EconS Cost Functions

Transcription

1 EconS Cost Functions Eric Dunaway Washington State University eric.dunaway@wsu.edu October 7, 2015 Eric Dunaway (WSU) EconS Lecture 17 October 7, / 41

2 Introduction When we previously talked about pro t functions, we didn t have an easy substitution to make for the costs that the rm faces. This is because the nature of costs vary greatly between di erent types of rms. Today, we will discuss di erent types of costs building up to next time where we will determine e cient production. Eric Dunaway (WSU) EconS Lecture 17 October 7, / 41

3 Costs In economics, as opposed to the rest of the world, we talk about economic costs. This is a combination of both explicit and implicit costs. Almost everyone else only considers explicit costs, which are the actual costs of purchasing something. For example, labor costs $10 per day and capital costs $5 per day. That s an explicit cost of $15 per day. Implicit costs include foregone bene ts that could be obtained by doing something else. We call them opportunity costs. Eric Dunaway (WSU) EconS Lecture 17 October 7, / 41

4 Opportunity Costs Let s elaborate a little bit more on opportunity costs. The opportunity costs is the "best" alternative that a person gives up in order to do their current action. For example, your opportunity cost for attending this class could be sleep, wages from working, or taking another (less interesting) class, among several other alternatives. Whichever of the three (or more) options would give you the highest bene t would serve as your opportunity cost. My opportunity cost for teaching this class is time that I could be spending on either research, family time, or playing video games (It varies day to day). Eric Dunaway (WSU) EconS Lecture 17 October 7, / 41

5 Opportunity Costs Example Here s a more concrete example: Jules has chosen to spend the rest of his life wandering the Earth. If he were not wandering, he could have three other opportunities. He could work at a prehistoric wildlife preserve, making sure no expenses are spared, earning a yearly salary of $50,000. He could become an FBI agent who specializes at escorting people on plane ights, earning a yearly salary of $40,000. He could operate a secret international organization that manages superheroes, earning a salary of $55,000 per year. What is Jules opportunity cost if he continues to wander the Earth? Eric Dunaway (WSU) EconS Lecture 17 October 7, / 41

6 Opportunity Costs Example Since Jules highest valued option is operating a secret international organization at $55,000 per year, that is his opportunity cost. It is his best alternative. Jules is choosing not to do this in order to wander the Earth. Eric Dunaway (WSU) EconS Lecture 17 October 7, / 41

7 Costs For simplicity, we lump implicit and explicit costs together in our cost functions. Why would we even di erentiate them then? This will make sense when we start talking about competitive markets later. We also break down costs as to how they behave. Let s go over some of those now. Eric Dunaway (WSU) EconS Lecture 17 October 7, / 41

8 Costs First, there are sunk costs. These are costs that once they are paid, cannot be recovered even if the item is not used. Think of a customized jacket that nobody else would want. You pay the money for the jacket and then you own it forever. Another example would be stocks. If you purchased a stock at $300 per share and now it s worth $200 per share, you pay a sunk cost of $100 per share. The key element for sunk costs is that regardless of whether the rm is operating, that cost has already been paid. This means that sunk costs do not play a role in the decision making process because once they re done, they re done. Eric Dunaway (WSU) EconS Lecture 17 October 7, / 41

9 Costs Next, we have xed costs. These costs are similar to sunk costs, but are only incurred if the rm decides to operate. They do not, however, vary depending on the level of output of the rm. A good example for this is capital investment. The capital investment is a xed cost because the rm can decide whether to use the capital or rent it to someone else (which is why it is not a sunk cost). By capital investment, think of a barn or a new combine for a farmer. Both of those are large capital investments. Eric Dunaway (WSU) EconS Lecture 17 October 7, / 41

10 Costs Lastly, we have variable costs. These costs depend on the level of output of the rm. Think of these as raw materials or labor. To get more output, the rm has to pay a some cost for each unit it produces. Eric Dunaway (WSU) EconS Lecture 17 October 7, / 41

11 Costs We can summarize the total cost function as Total Cost = Variable Costs + Fixed Costs TC = VC + F Eric Dunaway (WSU) EconS Lecture 17 October 7, / 41

