1. AAEC 3315 Agricultural Price Theory CHAPTER 6 Cost Relationships The Case of One Variable Input in the Short-Run

2. Objectives To gain understanding of:  Cost Relationships Fixed Costs, Variable Costs,& Total Costs Average and Marginal Costs  Cost Functions  Relationships between product and cost curves

3. Cost Relationships  A manager’s goal is to determine how much to produce to maximize profits.  We established earlier that Stage II is the rational stage of production, but realized that cost and revenue information are necessary to determine at which point in Stage II to produce.  Now, let’s introduce cost relationships into production.

4. Cost Definitions Costs of Production or Economic Costs: The payments that a firm must make to attract inputs and keep them from being used to produce other products. A firm’s cost functions show various relationships between its costs and output rate. Thus, the firm’s cost functions are determined by the firm’s production function and input prices. Since the production function can pertain to the short run or the long run, it follows that the cost functions can also pertain to the short run or the long run.

5. Cost Functions in the Short Run  Fixed Costs: Costs which do not vary with the level of production - These costs are associated with the fixed factors of production. Incurred regardless whether any output is produced  Variable Costs: Costs that vary as the output level changes - These costs are associated with variable factors of production.

6. Short-Run Cost Relationships The Case of One Variable Input  Costs Based on Total Output Total Fixed Costs (TFC): costs of inputs that are fixed in the SR & do not change as the output level changes. Total Variable Costs (TVC): costs of inputs that are variable in the Short Run, and change as output level changes, i.e., TVC = P X X Total Costs (TC): TFC + TVC

7. Total Cost Curves (Assume TFC = $80 and P x = $25 X Y TFC TFC

8. Total Cost Curves (Assume TFC = $80 and P x = $25 XY TFC TVC = XY TFC TVC = P X X TFC TVC

9. Total Cost Curves (Assume TFC = $80 and P x = $25 X Y TFC TVC TC = TFC+TVC X Y TFC TVC TC = TFC+TVC TVC TFC TC

10. Total Cost Curves TFC TVC TC

11. Total Cost Curve Functions  TFC = 100  TVC = 6Q – 0.4Q 2 + 0.02Q 3  TC = TFC + TVC = 100 + 6Q – 0.4Q 2 + 0.02Q 3

12. Short-Run Cost Relationships  Costs Based on Per Unit of Output Average Fixed Costs (AFC): Total fixed costs per unit of output, i.e., AFC = TFC / Q Average Variable Costs (ATC): Total variable cost per unit of output, i.e., AVC = TVC/Q Average Total Costs (ATC): Average total cost per unit of output, i.e., ATC = TC / Y = AFC + AVC Marginal Cost (MC): The increase in cost necessary to increase output by one more unit, i.e., MC = ∆TC/∆Q MC = (∆TVC + ∆ TFC) / ∆Q MC = ∆TVC / ∆Q MC = ∂TC/ ∂Q = ∂TVC/ ∂Q

13. Costs Based on Per-Unit Output  Average Fixed Costs (AFC): Average cost of fixed inputs per unit of output, i.e., AFC = TFC / Q YTFCAFC 080 10808.00 25803.20 50801.60 70801.14 85800.94 95800.84 100800.80 101800.79 95800.84 85800.94 AFC

14. Costs Based on Per-Unit Output Average Variable Costs (ATC): Total Variable cost per unit of output, i.e., AVC = TVC / Q YTVCAVC 00 10252.50 25502.00 50751.50 701001.43 851251.47 951501.58 1001751.75 1012001.98 952252.37 852502.94 AFC AVC

15. Costs Based on Per-Unit Output  Average Total Costs (ATC): Average total cost per unit of output, i.e., ATC = TC / Q = AFC + AVC YTCATC 080 1010510.50 251305.20 501553.10 701802.57 852052.41 952302.42 1002552.55 1012802.77 953053.21 853303.88 AFC AVC ATC

