Single Input Production Economics for Farm Management

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

2. Production EconomicsLearning 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 • Application of basic production economics to farm management • This will take a few weeks

3. Production • Definition: Using inputs to create goods and services having value to consumers or other producers • Production is what firms/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(s) that can be produced for the given input(s) • Generally two types: • Tabular Form (Production Schedule) • Mathematical Function

5. 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(s) • Single Input, Single Output • Milk = f(TDN) • Multiple Input, Single Output • Milk = f(TDN, Labor, Equipment) • Multiple Input, Multiple Output Implicit Function • F(Milk, Meat, TDN, Labor, Equipment) = 0

7. Examples • Polynomial: Linear, Quadratic, Cubic • Milk = b0 + b1TDN + b2TDN2 • Milk = -2261 + 2.535TDN – 0.000062TDN2 • Are many functions 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 methods 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 • AP: slope of line btwn origin and TP curve • Marginal Product (MP) = DQ/DX or derivative dQ/dX: output generated by the last unit of input used • MP: Slope of TP curve

10. Graphics Output Q • MP = 0 when Q at maximum, i.e. slope = 0 • 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

11. MP and AP: Tabular Form MP = DQ/DX = (Q2 – Q1)/(X2 – X1) AP = Q/X MP: 6 = (6 – 0)/(1 – 0) AP: 8.0 = 16/2 MP: 5 = (60 – 55)/(6 – 5) AP: 8.9 = 62/7

12. Same Data: Graphically

13. Think Break #1 • 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

14. Law of Diminishing Marginal Product • Diminishing MP: Holding all other inputs fixed, as use more and more of an input, eventually the MP will start decreasing, i.e., the returns to increasing the input eventually start decreasing • 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 decrease and X decreases means MP increase

15. Economics of Input UseHow Much Input to Use? Mathematically: 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

16. Economics of Input Use • Find X to Maximize 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 = the price of the input

17. 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 • 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

18. 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

19. 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

20. 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

21. Milk Cow Example Q X r 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

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

23. 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

24. Think Break #2 • 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?

25. Using MP = r/p Price Changes • Can use MP = r/p to find optimal X • Can also use MP = r/p to examine effect of price changes: what happens to profit maximizing X if output price and/or input price change? • Use MP = r/p and the Law of Diminishing MP • Output price p increases → r/p decreases • Input price r increases → r/p increases • X increases → MP decreases • X decreases → MP increases

26. Optimal X for Output Price Change • Output price p increases → r/p decreases • Need to change use of X so that the MP equals this new, lower, price ratio r/p • Law of Diminishing MP implies that to decrease MP, use more X • Intuition: p increase means output more valuable, so use more X to increase output • Everything reversed if p decreases

27. Optimal X for Input Price Change • Input price r increases → r/p increases • Need to change use of X so that the MP equals this new, higher, price ratio r/p • Law of Diminishing MP implies that to increase MP, use less X • Intuition: r increase means input more costly, so use less X • Everything reversed if r increases

28. Think Break #2 Example • Corn price = $2.00/bu • N price = $0.20/lb • Optimal N where VMP = r, or VMP = 0.20 • Alternative: MP = r/p, or MP = 0.2/2 = 0.1 • What if p = $2.25/bu and r = $0.30/lb, r/p = 0.133? • Relative price of N has increased, so reduce N, but where is it on the Table?

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

33. Calculus of Optimization • Problem: Choose X to Maximize f(X) • First Order Condition (FOC) • Set f’(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 f’’(X) at each X* identified • Condition for a maximum is f’’(X *) < 0 • Condition for a minimum is f’’(X *) > 0 • f’’(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, f(X) = – 5 + 6X – X2 • FOC: f’(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: f’’(X) = – 2 < 0 • Negative, satisfies SOC for a maximum

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

38. Example 2 • Maximize, wrt X, f(X) = 10 – 6X + X2 • FOC: f’(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: f’’(X) = 2 > 0 • Positive, does not satisfy SOC for maximum

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

40. Think Break #4 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 = pf(X) – rX – K • Suppose your production function is Q = f(X) = 30 + 5X – 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. SummarySingle 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? • Effect of price changes • Know how to use condition to find optimal input use • 1) with a production schedule (table) • 2) with a production function (calculus)