1. Single Input Production Economics for Farm Management
AAE 320
Paul D. Mitchell

2. Production Economics Learning Goals
Single and Multiple Input Production Functions
What are they and how to use them in production economics and farm management
Economics to identify optimal input use and output combinations
How much nitrogen fertilizer do I use for my corn?
How much corn will I get if I use this much nitrogen?
Application of basic production economics to farm management

3. Production
Definition: Using inputs to create goods and services having value to consumers or other producers
Production is what farms do!
Using land, labor, time, chemicals, animals, etc. to grow crops, livestock, milk, eggs, etc.
Can further process outputs: flour, cheese, ham
Can produce services: dude ranch, bed and breakfast, orchard/pumpkin farm/hay rides, etc. selling the “fall country experience”

4. Production Function
Production Function: gives the maximum amount of output that can be produced for the given input(s)
Generally two types:
Tabular Form (Production Schedule)
Mathematical Function

5. TDN (1000 lbs/yr)
Milk (lbs/yr) 1
800 2
1,700 3
3,000 4
5,000 5
7,500 6
10,200 7
12,800 8
15,100 9
17,100 10
18,400 11
19,200 12
19,500 13
19,600 14
19,400
Tabular Form
A table listing the maximum output for each given input level
TDN = total digestible nutrition (feed)

6. Production Function
Mathematically express the relationship between input(s) and output
Single Input, Single Output
Milk = f(TDN)
Milk = TDN – 0.2TDN2
Multiple Input, Single Output
Milk = f(Corn, Soy)
Milk = Corn – 0.2Corn2 + 2Soy – 0.1Soy CornSoy

7. Examples Polynomial: Linear, Quadratic, Cubic
Milk = b0 + b1TDN + b2TDN2
Milk = – TDN – TDN2
Many functions are used, depending on the process: Cobb-Douglas, von Liebig (plateau), Exponential, Hyperbolic, etc.

8. Why Production Functions?
More convenient & easier to use than tables
Estimate via regression with the tables of data from experiments
Increased understanding of production process: identify important factors and how important factor each is
Allows use of calculus for optimization
Common activity of agricultural research scientists

9. Definitions Input: X , Output: Q Total Product = Output Q
Average Product (AP) = Q/X: average output for each unit of the input used
Example: you harvest 200 bu/ac corn and applied 100 lbs of nitrogen
AP = 200/100 = 2, means on average, you got 2 bu of corn per pound of nitrogen
Graphics: slope of line between origin and the total product curve

10. Definitions Marginal Product (MP) = DQ/DX or derivative dQ/dX:
Output Q generated by the last unit of input used
Example: corn yield increases from 199 to 200 bu/ac when you increase nitrogen applied from 99 lbs to 100 lbs
MP = 1/1 = 1, meaning you got 1 bu of corn from last 1 pound of nitrogen
MP: Slope of total product curve

11. Graphics Output Q MP = 0 when Q at maximum, i.e. slope = 0 Q
AP = MP when AP at maximum, at Q where line btwn origin and Q curve tangent
MP > AP when AP increasing
AP > MP when AP decreasing Q
Input X
MP AP AP
Input X MP

12. MP and AP: Tabular Form MP = DQ/DX = (Q2 – Q1)/(X2 – X1) AP = Q/X
Input TP MP AP 1 6
6.0 2 16 10
8.0 3 29 13
9.7 4 44 15
11.0 5 55 11 60
10.0 7 62
8.9 8
7.8 9 61 -1
6.8 59 -2
5.9
MP: 6 = (6 – 0)/(1 – 0)
AP: 8.0 = 16/2
MP: 5 = (60 – 55)/(6 – 5)
AP: 8.9 = 62/7

13. Same Data: Graphically

14. Think Break #2 N
Yield AP MP 30
--- 25 45
1.8
0.6 50 75
1.2
105
1.4
100
135
1.35
125
150
165
1.1
200
170
0.85
0.1
250
160
0.64
-0.2
Fill in the missing numbers in the table for Nitrogen and Corn Yield
Remember the Formulas
MP = DQ/DX
= (Q2 – Q1)/(X2 – X1)
AP = Q/X

15. Law of Diminishing Marginal Product
Diminishing MP: Holding all other inputs fixed, as use more and more of one input, eventually the MP will start decreasing, i.e., the returns to increasing that input eventually start to decrease
For example, as make more and more feed available for a cow, the extra milk produced eventually starts to decrease
Main Point: X increase means MP decreases and X decreases means MP increases

