EconS Cost Structures

Transcription

1 EconS Cost Structures Eric Dunaway Washington State University eric.dunaway@wsu.edu Industrial Organization Eric Dunaway (WSU) EconS 425 Industrial Organization 1 / 34

2 Introduction Today, we ll review some basic cost structures faced by rms. We ll also look at how costs determine entry into a market. Eric Dunaway (WSU) EconS 425 Industrial Organization 2 / 34

3 Technology Technology is a broad term employed in economics. In general, it pertains to how inputs are turned into outputs. For our purposes, technology is the production function used by a rm. If we had some output q which was created using k di erent types of inputs, we would have a production function f such that where z i represents an input. q = f (z 1, z 2,..., z k ) Eric Dunaway (WSU) EconS 425 Industrial Organization 3 / 34

4 Technology The most common type of production function is known as Cobb-Douglas. In its most basic form, it uses two inputs, capital and labor. Let z 1 represent capital while z 2 represents labor. The Cobb-Douglas production function is q = z α 1 z β 2 where α and β are positive constants. This production function is fairly accurate in the real world and makes a pretty good approximation. The relationship between α and β determine the returns to scale of the production function. Eric Dunaway (WSU) EconS 425 Industrial Organization 4 / 34

5 Returns to Scale Returns to scale occur when a proportional increase in the inputs leads to a proportional increase in the outputs. While the proportion of the inputs have to be the same, the output proportion can be di erent. For example, if I doubled all of the inputs I used and saw that my output tripled, I would say that the production function has increasing returns to scale. This is because the output increased more than the inputs. Eric Dunaway (WSU) EconS 425 Industrial Organization 5 / 34

6 Returns to Scale To calculate returns to scale, multiply all of the inputs by a common factor, t. f (tz 1, tz 2,..., tz k ) Using algebra, try to factor out the multiple. If you can obtain an expression where you have t raised to some exponent times the original production function, you have returns to scale, where r determines the type. t r f (z 1, z 2,..., z k ) If r > 1, the function has increasing returns to scale. If r = 1, the function has constant returns to scale, and if r < 1, the function has decreasing returns to scale. Eric Dunaway (WSU) EconS 425 Industrial Organization 6 / 34

7 Returns to Scale Returning to our Cobb-Douglas example, let s multiply both inputs by t, (tz 1 ) α (tz 2 ) β Next, we can distribute the exponents through the parenthesis to obtain t α z α 1 t β z β 2 and rearranging, t α+β z α 1 z β 2 Since we were able to reassemble the original production function, we have returns to scale (a property of Cobb-Douglass) and r = α + β in this case. Eric Dunaway (WSU) EconS 425 Industrial Organization 7 / 34

8 Technology We want to use the production function to derive a total cost function as a function of the output level, q. To get there, we need to make use of the cost minimization problem. Remember from consumer theory that there was both the utility maximization problem and the expenditure minimization problem. They represented opposite sides of the consumer choice problem, and each gave way to di erent perspectives of the consumer s optimal choice. The cost minimization problem is the duality to the pro t maximization problem. Both will give the same solution, they just reveal di erent parts of the problem. Eric Dunaway (WSU) EconS 425 Industrial Organization 8 / 34

9 Technology The total cost function is simply each input times its respective price, TC = w 1 z 1 + w 2 z 2 + F where in this case, w 1 is the price of capital (rent), w 2 is the price of labor (wage), and F is some non-negative xed cost. While this is useful, we want to know how the total cost changes with output level, q. Eric Dunaway (WSU) EconS 425 Industrial Organization 9 / 34

10 Cost Minimization Solution Strategy: Set up Lagrangian and calculate rst-order conditions. Use the rst-order conditions to get rid of λ, the lagrange multiplier. Solve this new expression for either z 1 or z 2. Substitute this into the third rst-order condition (the constraint) and solve for the remaining input (z 1 or z 2 ). From here, you can work backwards to solve for the other input level. Your expressions for z 1 and z 2 are the input demands for capital and labor. Lastly, you can substitute these input demands into the total cost function to get the cost as a function of output level. Eric Dunaway (WSU) EconS 425 Industrial Organization 10 / 34

