CDAE 254 - Class 18 Oct. 25

Last class: 5. Production functions

Today: 5. Production functions 6. Costs

Next class:6. CostsQuiz 5

Important date:Problem set 5: due Thursday, Nov. 1

Problem set 5-- Due at the beginning of class on Thursday, Nov. 1

-- Please use graph paper to draw graphs

-- Please staple all pages together before you turn them in

-- Scores on problem sets that do not meet the above requirements will be discounted.

Problems 5.1., 5.2., 5.4., 5.6. and 5.8.

5. Productions5. Productions

5.1. Production decisions

5.2. Production functions

5.3. Marginal physical product

5.4. Isoquant and isoquant map

5.5. Return to scale

5.6. Input substitution

5.7. Changes in technology

5.8. An example

5.9. Applications

5.4. Isoquant and isoquant map 5.4.4. Rate of technical substitution (RTS)

RTS = - (Change in K)/(Chang in L) - slope of the isoquant

Note that RTS is a positive number and this is similar to the marginal rate of substitution (MRS)

5.4.5. How to calculate & interpret RTS?

5.5. Returns to scale 5.5.1. Definition: The rate at which output

increases in response to proportional increases in all inputs

5.5.2. Graphical analysis (Fig. 5.3):

(1) Constant returns to scale

(2) Decreasing returns to scale

(3) Increasing returns to scale

5.6. Input substitution 5.6.1. General situations (Fig. 5.2.)

5.6.2. Fixed-proportions (Fig. 5.4.)

5.6.3. Perfect-substitution

5.7. Changes in technology 5.7.1. A graphical analysis with a handout

(1) The curve labeled by q0 = 100

represents the isoquant of the old

technology:

100 units of the output can be produced by different combinations of L and K.

e.g., Point B: L= 20 and K= 20

Point E: L= 10 and K= 40

Point F: L= 30 and K= 14

5.7. Changes in technology 5.7.1. A graphical analysis with a handout (2) The curve labeled by q0* = 100

represents the isoquant of the new

technology:

100 units of the output can be produced by different combinations of L and K.

e.g., Point A: L= 15 and K= 14

Point C: L= 20 and K= 9

Point D: L= 10 and K= 20

5.7. Changes in technology 5.7.1. A graphical analysis with a handout

(3) Comparison of the two technologies in producing 100 units of the output:

From B to A:

From B to D:

From B to C:

From E to D:

From F to A:

5.7. Changes in technology 5.7.2. Technical progress vs. input

substitution (1) Input substitution (move along q0 = 100)

e.g., from Point B to Point E:

L reduced from ( ) to ( )

K increased from ( ) to ( )

APL increased from ( ) to ( )

APK reduced from ( ) to ( )

5.7. Changes in technology 5.7.2. Technical progress vs. input substitution

(2) Technical progress (move from q0 = 100 to q0* = 100)

e.g., from Point B to Point D:

L reduced from ( ) to ( )

K has no change

APL increased from ( ) to ( )

APK has no change

5.8. An example 5.8.1. Production function:

where q = hamburgers per hour

L = number of workers

K = the number of grills

5.8.2. What is the returns to scale of this function?

When L = 1 and K = 1, q =

when L = 2 and K = 2, q =

when L = 3 and K = 3, q =

LKq 10

5.8. An example 5.8.3. How to construct (graph) an isoquant?

-- For example q = 40

-- Simplify this function:

4010 LKq

4010 LK

4LK

16LK

5.8. An example 5.8.4. How to construct (graph) an isoquant?

-- Calculate K for each value of L (Table 5.3):

when L=1, K= ( )

when L=2, K= ( )

……

when L=10, K= ( )

-- Draw the isoquant of q=40

16LK LK 16

5.8. An example 5.8.5. Technical progress

-- A new production function:

-- Construct the new isoquant of q=40

when L=1, K= ( )

when L=2, K= ( )

when L=3, K= ( ) ……

-- Draw the new isoquant of q=40

LKq 20

LK2040 LK4

Take-home exercise(Thursday, Oct. 25)

For production function Q = 40 (LK)0.5, draw the isoquant for Q=50

6. Costs6. Costs

6.1. Basic concepts of costs

6.2. Cost minimizing input choice

6.3. Cost curves

6.4. Short-run and long-run costs

6.5. Per unit short-run cost curves

6.6. Shifts in cost curves

6.7. An example

6.8. Applications

6.1. Basic concepts of costs6.1.1. Opportunity cost, accounting cost, and economic

cost:

-- Opportunity cost: the cost of a good or

service as measured by the alternative

uses of the resources that are foregone by producing the good or service.

e.g., one acre of land, 10 hours of labor

and $20 of capital can be used to produce 800 lb. of hay OR 60 bu. of soybeans. What is the opportunity cost of producing 60 units of soybeans?

6.1. Basic concepts of costs6.1.1. Opportunity cost, accounting cost, and economic cost:

-- Accounting cost: the cost of a good or service as measured by what was

paid for it (i.e., out-of-pocket expenses, historical costs of machines and depreciation related to them, and other bookkeeping entries).

e.g., accounting cost of producing 60 bu. of soybeans:

6.1. Basic concepts of costs6.1.1. Opportunity cost, accounting cost, and

economic cost:

-- Economic cost: the payment required to keep a resource in its present use or the amount that the resource would be worth in its next best alternative use.

e.g., Mr. Smith is making 60k a year with IBM. If the next best offer in this region is also 59K, what is the economic cost?

6.1. Basic concepts of costs6.1.2. Labor costs, capital costs, and

entrepreneurial costs:

Labor costs = w L

Capital costs = v K

Entrepreneurial cost e.g., Phil has a flexible job with a wage rate of $10 per hour.

He also has his own roofing business and has just completed a project:

Revenue: $3000 Materials: $1100

Hired labor: $500 His labor: 50 hrs

Accounting profit = Economic profit =

Entrepreneurial cost =

6.1. Basic concepts of costs6.1.3. Two simplifying assumptions:

(1) All the inputs are aggregated into labor and capital inputs (L and

K)

(2) The inputs are hired in perfectly

competitive markets

6.1. Basic concepts of costs6.1.4. Costs and profits Total economic costs = TC = wL + vK

Total revenues = TR = Pq = P f (L, K)

Total economic profits

= = TR – TC

= Pq – wL – vK

= P f(L, K) – wL – vK

where q = f (L, K) is the production function

6.2. Cost-minimizing input choice 6.2.1. A graphical analysis (Fig. 6.1)

6.2.2. What is the condition for the best point? Slope of the cost line = slope of the isoquant

-w/v = -RTS w/v = RTS

What will happen if the two are not equal? e.g., if w/v = 0.5 and RTS = 0.8, the producer

will increase ? and decrease ? to minimize cost.

Note that this is similar to the analysis of utility maximization in Chapter 3.

Class exercise (Thursday, Oct. 25)

If the cost is TC = 4L + 5K and the rate of

technical substitution (RTS) is equal to 1.2 at the current production point, what will be the directions of change in L and K to minimize the cost? Why?