12 Costs Example It costs a lawn mowing rm $1,000 to rent lawn mowers (capital) and $10 in labor for every lawn mowed. As a function of lawns mowed (output, q), what is the total cost function? Eric Dunaway (WSU) EconS Lecture 17 October 7, / 41

13 Costs Example The cost of capital, $1,000 is a xed cost. They either rent the capital or they do not. For every unit of output, q, it costs the rm $10 in labor. Putting these two factors together, we have TC = 10q where q is the number of lawns mowed. Eric Dunaway (WSU) EconS Lecture 17 October 7, / 41

14 Costs TC = 10q Just like with the production function, we are interested in both the marginal and average costs. To get the marginal cost, we simply apply the power rule to the total cost function. In our before example, the marginal cost would be MC = 10 this tells us that for one additional lawn mowed, the cost is 10. Eric Dunaway (WSU) EconS Lecture 17 October 7, / 41

15 Costs Also, similar to our production function last time, we are interested in average costs. To get them, we divide the cost function by the total level of output, q We like to break average costs into their individual variable and xed components. The process for nding them, however, is the same. Eric Dunaway (WSU) EconS Lecture 17 October 7, / 41

16 Costs TC = VC + F = 10q Starting with the average variable cost, we take the variable cost of our total cost function and divide it by the total output level, q, Average Variable Cost = AVC = VC q = 10 Quick note: Marginal and Average Variable Costs are not the same thing. This is a special case that only happens then the variable cost is linear. We follow the same process to derive the average xed costs. Using our example. Average Fixed Cost = AFC = F q = 1000 q Eric Dunaway (WSU) EconS Lecture 17 October 7, / 41

17 Costs Adding them together, we get the Average Cost function A few things to note: AC = AVC + AFC = q The average variable costs shows the level of cost per unit of output. This number will normally vary with q (except when it is linear, as in our example). This means that the cost per unit could either go up or go down as more output is produced. We will see later that initially, the average variable costs go down, then quickly turn upward. The average xed costs show how large the xed cost is relative to the output level. Intuitively, since the xed costs don t matter on the level of output, producing more should lessen their burden on the rm. Eric Dunaway (WSU) EconS Lecture 17 October 7, / 41

18 Short Run Costs All of these concepts can be used to explain short run costs. Remember that at lease one input (capital) is xed. This represents our xed cost. The rest of the inputs (labor) are not xed. These become our variable costs. Let s look at some typical gures. Eric Dunaway (WSU) EconS Lecture 17 October 7, / 41

19 Short Run Costs Costs F q Eric Dunaway (WSU) EconS Lecture 17 October 7, / 41

20 Short Run Costs Costs VC F q Eric Dunaway (WSU) EconS Lecture 17 October 7, / 41

21 Short Run Costs Costs TC VC F q Eric Dunaway (WSU) EconS Lecture 17 October 7, / 41

22 Short Run Costs As we can see, variable costs grow slowly at rst, then begin to accelerate rapidly. When the output level is low, the xed inputs are likely being underutilized, and large increases in output can be obtained for small increases in the variable input. When the output level is high, the xed inputs are likely being overutilized, giving small return on increasing the variable input. The gap between the total and variable costs is exactly the xed costs. Let s look at our measurements of change. Eric Dunaway (WSU) EconS Lecture 17 October 7, / 41

23 Short Run Costs Costs per unit AFC q Eric Dunaway (WSU) EconS Lecture 17 October 7, / 41

24 Short Run Costs Costs per unit AVC AFC q Eric Dunaway (WSU) EconS Lecture 17 October 7, / 41

25 Short Run Costs Costs per unit AC AVC AFC q Eric Dunaway (WSU) EconS Lecture 17 October 7, / 41

26 Short Run Costs Costs per unit MC AC AVC AFC q Eric Dunaway (WSU) EconS Lecture 17 October 7, / 41

27 Short Run Costs These curves re ect the changes we saw from the total curves. Note that the average variable and average total cost curves start farther apart and get closer as output increases. This is because of the diminishing nature of the average xed costs. Like before in production, the marginal cost curve will always intersect at the minimum of both the average variable and average cost curves. This is due to the "pulling" e ect I described last time. Eric Dunaway (WSU) EconS Lecture 17 October 7, / 41

