Chapter 7. The Cost of Production. Topics to be Discussed. Introduction. Introduction. Measuring Cost: Which Cost Matter?

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1 Chapter 7 The of Production Topics to be Discussed Measuring : Which s Matter? s in the Short Run & Long Run Production with Two s-- Introduction The production function measures the relationship between input and output. Given the production technology, managers must choose how to produce. (I.e., how many units? ) Introduction To determine the optimal level of output and the input combinations, we must convert from the unit measurements of the production function to dollar measurements or costs. Measuring : Which Matter? ccounting Consider only explicit cost, the out of pocket cost for such items as wages, salaries, materials, and proty rentals Measuring : Which Matter? Economic Considers explicit and opportunity cost. Opportunity cost is the cost associated with opportunities that are foregone by not putting resources in their highest valued use. Sunk n expenditure that has been made and cannot be recovered--they should not influence a firm s decisions.

2 Total output is a function of variable inputs and fixed inputs. Therefore, the total cost of production equals the fixed cost (the cost of the fixed inputs) plus the variable cost (the cost of the variable inputs), or TC = FC + VC Marginal (MC) is the cost of expanding output by one unit. Since fixed cost have no impact on marginal cost, it can be written as: VC MC = Q TC = Q verage Total (TC) is the cost unit, or average fixed cost (FC) plus average variable cost (VC). This can be written: TFC TVC TC = + Q Q TC = FC + VC or TC Q The Determinants of Short-Run The relationship between the production function and cost can be exemplified by either increasing returns and cost or decreasing returns and cost. The Determinants of Short-Run Increasing returns and cost With increasing returns, output is increasing relative to input and variable cost and total cost will fall relative to output. Decreasing returns and cost With decreasing returns, output is decreasing relative to input and variable cost and total cost will rise relative to output. For Example: ssume the wage rate (w) is fixed relative to the number of workers hired. Then: VC MC = Q VC = wl

3 Continuing: VC = w L L MC = w Q Continuing: Q MPL = L L 1 L for a1unit Q = = Q MPL In conclusion: w MC = MPL and a low marginal product (MP) leads to a high marginal cost (MC) and vise versa. Consequently (from the table): MC decreases initially with increasing returns 0 through 4 units MC increases with decreasing returns 5 through 11 units VC and the Production Function VC VC = Q VC = wl VC = wl Q VC and the Production Function Q PL = L w VC = P L

4 Observations If a firm is exiencing increasing returns, P is increasing and VC will decrease. If a firms is exiencing decreasing returns, P is decreasing and VC will increase. Summary The production function (MP & P) shows the relationship between inputs and output. The cost measurements show the impact of the production function in dollar terms. Price ($ ) Curves for a Firm VC FC (units ) B TC Price ($ unit) Curves for a Firm MC TC VC FC Marginal cost decreases initially then increases (units ) Curves for a Firm The line drawn from the origin to the tangent of the variable cost curve: Its slope equals VC The slope of a point on VC equals MC Therefore, MC = VC at 7 units (point ) P VC 100 F C B TC Curves for a Firm The ray drawn from the origin to the tangent of the total cost curve: The slope of a tangent equals the slope of the point. TC at 8 units = MC = 8 units. P B TC VC 100 F C

5 Curves for a Firm Unit s FC falls continuously When MC < VC or MC < TC, VC & TC decrease When MC > VC or MC > TC, VC & TC increase P MC TC VC FC Curves for a Firm Unit s MC = VC and TC at minimum VC and TC Minimum VC occurs at a lower output than minimum TC due to FC P MC TC VC FC in the Long Run Choosing Inputs ssumptions Two Inputs: Labor (L) & capital (K) Wage rate for labor (w) and rental rate for capital (r) are determined in competitive markets in the Long Run Choosing Inputs Decision Model C = wl + rk Isocost: line showing all combinations of L & K that can be purchased for the same cost in the Long Run Choosing Inputs Rewriting C as linear: K = C/r - (w/r)l K w Slope of the isocost: L = r ( ) Choosing Inputs We will address how to minimize cost for a given level. We will do so by combining isocosts with isoquants is the ratio of the wage rate to rental cost of capital. This shows the rate at which capital can be substituted for labor with no change in cost.