16. Law of Diminishing Marginal Product
Happens all the time in biological, physical and social systems: eventually the marginal product (MP) will start decreasing
Farm example: Acres farmed is your input, total corn produced your output: as add more and more acres, eventually your MP per acre (per acre yield) will decrease: less time to manage, more travel time to get to fields, poorer land available, …

17. Transition
We spent time explaining production functions Q = f(X) and their slope = MP, and AP = Q/X
Now we can ask:
How do I decide how much input to use?
How much nitrogen should I use for my corn?
How many seeds should I plant per acre?
Choose each input to maximize farmer profit
Set it up as an economic problem

18. Intuition
MP is the extra output generated when increasing X by one unit
If you add one more pound of N, how much more corn do you get? The MP
How much is this extra corn worth?
Output Price x MP = value of the marginal product or the VMP
What does this last pound of N cost?
Input Price x Extra N added = Input Price x 1

19. Economics of Input Use How Much Input to Use?
Mathematical Model: Profit = Revenue – Cost
Profit = price x output – input cost – fixed cost
p = pQ – rX – K = pf(X) – rX – K
p = profit Q = output X = input
p = output price r = input price
f(X) = production function K = fixed cost
Learn this model, we will use it a lot!!!

20. Economics of Input Use Find X to Maximize profit p = pf(X) – rX
Calculus: Set first derivative of p with respect to X equal to 0 and solve for X, the “First Order Condition” (FOC)
FOC: pf’(X) – r = 0 p x MP – r = 0
Rearrange: pf’(X) = r p x MP = r
p x MP is the “Value of the Marginal Product” (VMP), what would get if sold the MP
FOC: Increase use of input X until p x MP = r, i.e., until VMP = r, the price of the input

21. Intuition
Remember, MP is the extra output generated when increasing X by one unit
The value of this MP is the output price p times the MP, or the extra income you get when increasing X by one unit: VMP = p x MP
The rule, keep increasing use of the input X until VMP equals the input price (p x MP = r), means keep using X until the income the last bit of input generates just equals the cost of buying the last bit of input

22. Another Way to Look at Input Use
Have derived the profit maximizing condition defining optimal input use as:
p x MP = r or VMP = r
Rearrange this condition to get an alternative: MP = r/p
Keep increasing use of the input X until its MP equals the price ratio r/p
Both give the same answer!
Price ratio version useful to understand effect of price changes

23. MP=r/p: What is r/p?
r/p is the “Relative Price” of input X, how much X is worth in the market relative to Q
r is $ per unit of X, p is $ per unit of Q
Ratio r/p is units of Q per one unit of X
r/p is how much Q the market place would give you if you traded in one unit of X
r/p is the cost of X if you were buying X in the market using Q in trade

24. MP = r/p Example: N fertilizer
r = $/lb of N, p = $/bu of corn, so
r/p = ($/lb)/($/bu) = bu/lb, or the bushels of corn received if “traded in” one pound of N
MP = bushels of corn generated by the last pound of N
Condition MP = r/p means: Find N rate that gives the same conversion between N and corn in the production process as in the market, or find N rate to set the
Marginal Benefit of N = Marginal Cost of N

25. Milk Cow Example Milk Price = $12/cwt or p = $0.12/lbQ r
TDN
Milk MP
VMP
price TDN
profit $0
$150
-$400 1
800
$96
-$454 2
1,700
900
$108
-$496 3
3,000
1300
$156
-$490 4
5,000
2000
$240 5
7,500
2500
$300
-$250 6
10,200
2700
$324
-$76 7
12,800
2600
$312
$86 8
15,100
2300
$276
$212 9
17,100
$302 10
18,400
$308 11
19,200
$254 12
19,500
300
$36
$140 13
19,600
100
$12 $2 14
19,400
-200
-$24
-$172
Milk Price = $12/cwt
or p = $0.12/lb
TDN Price = $150 per 1,000 lbs
Fixed Cost = $400/yr
Price Ratio r/p = $150/$0.12 = 1,250
VMP = r
Optimal TDN = 10+
MP = r/p
Maximum Production

26. Profit Max is where MP = r/p
Output max is where MP = 0
Profit Max is where MP = r/p
r/p Q
TDN MP
TDN

27. Milk Cow Example: Key Points
Profit maximizing TDN is less than output maximizing TDN, which implies profit maximization ≠ output maximization
Profit maximizing TDN occurs at TDN levels where MP is decreasing, meaning will use TDN so have a diminishing MP
Profit maximizing TDN depends on both the TDN price and the milk price
Profit maximizing TDN same whether use VMP = r or MP = r/p

28. Think Break #3
N lbs/A
Yield
bu/A MP
VMP 30
--- 25 45
0.6 50 75
1.2
105
100
135
125
150
165
200
170
0.1
250
160
-0.1
Fill in the VMP column in the table using $2/bu for the corn price.
What is the profit maximizing N fertilizer rate if the N fertilizer price is $0.2/lb?

