Total, average and marginal quantities

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MICROECONOMICS I.

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ELTE Faculty of Social Sciences, Department of Economics

Microeconomics I.

week 4

WORKING TOOLS, PART 2 Authors:

Gergely K®hegyi, Dániel Horn, Klára Major Supervised by

Gergely K®hegyi

June 2010

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K®hegyi-Horn-Major

Optimization Total, average and marginal quantities Relationship between quantities

The course was prepaerd by Gergely K®hegyi, using Jack Hirshleifer, Amihai Glazer and David Hirshleifer (2009) Mikroökonómia. Budapest: Osiris Kiadó, ELTECON-books (henceforth HGH), and Gábor Kertesi (ed.) (2004) Mikroökonómia el®adásvázlatok.

http://econ.core.hu/ kertesi/kertesimikro/ (henceforth KG).

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K®hegyi-Horn-Major

1 Optimization

Total, average and marginal quantities Relationship between quantities

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Total, average and marginal quantities

Sold quantity: Q Price: P

Revenue: R =PQ

Average revenue: AR =_Q^R =^PQ_Q =P Marginal revenue: MR =_∆^∆^R_Q

Note

The∆shows a small or unit change.

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Total, average and marginal quantities (cont.)

Total, average, and marginal revenues

Quantity Price or average revenue Total revenue Marginal revenue

(Q) (P=AR) (R=PQ) (MR)

0 10 0

1 9 9 9

2 8 16 7

3 7 21 5

4 6 24 3

5 5 25 1

6 4 24 −1

7 3 21 −3

8 2 16 −5

9 1 9 −7

10 0 0 −9

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K®hegyi-Horn-Major

Total, average and marginal quantities (cont.)

The top graph shows the total revenue R, the bottom graph the average revenue AR and the marginal revenue MR. If Q =4 then R=24. The height of the AR curve equals the slope of the ON line on the top graph if Q=4, that is AR=R/Q=24/4=6. The height of the MR curve equals the slope of the total revenue curve. At Q=4 we approximate it with the average of the two slopes of LN and NM.

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K®hegyi-Horn-Major

Total, average and marginal quantities (cont.)

Note

ATTENTION: total quantities (such as the total revenue on the upper part of the graph) should NEVER be depicted on the same graph with the average and marginal quantities (see bottom part of the graph)!!! Their measures are dierent. While the total units are measured in money (e.g. dollar) the average and marginal quantities are measured in dollar/unit.

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K®hegyi-Horn-Major

Total, average and marginal quantities (cont.)

The AC average cost function and MC marginal cost function can be deduced from C total cost function. At the quantity, where the slope of the total cost function is the smallest, the MC is minimal (K on the upper graph).

Where the slope of the line drawn from the origin to the graph is the smallest, AC is minimal (L on the upper graph).

Where AC is declining MC is below AC; where AC is increasing MC is above AC.

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E.g. Foraging

The optimal stay time s∗, at any single resource patch with yield, occurs when the marginal yield in the patch equals the average yield y/t taken over the entire period - dividing the yield per patch y by the overall time per patch t, where t =d+s.That is, the average time per patch includes not only the stay time s but the dead time d spent traveling from one patch to the next.

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K®hegyi-Horn-Major

Discrete quantities

Note

If only discrete choices are possible, then the optimum quantity is where the marginal revenue is smaller than the marginal cost in the "next step", while the marginal revenue is larger than the marginal cost in the "earlier step".

Number of Average Marginal

articles salary gain (dollar) salary gain (dollar)

1 543 543

5 295 191

10 227 153

15 194 120

20 174 109

25 160 100

30 149 93

35 150 49

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A little more precise, mathematically

One variable:

Endogenous variable: x

Total quantity as a function of the endogenous variable:

G =f(x),f :R→R Average quantity: AG= ^f⁽_x^x⁾

Marginal quantity: MG=lim∆x→0∆f(x)

∆x =^df_dx^(x) =f⁰ Two variables:

Endogenous variables: x1,x2

Total quantity as a function of the endogenous variables:

G =f(x1,x2),f :R²→R Average quantities:AG1= _x^G

1 =^f^(x_x¹^,x²⁾

1 ,AG2=_x^G

2 =

f(x1,x2)

x2 ,AGi :R²→R;i =1,2 Marginal quantities:

MG1= ^∂^f⁽_∂^x_x¹^,^x²⁾

1 ,MG2=^∂^f⁽_∂^x_x¹^,^x²⁾

2 ;MGi :R²→R;i=1,2 n variables:

Endogenous variables: x1,x2, . . . ,xi, . . . ,xn

G =f(x1,x2, . . . ,xi, . . . ,xn),f :Rⁿ→R

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K®hegyi-Horn-Major

A little more precise, mathematically (cont.)

Average quantities:

AG1=_x^G

1,AG2=_x^G

2, . . . ,AGi =_x^G

i, . . . ,AGn=_x^G Marginal quantities: n

MG1= _∂^∂f_x

1,MG2= _∂^∂f_x

2, . . . ,MGi =_∂^∂f_x

i, . . . ,MGn= _∂^∂f_x With vectors: n

Endogenous variables: x=





 x1

x2

... xi

... xn







G =f(x),f :Rⁿ →R

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week 4

K®hegyi-Horn-Major

A little more precise, mathematically (cont.)