11 Cost Minimization Setting up the Lagrangian, TC = w 1 z 1 + w 2 z 2 + F q = z α 1 z β 2 L = w 1 z 1 + w 2 z 2 + F + λ And calculating rst-order conditions, q z1 α z β 2 L = w 1 λαz1 α 1 z β 2 z = 0 1 L = w 2 λβz1 α z β 1 2 = 0 z 2 L λ = q z 1 α z β 2 = 0 Eric Dunaway (WSU) EconS 425 Industrial Organization 11 / 34

12 Cost Minimization L = w 1 λαz1 α 1 z β 2 z = 0 1 L = w 2 λβz1 α z β 1 2 = 0 z 2 Using the rst two rst-order conditions, we can solve them both for λ to obtain λ = w 1 w 2 αz1 α 1 z β = 2 βz1 αz β 1 2 Using algebra, we can solve this new expression for z 1, z 1 = αw 2 βw 1 z 2 Eric Dunaway (WSU) EconS 425 Industrial Organization 12 / 34

13 Cost Minimization z 1 = αw 2 βw 1 z 2 L λ = q z α 1 z β 2 = 0 Now, we substitute this expression into the third rst-order condition, α αw2 q z 2 z β 2 βw = 0 1 Rearranging terms, α αw2 q = z α+β 2 βw 1 solving for z 2, we have the input demand for labor, z2 = q α+β 1 βw1 αw 2 α α+β Eric Dunaway (WSU) EconS 425 Industrial Organization 13 / 34

14 Cost Minimization z 1 = αw 2 βw 1 z 2 z2 = q α+β 1 βw1 αw 2 α α+β Now, we can solve for the input demand for capital, z 1, z 1 = αw 2 βw 1 z 2 = αw 2 βw 1 q 1 = q α+β 1 βw1 αw 2 = q α+β 1 αw2 βw 1 α α+β 1 β α+β βw1 α+β αw 2 α α+β Eric Dunaway (WSU) EconS 425 Industrial Organization 14 / 34

15 Cost Minimization z1 = q α+β 1 αw2 βw 1 z2 = q α+β 1 βw1 αw 2 β α+β α α+β All that s left to do now is substitute these back into the total cost function, to get total costs as a function of output level. TC = w 1 z1 + w 2 z2 + F = β w 1 q α+β 1 αw2 α+β + w2 q 1 βw1 α+β βw 1 αw 2 = (α + β)q α+β 1 w1 α α+β w2 α β β α+β + F α α+β + F Eric Dunaway (WSU) EconS 425 Industrial Organization 15 / 34

16 Cost Minimization TC q Eric Dunaway (WSU) EconS 425 Industrial Organization 16 / 34

17 Total Cost Function We ve encountered the total cost function before. This is how it is derived. In general, I won t have you derive these complex forms. We ll stick to mathematically simple examples like what you have seen in homework thus far. As you have encountered before, there are several useful calculations that can be derived from the total cost function. Eric Dunaway (WSU) EconS 425 Industrial Organization 17 / 34

18 Total Cost Function Consider the total cost function TC = q + q 2 We can break this total cost function up into xed costs and variable costs. The xed cost is any term that isn t dependent on quantity, i.e., 50 in this case. FC = 50 Everything else is a variable cost. VC = 10q + q 2 Eric Dunaway (WSU) EconS 425 Industrial Organization 18 / 34

19 Total Cost Function TC = FC + VC = q + q 2 Also, averages of both of these functions are useful in our analysis. We obtain all of them simply by dividing by quantity, q. AC = 50 q q AFC = 50 q AVC = 10 + q Lastly, we have marginal cost. This is obtained by di erentiating the total cost function by quantity. MC = dtc dq = q Eric Dunaway (WSU) EconS 425 Industrial Organization 19 / 34

20 Total Cost Function The Average Fixed Cost is always decreasing in quantity. This should be quite intuitive. In the short run, a rm makes a one time payment for their xed cost (e.g., rent for their building). After that, it doesn t matter how much the rm produces, the xed cost will never go up. Producing a higher quantity e ectively spreads the xed cost over the quantity produced. Eric Dunaway (WSU) EconS 425 Industrial Organization 20 / 34

21 Total Cost Function C/q AFC q Eric Dunaway (WSU) EconS 425 Industrial Organization 21 / 34

22 Total Cost Function The Average Variable Cost shows how much is spent per unit produced, less the xed costs. Primarily, we use it to determine whether a rm should operate. If price is below Average Variable Cost, the rm should shut down in the short run. Average Variable Cost usually decreases at rst, due to a rm better utilizing its xed inputs (capital). Later, it will increase, as the capital is overutilized. Eric Dunaway (WSU) EconS 425 Industrial Organization 22 / 34

