ital

Labor

Cost Curve


7.4.6 Step 5: marginal and average cost curves We can now derive our marginal and average cost curves.

C w, r, x( ) = x

20!

"#

$

%&

542 rw( )

12

MC =δC x, r,w( )

δx

=12054

x20!

"#

$

%&

142 rw( )

12

=x20!

"#

$

%&

14 rw( )

12

8

AC =C x,c,w( )

x

=

x20!

"#

$

%&

542 rw( )

12

x

=1x

x20!

"#

$

%&

542 rw( )

12 =

120

x20!

"#

$

%&−1 x20!

"#

$

%&

542 rw( )

12

=x20!

"#

$

%&

14 rw( )

12

10


7.4.7 Profit maximizing The above has shown us how to minimize the costs of production, and what the minimum costs of production are for each level of output. However, we still haven’t shown how much a producer must produce to maximize his profit at each price-‐point: creating the Output Supply Curve: x p,w, r( ) . This is the same as in the short run model, our output supply curve is the Marginal Cost Curve where MC>AC (i.e. above the breakeven point).

x p,w, r( ) =MC−1 p,w( ) | p ≥minAC0 | p <minAC

#$%

&%

'(%

)%

Why the inverse MC? Is it because we are making x the subject?

7.4.7.1 Example

Where, C w, r, x( ) = x20!

"#

$

%&

542 rw( )

12

We have the following cost curves (as above):

MC = x

20!

"#

$

%&

14 rw( )

12

8

AC = x20!

"#

$

%&

14 rw( )

12

10

Since the equations are equal, but for the denominator on the “rw” term, and the AC curve has a higher denominator, the MC curve is necessarily above the AC curve for all values of “x”. Thus, the Output Supply Curve is equal to the MC curve.

p =MC w, r, x( )

p = x20!

"#

$

%&

14 rw( )

12

8

p• 8

rw( )12

=x20!

"#

$

%&

14

p4 84

rw( )12•4= 84 p4

rw( )2=x20

x p, r,w( ) = 20•84 • p4

rw( )2


7.5 Labor and Capital Demand Curves We used to derive the Capital and Labor Demand curves via a function of wages, interest rates and level of output (itself a function of wages, interest rates and prices). By substituting the Output Supply Curve into the Demand Functions, we can make them a function of Interest Rates, Wages and Prices, a simpler equation to solve.

l w, r, x w, r, p( )( )→ l w, r, p( )k w, r, x w, r, p( )( )→ k w, r, p( )

Documents

Intermediate Microeconomics (Producer Model) · PDF fileIntermediate*Microeconomics*(ii)* * 6—1* * IntermediateMicroeconomics!(ii)! ProducerModel(* * 6*The*producer*model*in*the*short*term

Intermediate Microeconomics (Producer Model) · PDF fileIntermediateMicroeconomics(ii)* * 6—1* * IntermediateMicroeconomics!(ii)! ProducerModel(* * 6Theproducermodelintheshort*term