Chapter5-LaTech

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Introduction

Factor-factor decision: The manager must decide which combinations of inputs to use in order to achieve the firm’s economic objective.

Cost minimization: The factor-factor decision involves finding the least-cost combination of inputs, from the different input combinations possible, that will produce a given level of output.

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Firm produces a single product Firm has two variable inputs; other are fixed

(The production function can be expressed as:

Y = ƒ(X1, X2 | X3 ……. Xn) )

Firm is a price taker.

Assumptions

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Physical Relationships

Production Surface: describes the “hill” of increased production obtained as we increase use of the variable inputs.– A three dimension production surface;

(1) Each elevation on the “hill” has the same level of output (such as line A, B, and C);

(2) Production level C > B > A.

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• Isoquant: line indicating all combinations of two variable inputs that will produce a given, or constant, level of output.

– How do you derive an isoquant?

Example: (1) Daily farm combine grain (X1) and hay (X2) for his herd’s feed ration. Different combinations that can produced the same level of output (Y = 11000 lbs of milk)

Physical Relationships

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• distance between the isoquant and the origin indicates the level of output, or height of the production surface.

• closer the isoquant is to the origin, the lower the level of output. The farther away from the origin, the higher the level of output.

• Isoquants can never intersect.

Characteristics of Isoquants

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Marginal Rate of Substitution (MRS): rate at which one variable input can physically substituted for another variable input in the physical production process– calculated by the dividing the change in the replaced

input ( replaced) by the change in the added input ( added).

Marginal Rate of Substitution (MRS)

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X1 -- the variable input being replaced

X2 -- the variable input being added

MRSX2X1 = X1 / X2 .X1

X2

I0

X1

X2

Marginal Rate of Substitution (MRS)

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X2 -- the variable input being replaced

X1 -- the variable input being added

MRSX1X2 = X2 / X1.X2

X1

I0

X2

X1

Marginal Rate of Substitution (MRS)

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• Input being replaced: vertical axis• Input being added: horizontal axis.• Slope of the isoquant at any point is equal to MRS at

that point.

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– Calculate the MRS for the dairy example.

Note:

|MRSX2X1| > 1, X2 is good substitute for X1.

|MRSX2X1| < 1, X2 is less efficient substitute for X1.

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(1) Perfect Substitutability: It occurs when one unit of an input can be exchanged for another input on a consistent basis (or unchanged ratios).

MRSX2X1 = X1 / X2 =

constant.X1

X2

I0

X1

X2

Possible Rates of Input Substitution

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(2) Imperfect Substitutability: one unit of an input can be exchanged for another input on a inconsistent basis (or changed ratios).

X1

X2

I0

X1

X2

Possible Rates of Input Substitution

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(3) Fixed proportions (or Perfect

complementarity): inputs must be used in a fixed ratio. Note: The substitutability is zero.

X1

X2

I0

Possible Rates of Input Substitution

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Economic Relationships

Isocost line: line indicating all combinations of two variable inputs that can be purchased for a given, or same level of expenditure.

TE = (PX1* X1) + (PX2* X2)

(1) Example: Daily farmer TE = $100;

Price of grain: PX1=$0.20/lb;

Price of hay: PX2 = $0.10/lb.

Question: how many units of each input could be purchase?

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X1

X2

TE=$100

TE=$200

Isocost Curve

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• The closer to the origin, the lower the level of total expenditure and the fewer inputs can be purchased; the farther away from the origin the higher the level of total expenditure and the more inputs can be purchased.

• Inverse Price Ratio(IPR): slope of the isocost line is - PX2 /PX1; the price of the input on the horizontal axis

divided by the price of the input on the vertical axis.• indicates the relative value of the input, or how they

substitute for one another economically.

Isocost Curves

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Price Changes

If the price of the one of the variable input changes, the slope of the isocost line also changes.

If PX2 , TE0 TE1

If PX2 , TE0 TE2

X1

X2TE0

TE2TE1

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If PX1 , TE0 TE1

If PX1 , TE0 TE2

X1

X2

TE0

TE2

TE1

Price Changes

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(1) The end points of the isocost line show how many units of the variable input could be purchased if all the expenditure were spend on one input.

(2) A change in total outlay causes a parallel shift in the isocost line.

(3) A change in one input price causes the isocost line to rotate around the intersection of the isocost line and the axis of the other input.

(4) A reduction in the use of one input is necessary to increase use of the other input.

(5) The slope of the isocost, also called IPR, is the price of the input on the horizontal axis divided by the price of the input on the vertical axis

Isocost Summary

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Factor-Factor Decision

Two equivalent approaches (Numerical and Graphical) can be used to determined the least-cost combination of inputs.