Average quantities:

AG=





 AG1

AG2

... AGi

... AGn







=







xG1 xG2

...

Gx_i

...

xGn







;AG:Rⁿ→Rⁿ

Marginal quantities:

MG=





 MG1

MG2

... MGi

... MGn







=







∂G

∂x1

∂G

∂x2

...

∂G

∂x_i

...

∂G

∂xn







;MG:Rⁿ→Rⁿ

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K®hegyi-Horn-Major

Repeating the math

Let us assume that the relationship between x and y endogenous variables is described by y =x³−6x+x² function. What are the x values where y is maximal/minimal? How large is y?

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Relationship between the average and the marginal quantities

The marginal value is the slope of the function of total quantity.

The average value is the slope of the line drawn from the origin to the function of total quantity.

Statement

If total quantity is increasing the marginal quantity is positive.

(frequent mistake!)

If total quantity is decreasing the marginal quantity is negative.

Where total quantity is minimal or maximal, marginal quantity is zero.

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Relationship between the average and the marginal quantities (cont.)

Statement

Where average quantity is decreasing, marginal quantity has to be under the average quantity.

Where average quantity is increasing, marginal quantity has to be over the average quantity.

Where average quantity is neither decreasing nor increasing (its minimal or maximal), marginal quantity equals average quantity.

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K®hegyi-Horn-Major

Repeating the math

Let us assume that the relationship between x and y endogenous variables is described by y =x³−6x+x² function. What are the x values where y is maximal/minimal? How large is y if we look at the [0;2] closed interval?

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K®hegyi-Horn-Major

Repeating the math

Denition

Let's assume f(x)S →Rcan be dierentiated at S⊆Rⁿ! Let c∈S be an inside value of S subset! Then c is a stationer point of f(x)function, if

f_i⁰(c) =0 i=1,2, . . . ,n m

f⁰(c) =0.

Theorem

Let's assume f(x)S →Rcan be dierentiated at S⊆Rⁿ! Let c∈S be an inside value of S subset! If c is a minimum or maximum value of az f(x)function in S subset, then c is a stationer point of f(x)function.

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K®hegyi-Horn-Major

Repeating the math (cont.)

Theorem

Let f(x,y)be a function dened in S ⊆R²set. Let's assume that the rst and second partial derivatives are also continuous! Let moreover(x₀,y₀)be an inside value of S set, which is a stationer point of f(x,y)function. Then

f₁₁⁰⁰(x₀,y₀)<0 és

f₁₁⁰⁰(x₀,y₀)f₂₂⁰⁰(x₀,y₀)−f₁₂⁰⁰²(x₀,y₀)>0⇒(x₀,y₀)local maximum;

f₁₁⁰⁰(x0,y0)>0 és

f₁₁⁰⁰(x0,y0)f₂₂⁰⁰(x0,y0)−f₁₂⁰⁰²(x0,y0)>0⇒(x0,y0)local minimum;

f₁₁⁰⁰(x₀,y₀)f₂₂⁰⁰(x₀,y₀)−f₁₂⁰⁰²(x₀,y₀)<0⇒(x₀,y₀)saddle point;

f₁₁⁰⁰(x₀,y₀)f₂₂⁰⁰(x₀,y₀)−f₁₂⁰⁰²(x₀,y₀) =0⇒(x₀,y₀)can be local minimum, local maximum or saddle point as well.

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week 4

K®hegyi-Horn-Major

Repeating the math

E.g.: Let y,x₁és x₂be endogenous variables, and let

y =x₁²−6x₁+x₂²−4x₂+113 function describe the relationship in between them. What will be the x₁ and x₂ values, where y is minimal or maximal? And what is this y value??

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K®hegyi-Horn-Major

Repeating the math

Theorem

Let's assume that both f(x,y)and g(x,y)have continuous partial derivatives within A range of xy plane, and also that(x₀,y₀)is an inside value of A, and that it is the local minimum/maximum value of f(x,y)if g(x,y) =0. Let us assume moreover that one of the g₁⁰(x₀,y₀)or g₂⁰(x₀,y₀)is not zero. Then there exists one and only oneλnumber that(x₀,y₀)pair is a stationer point of

L(x,y) =f(x,y)−λg(x,y) Lagrange-function.

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K®hegyi-Horn-Major

Repeating the math

Theorem

Let f(x,y)and g(x,y)R²→Ra continuously dierentiable function, and let's assume that we look for the optimal solution of

max(min)f(x,y) g(x,y) =0

)

Let us assume moreover, that(x₀,y₀)is the stationer point of L(x,y) =f(x,y)−λg(x,y)

Lagrange-function and that g(x₀,y₀) =0.

Then

L(x,y)concave ⇒(x0,y0)the maximum solution; L(x,y)convex ⇒(x₀,y₀)the minimum solution.

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K®hegyi-Horn-Major

Repeating the math

Example:

E.g.: Let y,x₁and x₂endogenous variables and let

y =x₁²−6x₁+x₂²−4x₂+113 function describe the relationship in between. What are the values of x₁ and x₂ where y is minimal or maximal? And what are these y values assuming that

x1+x2=100?