M. Muniagurria Econ 464 Handout (Part 1)

I. : Function, Marginal of Inputs, Isoquants

(1) Case of One Input: L (Labor): q = f (L) • Let q equal so the relates L to q. (How much output can be produced with a given amount of labor?)

• Marginal productivity of labor = MPL is defined as

q = Slope of prod. Function L Small changes i.e. The change in output if we change the amount of labor used by a very small amount.

• How to find total output (q) if we only have information about the MPL: “In general” q is equal to the area under the MPL curve when there is only one input.

Examples: (a) Linear production functions. Possible forms:

q = 10 L| MPL = 10 q = ½ L| MPL = ½ q = 4 L| MPL = 4

The production function q = 4L is graphed below.

-1- Notice that if we only have diagram 2, we can calculate output for different amounts of labor as the area under MPL:

If L = 2 | q = Area below MPL for L Less or equal to 2 = = in Diagram 2 8

Remark: In all the examples in (a) MPL is constant.

(b) Production Functions With Decreasing MPL.

Remark: Often this is thought as the case of one variable input (Labor = L) and a fixed factor (land or entrepreneurial ability)

(2) Case of Two Variable Inputs: q = f (L, K)

L (Labor), K ()

• Production function relates L & K to q (total output)

• Isoquant: Combinations of L & K that can achieve the same q

-2- • Marginal Productivities

)q MPL ' Small changes )L K constant

)q MPK ' Small changes )K L constant

)K • MRTS = - Slope of Isoquant = Absolute of Along Isoquant )L

Examples

(a) Linear (L & K are perfect substitutes) Possible forms:

q = 10 L + 5 K Y MPL = 10 MPK = 5 q = L + K Y MPL = 1 MPK = 1 q = 2L + K Y MPL = 2 MPK = 1

• The production function q = 2 L + K is graphed below.

MRTS = 2

-3- Marginal Productivities are constant in all these examples, then MRTS = constant

(b) Fixed coefficients (i.e. no factor substitution) Possible Forms

• Example (i)

Unit Labor requirement = aL = 2 Unit Capital requirement = aK = 1

aL= Amount of L needed to produce one unit of output

aK = Amount of K needed to produce one unit of output

-4- q = L

Average Productivity of Labor =

Average Productivity of Capital =

Example (ii)

Average Productivity of Labor = APL = q/L = 1

Average Productivity of Capital = APK = q/K = ½

Notice that the APL = 1/aL and APK = 1/aK

MRTS not easy to calculate.

-5- (c) “Nice Isoquants”

Possible Forms:

Slope of Isoquants in absolute value 9 as L8 ( i.e. 9MRTS as L8, as L8it becomes more and more costly to replace capital.)

II. MINIMIZATION

How to choose inputs to minimize cost:

(1) One input: Trivial: of labor Total Cost = w. L , where w is the price of labor

(2) Two variable inputs (a) Fixed coefficients: No substitution (still easy) Let r be the price of capital, then the cost of producing one unit of output (Average cost: AC) is:

' % AC aL . w aK . r

Notice that this does not change for different levels of output:

The AC is constant.

-6- If either w or r change, the AC changes but the firm still uses aL units of L and aK units of K to produce one unit of output.

(b) Linear Isoquants (perfect substitutes) (Hard - we will skip this)

(c) “Nice Isoquants” - Define Isocost lines - Find tangency between Isoquant and Isocost lines

Isocost = c = w.L + r.k (Combinations of L & K that cost same amount of money)

Example: w = 5 , r = 2 Find the Isocost for $10

5 Slope of Isocost ' 2

In General: cw w Slope of Isocost ' r

Example: Cost minimizing combination of K & L to produce q = 10

w slope ' r

-7- Effect of changes in :

•8w, r constant (i.e. 8w/r) Y 9 L* &8 K*

New Combination: K$,L$

K$

•8 r w constant (i.e. 9 w/r) Y 8 L* & 9 K*

III. PROFIT MAXIMIZING FACTOR DEMANDS

We assume that each firm maximizes profits - taking the product price as given (P) and - taking input prices as given (w, r)

How many units of L & K should the firm hire?

(1) Case of One Variable Input: Labor (L) A profit maximizing firm will hire L up to the point at which: Marginal product x output price = input price In our case: MPL . p = w Marginal Marginal cost benefit for of hiring an extra the firm unit of labor Y MPL = w/p

-8- Example: Production function with decreasing MPL (Think about a fixed factor).

(2) Case of Two Variable Inputs: L & K “Nice Isoquants”

IV. OF INCOME AMONG PRODUCTIVE FACTORS (FIXED & VARIABLES)

(1) Case of one variable input with decreasing marginal productivity due to the existence of a fixed factor.

Example: Think of the fixed factor as land or entrepreneurial ability. Labor is the variable factor.

L* is the profit maximizing amount of labor. Remember: Area below MPL to the left of L* is total output produced with L* (Let’s call it q*)

-9- Total Revenue p . q* = w . L* + Income to the Fixed Factor 99 Labor Residual Income

OR

q* = w/p . L* + Income to Fixed Factor/P 99 Real Labor Real Income of Income The Fixed Factor

(yL) (yF) Then yF = q* - yL

Notice in Diagram that:

yF =

•What happens if real wage (w/p) goes up?

Y 9 Profit max. amount of labor AND The surplus that the fixed factor goes down (i.e. yF 9 )

-10- • What happens if p8?

Y w/p 9Y More labor is hired Y Surplus to the fixed factor 8 i.e. yF 8

“The owners of the fixed factor, say land, benefit when the price of corn 8”.

464part1.wpd

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