29. Why We Need Calculus
What do you do if the relative price ratio of the input is not on the table? What do you do if the VMP is not on the table?
If you have the production function Q = f(X), then you can use calculus to derive an equation for the MP = f’(X)
With an equation for MP, you can “fill in the gaps” in the tabular form of the production schedule

30. Calculus and AAE 320 I will keep the calculus simple!!!
Production Functions will be Quadratic Equations: Q = b0 + b1X + b2X2
First derivative = slope of production function = Marginal Product
3 different notations for derivatives
dy/dx (Newton), f′(x) and fx(x) (Leibniz)
2nd derivatives: d2y/dx2, f′’(x), fxx (x)

31. Quick Review of Derivatives
Constant Function
If Q = f(X) = K, then f’(X) = 0
Q = f(X) = 7, then f’(X) = 0
Power Function
If Q = f(X) = aXb, then f’(X) = abXb-1
Q = f(X) = 7X = 7X1, then f’(X) = 7(1)X1-1 = 7
Q = f(X) = 3X2, then f’(X) = 3(2)X2-1 = 6X
Sum of Functions
Q = f(X) + g(X), then dQ/dX = f’(X) + g’(X)
Q = 3 + 5X – 0.1X2, dQ/dX = 5 – 0.2X

32. Think Break #4
What are the 1st and 2nd derivatives with respect to X of the following functions?
Q = X – 7X2
p = 2(5 – X – 3X2) – 8X – 15
p = p(b0 + b1X + b2X2) – rX – K

33. Calculus of Optimization
Problem: Choose X to Maximize some function g(X)
First Order Condition (FOC)
Set g’(X) = 0 and solve for X
May be more than one
Call these potential solutions X*
Identifying X values where the slope of the objective function is zero, which occurs at maximums and minimums

34. Calculus of Optimization
Second Order Condition (SOC)
Evaluate g’’(X) at each X* identified
Condition for a maximum is g’’(X*) < 0
Condition for a minimum is g’’(X*) > 0
g’’(X) is function's curvature at X
Positive curvature is convex (minimum)
Negative curvature is concave (maximum)

35. Calculus of Optimization: Intuition
FOC: finding the X values where the objective function's slope is zero, candidates for minimum/maximum
SOC: checks the curvature at each candidates identified by FOC
Maximum is curved down (negative)
Minimum is curved up (positive)

36. Example 1 Maximize, wrt X, g(X) = – 5 + 6X – X2
FOC: g’(X) = 6 – 2X = 0
FOC satisfied when X = 3
Is this a maximum or a minimum or an inflection point? How do you know?
Check the SOC: g’’(X) = – 2 < 0
Negative, satisfies SOC for a maximum

37. Example 1: Graphics
Slope = 0
g’(X) = 0

38. Example 2 Maximize, wrt X, g(X) = 10 – 6X + X2
FOC: g’(X) = – 6 + 2X = 0
FOC satisfied when X = 3
Is this a maximum or a minimum or an inflection point? How do you know?
Check the SOC: g’’(X) = 2 > 0
Positive, does not satisfy SOC for maximum

39. Example 2: Graphics What value of X maximizes this function? Slope = 0
g’(X) = 0

40. Think Break #5 Find X to Maximize: p = 10(30 + 5X – 0.4X2) – 2X – 18
1) What X satisfies the FOC?
2) Does this X satisfy the SOC for a maximum?

41. Calculus and Production Economics
In general, p(X) = pf(X) – rX – K
Suppose your production function is
Q = f(X) = X – 0.4X2
Suppose output price is 10, input price is 2, and fixed cost is 18, then
p = 10(30 + 5X – 0.4X2) – 2X – 18
To find X to maximize p, solve the FOC and check the SOC

42. Calculus and Production Economics
p = 10(30 + 5X – 0.4X2) – 2X – 18
FOC: 10(5 – 0.8X) – 2 = 0
10(5 – 0.8X) = 2
p x MP = r
5 – 0.8X = 2/10
MP = r/p
When you solve the FOC, you set VMP = r and/or MP = r/p

43. Summary Single Input Production Function
Condition to find optimal input use:
VMP = r or MP = r/p
What does this condition mean?
What does it look like graphically?
Know how to use condition to find optimal input use
1) with a production schedule (table)
2) with a production function (calculus)