16. Costs Based on Per-Unit Output Marginal Cost (MC): The increase in cost necessary to increase output by one more unit, i.e., MC = ∆TC/∆Q= ∆TVC / ∆Q = ∂TC/ ∂Q = ∂TVC/ ∂Q YTCMC 080 101052.50 251301.67 501551.00 701801.25 852051.67 952302.50 1002555.00 10128025.00 95305-4.17 85330-2.50 ATC AFC AVC MC

17. Summary of Relationships Between Short-Run Cost Curves  AFC is a continuously decreasing function  AVC & ATC curves are U- shaped  The vertical distance between ATC & AVC at each output level is equal to AFC  MC crosses both AVC & ATC from below at their respective minimums  MC is not affected by fixed costs

18. Relationship Among Cost Curves... ATC AVC MC TC TVC TFC AFC Costs/unit Output Inflection Point

19. Changes in Input Price  Price of Variable Input Increases The cost of producing each output level increases  VC & TC shift upward & left; TFC remains unchanged  AVC, AC, & MC shift upward & left  Price of variable Input Decreases The cost of producing each output level decreases  TVC & TC shift downward & right; TFC remains unchanged  AVC, ATC, & MC shift downward & right

20. Working With Cost Functions  Given the total cost functions: TC = 100 + 6Q – 0.4Q 2 + 0.02Q 3, TFC = 100, TVC = 6Q – 0.4Q 2 + 0.02Q 3,  Average and Marginal costs functions can be derived. ATC = TC/Q = 100/Q + 6 – 0.4Q + 0.02Q 2, AFC = TFC/Q = 1000/Q, AVC = TVC/Q = 6 – 0.4Q + 0.02Q 2, and MC = = ∂TC/ ∂Q = ∂TVC/ ∂Q = 6 – 0.8Q + 0.06Q 2  With these given: Can you calculate the level of output at the minimum of AVC, and MC?. ATC AVC MC TC TVC Costs/unit Output Inflection Point AFC TFC

21. Working With Cost Functions  Given the cost functions: TC = 100 + 6Q – 0.4Q 2 + 0.02Q 3, TFC = 100, TVC = 6Q – 0.4Q 2 + 0.02Q 3, ATC = TC/Q = 100/Q + 6 – 0.4Q + 0.02Q 2, AFC = TFC/Q = 100/Q, AVC = TVC/Q = 6 – 0.4Q + 0.02Q 2, and MC = = ∂TC/ ∂Q = ∂TVC/ ∂Q = 6 – 0.8Q + 0.06Q 2  Level of output at the minimum of AVC: ∂ AVC/ ∂Q = -0.4 + 0.04Q = 0 Q = 10. ATC AVC MC TC TVC Costs/unit Output Inflection Point AFC TFC 10

22. Working With Cost Functions  Given the cost functions: TC = 100 + 6Q – 0.4Q 2 + 0.02Q 3, TFC = 100, TVC = 6Q – 0.4Q 2 + 0.02Q 3, ATC = TC/Q = 100/Q + 6 – 0.4Q + 0.02Q 2, AFC = TFC/Q = 100/Q, AVC = TVC/Q = 6 – 0.4Q + 0.02Q 2, and MC = = ∂TC/ ∂Q = ∂TVC/ ∂Q = 6 – 0.8Q + 0.06Q 2  Level of output at the minimum of MC: ∂ MC/ ∂Q = -0.8 + 0.12Q = 0 Q = 6.67. ATC AVC MC TC TVC Costs/unit Output Inflection Point AFC TFC 106.67

23. Relationships among Product Curves and Cost Curves  The cost curves are derived directly from the production process.  TPP & TVC, APP & AVC and MPP & MC are mirror images of each other Therefore, the production function can be transferred directly to the cost curves The three stages of a production function can be transferred directly to the cost curves

24. Relationship Between TPP and TVC A A*A* B B* The TVC is derived from the TPP: At “A” on TPP, 25 units of the output is being produced with 2 units of the input. The corresponding point “A * ” on the TVC shows that the variable cost of producing 25 units of output is $50 (P X :$25 * 2 units of input =$50). Note similar linkage between point “B” on TPP and point “B * ” on TVC. Similar relationships can be derived between AVC & APP and between MPP & MC. 25