28 Short Run Costs How did you get the shape of the variable cost curve? This is actually directly related to the short run production function we looked at last time. Eric Dunaway (WSU) EconS Lecture 17 October 7, / 41

29 Short Run Costs q L Eric Dunaway (WSU) EconS Lecture 17 October 7, / 41

30 Short Run Costs All that needs to be done is re ect the curve over the 45-degree line. Intuitively, we just switch the axes, making output the independent variable and labor (the variable input) the dependent variable. This should make sense, because with a production function, it was telling us how much output more labor will give us, and with a variable cost funtion, it tells us how much labor we need for more output. Eric Dunaway (WSU) EconS Lecture 17 October 7, / 41

31 Short Run Costs L VC q Eric Dunaway (WSU) EconS Lecture 17 October 7, / 41

32 Long Run Costs Now, let s look at long run cost functions. They re similar, but we have to take into account that all inputs are now variable. This implies that there are no xed costs in the long run. (F = 0). Our cost function becomes C = VC Eric Dunaway (WSU) EconS Lecture 17 October 7, / 41

33 Long Run Costs Recall from last time that the goal of rms is to maximize pro t. This can be achieved by minimizing the costs. The rms do this by choosing the optimal level of inputs to get their desired level of outputs. We were able to show which combinations led to each level of output with isoquant lines. Eric Dunaway (WSU) EconS Lecture 17 October 7, / 41

34 Long Run Costs Now, we introduce isocost lines. These lines pick a given level of total cost and show all the possible combinations of inputs that cost that amount. This de nition is a bit tough to grasp, so let s look at an example. A rm has $50 to spend on labor and capital. Labor costs $2 per unit and capital costs $10 per unit. The rm s isocost line would pass through the following points (among many) Labor Capital $ Spent on Labor $ Spent on Capital Eric Dunaway (WSU) EconS Lecture 17 October 7, / 41

35 Long Run Costs K L Eric Dunaway (WSU) EconS Lecture 17 October 7, / 41

36 Long Run Costs Isocosts look a lot like budget lines, and we will be using mathematical techniques similar to them. Behavior-wise, they re more like indi erence curves. Firms was to pick the lowest cost level possible to achieve a given level of production. As opposed to consumers wanting to pick the highest utility level given a level of income. If we let the price of labor be w (wages) and the price of capital be r (rent), we can express the equation for the isocost as C = wl + rk Eric Dunaway (WSU) EconS Lecture 17 October 7, / 41

37 Long Run Costs Like a budget line, the rate of change is just the slope of the line, which is the ratio of prices (we don t have a cool marginal term for this one). w r The solution of the rm s cost minimization problem will be on the smallest isocost line that is tangent to the isoquant. Which we will talk about next time. Eric Dunaway (WSU) EconS Lecture 17 October 7, / 41

38 Long Run Costs Let s look at our earlier example where labor cost $2 per unit (w = 2) and capital costs $10 per unit (r = 10) We can calculate the slope of the isocost line as just the negative ratio of the prices. w r = 2 10 = 1 5 This will be useful when we nd our optimal capital and labor levels next time. Eric Dunaway (WSU) EconS Lecture 17 October 7, / 41

39 Summary Costs can take on several di erent forms. We need to know how they work together to solve for the rm s cost minimization problem. Eric Dunaway (WSU) EconS Lecture 17 October 7, / 41

40 Preview for Friday Solving the Firm s optimization problem! Eric Dunaway (WSU) EconS Lecture 17 October 7, / 41

41 Assignment 4-2 (1 of 1) 1. A golf ball manufacturer faces a xed cost of F = 20 for operating and a variable cost of VC = 2q 2 where q is the quantity of golf balls it produces. a. Derive the total cost function. b. Derive the marginal and average cost functions. c. Are these short run or long run cost functions? Eric Dunaway (WSU) EconS Lecture 17 October 7, / 41