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Manager equates the the MRS (the slope of the isoquant) to the inverse price ratio (the slope of the isocost) to determine the least-cost combination of inputs. That is,

MRSX2X1 = - PX2 /PX1

Numerical Approach

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The least-cost combination of inputs is located at the point of tangency between the isocost line and the isoquant.

(Recall that at the point of tangency of two curves, the slope of two curves are equal to each other.)

(1) Isoquant is known, and the slope of isocost curve

-PX2 /PX1

(2) Parallel move the isocost curve and find TE1.

TE0 is too lower to reach the isoquant (optimal Y);

TE2 is too higher b/c it has the same effect as TE1.

Graphical Approach

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Expansion of Production Level

• Isocline: is a line connecting the least-cost combinations of inputs for all output levels at a specific price ratio; shows how, at a given price ratio, inputs should be added as production increases.

• Expansion path: is a line connecting least-cost combinations of inputs for each level of output at “the specific price ratio the manager believes relevant.”

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Product-Product Decision

situations in which the firm produces multiple products from a given set of resources.

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The firm produces two products The firm has a fixed set of resources The firm is a price taker (both inputs and

outputs)

– Question: How to mathematically express the above assumptions as a production function?

Assumptions

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The product-product decision involves determining how the two products substitute for one another in the firm’s production process.

– Because the firm will act as a rational producer with fixed amounts of inputs, the objective is to combine outputs produced in the proportion that results in higher revenue, given a resource constraint.

Product-Product Decision

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Physical Relationships

Production possibilities frontier (PPF): curve depicting all the combinations of two products that can be produced using a given level of inputs (or expenditure).

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Example: A farmer with the following inputs and output:

Input (X): Nitrogen fertilizer Output 1 (Y1): Corn

Output 2 (Y2): Grain Sorghum

– Step 1: Define TPP curve for each output;– Step 2: Derive PPF by specifying the level of input

use.

Note: The relationship that exists between the two crops is considered competitive.

Deriving a Production Possabilities Frontier

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• Economic convention places the output that is being replaced on the vertical axis, and the output being added on the horizontal axis.

The distance from origin is an indication of the level of input being used.

PPF

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Marginal Rate of Product Substitution (MRPS): describes the rate at which one output must be decreased as production of the other product is increased, and, by definition, is the slope of the PPF;

Calculated by dividing the change in the product being replaced by the change in the product being added.

Marginal Rate of Product Substitution (MRPS)

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Isorevenue Line

Isorevenue line: is a line depicting all combinations of two products that will generate a given, or same, level of total revenue.

TR = (PY1* Y1) + (PY2* Y2)

– Example: Corn: PY1 = $2.50/bu.; Sorghum: PY2 = $5.00/bu.

Question: If TR = $150, how many units of each crop will be sold?

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– distance from the origin from the origin indicates the level of revenue. The closer to the origin, the lower the TR; the farther away from the origin, the higher the TR.

– slope of isorevenue line is - PY2/PY1, also referred to as

the Inverse Price Ratio (IPR). – end points of the isorevenue line show how many

units of each output would have to be sold if all the TR came from selling only that output.

Isorevenue Curve

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Price Changes

• A change in an output price causes the isorevenue line to rotate around the intersection of the isorevenue line and axis of the other output.

• Question: If price of one product increase, what will happen to isorevenue curve?

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Product-Product Decision

Two equivalent approaches can be used to determine the revenue maximizing combination of a given level of input use: numerical and graphical approaches.

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The manager equates the MRPS to IPR to determine the output combination that maximize revenue. MRPSY2Y1 = - PY2 /PY1

Numerical (mathematical) Approach

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The revenue maximizing combination of product is located at the point of tangency between the isorevenue line and the PPF, or the point where the slopes of the two curves are equal to each other.

Graphical Approach

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Product Price Increase:PY2

increases to PY2 , and PY1

remain constant

TR (slope - PY2 /PY1) shifts to TR (slope - PY2 /PY1)

Producers use more resources to produce Y2 and less to produce Y1 until the slope of TR equals to the slope of PPF.

Product Price Decrease: Product Price Decrease: PY2

decreases to PY2 , and PY1

remain constant

TR (slope - PY2 /PY1) shifts to TR (slope - PY2 /PY1)

Producers use less resources to produce Y2 and more to produce Y1 until the slope of TR equals to the slope of PPF.

Price Changes

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Competitive Relationship Complementary Relationship

– An Increase in one product brings about an increase in the level of production of the other.

Supplementary Relationship – The amount of one product can be increased without

increasing or decreasing the amount of the other product.

Joint Product – It occurs when the production of one product actually

results in the production of another.

Possible Product-Product Relationships

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Expansion path:Expansion path: the line connecting the combination of products that will maximize revenue for any given level of resources, at the price ratio the manager considers relevant and probable.

Expansion Path