23 Total Cost Function C/q AVC AFC q Eric Dunaway (WSU) EconS 425 Industrial Organization 23 / 34

24 Total Cost Function Adding the Average Fixed and Average Variable costs together gives us the Average cost function. The Average Cost function tells us how much per unit it costs to make something, and is quite useful for pro t calculations. Price minus Average Cost is the pro t per unit. Notice that Average Cost has the same shape as the Average Variable Cost, and as quantity increases, the two curves approach one another. Eric Dunaway (WSU) EconS 425 Industrial Organization 24 / 34

25 Total Cost Function C/q AC AVC AFC q Eric Dunaway (WSU) EconS 425 Industrial Organization 25 / 34

26 Total Cost Function Lastly, we have the Marginal Cost. This tells us how much the next unit of output adds to the total cost. Marginal cost is one of our key decision parameters. The rm will produce until the amount of revenue from the next unit of output (the marginal revenue) is equal to the marginal cost. The Marginal Cost function is equivalent to the supply curve when price is above Average Variable Cost. Eric Dunaway (WSU) EconS 425 Industrial Organization 26 / 34

27 Total Cost Function C/q MC AC AVC AFC q Eric Dunaway (WSU) EconS 425 Industrial Organization 27 / 34

28 Total Cost Function C/q S AC AVC AFC q Eric Dunaway (WSU) EconS 425 Industrial Organization 28 / 34

29 Total Cost Function Remember that the Marginal Cost function has a unique relationship with the Average and Average Variable Costs. When Marginal Cost is below Average or Average Variable Cost, the Averages are decreasing. Consequently, Marginal Cost intersects with Average and Average Variable Cost at exactly their minimum. We can prove this by di erentiating the Average Cost Function with respect to quantity (The results for Average Variable Cost are similar) dac dq = d c(q) q = qc0 (q) c(q) dq q 2 = q c 0 c(q) (q) q q 2 = c0 (q) q c(q) q = MC AC q q is always positive, so the di erence between Average Cost and Marginal Cost determines the sign of this derivative. Eric Dunaway (WSU) EconS 425 Industrial Organization 29 / 34

30 Sunk Costs There is one more type of cost that is quite important to ms: the Sunk Cost. Sunk Costs are similar to Fixed Costs in that they are independent of the output level. The di erence is that Sunk Costs are not recoverable. Intuitively, a rm only pays a Fixed Cost if they produce a positive output level, i.e., q > 0. Sunk Costs are incurred whether or not a rm produces, so they can happen when no production occurs, i.e., q = 0. For example, a rm s capital rent would be a Fixed Cost. The Research and Development incurred by the rm would be a Sunk Cost. In the short run, once a Fixed Cost is paid, it is sunk. Eric Dunaway (WSU) EconS 425 Industrial Organization 30 / 34

31 Sunk Costs Sunk Costs are critical for a rm s entry decision. There are usually large costs associated with starting up a rm that are not recoverable. Licensing, Research and Development, etc. A rm will compare its lifetime pro ts versus the sunk cost it faces to determine whether it will enter a market or not. Remember that when dealing with lifetime pro ts, rms will discount future pro ts into their present value. We ll talk about that next time. Eric Dunaway (WSU) EconS 425 Industrial Organization 31 / 34

32 Summary We can derive cost functions through the cost minimization problem. Costs can be broken down into several components, each having a key role in the decision making process of the rm. Eric Dunaway (WSU) EconS 425 Industrial Organization 32 / 34

33 Next Time Firm Entry Decision Economies of Scale and Scope Reading: Section 3.2 I won t be covering section 3.4, but it has a great empirical example. If you like econometrics, see how they adapt a production function into something that can be estimated using available data. Eric Dunaway (WSU) EconS 425 Industrial Organization 33 / 34

34 Practice Problem (Optional) Consider the following production function consisting of two inputs, capital (z 1 ) and labor (z 2 ), q = z z 1 z 2 + z Does this production function have returns to scale? If so, what kind? 2. Set up the cost minimization problem and derive rst-order conditions. Eric Dunaway (WSU) EconS 425 Industrial Organization 34 / 34