6 Producing a Given at Minimum K 2 Isocost C 2 shows quantity can be produced with combination K 2 L 2 or K 3 L 3. However, both of these are higher cost combinations than. Producing a Given at Minimum Isocost C 1 shows quantity can be produced with combination.. This is the low cost combination because it is tangent to. K 3 C 0 C 1 C 2 C 0 C 1 C 2 L 2 L 3 Input Substitution When an Input Price Change If the price of labor changes, the isocost curve becomes stee due to the change in the slope -(w/l). Input Substitution When an Input Price Change K 2 B This yields a new combination of K and L to produce. Combination B is used in place of combination. The new combination represents the higher cost of labor relative to capital and therefore capital is substituted for labor. C 1 L 2 C 2 C 1 in the Long Run Isoquants and Isocosts and the Production Function MRTS = - K = MPL L MPK Slope of isocost line = K w L = r in the Long Run The minimum cost combination can then be written as: MP = L MPK w Minimum cost for a given output will occur when each dollar of input added to the production process will add an equivalent amount. r

7 in the Long Run Question: If w = $10, r = $2, and MP L = MP K, which input would the producer use more of? Why? minimization with Varying Levels firm s expansion path shows the minimum cost combinations of labor and capital at each level. Firm s Expansion Path B C D E The expansion path illustrates the least-cost combinations of labor and capital that can be used to produce each level of output in the long -run. Expansion Path What happens to average costs when both inputs are variable (long run) versus only having one input that is variable (short run)? The Inflexibility of Short-Run Production Begin with and isocost B which yields. The Inflexibility of Short-Run Production E ssume K is fixed (short-run) and output is increased to Q 2. Combination L 3 would have to be used on isocostef. P Q 2 B B L 3 F

8 The Inflexibility of Short-Run Production E C K 2 If K is flexible (long -run), isocost line CD is used yielding combination K 2 L 2. CD is a lower cost level than EF. In the long -run, the firm substitutes chea K for L. The Inflexibility of Short-Run Production E C K 2 Expansion Path The long -run expansion path is drawn as before.. Q 2 Q 2 L 2 B D F L 2 B D F Long-Run verage (LC) Constant Returns to Scale If input is doubled, output will double and average cost is constant at all levels. Long-Run verage (LC) Increasing Returns to Scale If input is doubled, output will more than double and average cost decreases at all levels. Long-Run verage (LC) Decreasing Returns to Scale If input is doubled, the increase in output is less than twice as large and average cost increases with output. Long-Run verage (LC) In the long-run: Firms exience increasing and decreasing returns to scale and therefor long-run average cost is U shaped.

9 Long-Run verage (LC) Long-run marginal cost leads long-run average cost: If LMC < LC, LC will fall If LMC > LC, LC will rise Therefore, LMC = LC at the minimum of LC Long-Run verage and Marginal LMC LC Question What is the relationship between long-run average cost and long-run marginal cost when long-run average cost is constant? Economies and Diseconomies of Scale Economies of Scale Increase in output is greater than the increase in inputs. Diseconomies of Scale Increase in output is less than the increase in inputs. Measuring Economies of Scale Ec = Elasticity = % in cost froma1%increase in output Measuring Economies of Scale Ec Ec = ( C / C)/( Q / Q) = ( C / Q) /( C / Q) = MC/C

10 Therefore, the following is true: E C < 1: MC < C verage cost indicate decreasing economies of scale E C = 1: MC = C verage cost indicate constant economies of scale E C > 1: MC > C verage cost indicate increasing diseconomies of scale The Relationship Between Short-Run and Long-Run We will use short and long-run cost to determine the optimal plant size Long-Run with Constant Returns to Scale Known: The SC for three plant sizes with constant returns to scale. Long-Run with Constant Returns to Scale SC 1 SMC 1 Long-Run with Constant Returns to Scale SC 1 SC 2 Long-Run with Constant Returns to Scale SC 1 SC 2 SC 3 SMC 1 SMC 2 SMC 1 SMC 2 SMC 3 Q 2 Q 2 Q 3

11 Long-Run with Constant Returns to Scale With many plant sizes with SC = $10 the LC = LMC and is a straight line SC 1 SC 2 SC 3 SMC 1 SMC 2 SMC 3 LC = LMC Long-Run with Constant Returns to Scale Observation The optimal plant size will depend on the anticipated output (e.g. choose SC 1,etc). The long-run average cost curve is the envelope of the firm s short-run average cost curves. Question What would happen to average cost if an output level other than that shown is chosen? Q 2 Q 3 Long-Run with Economies and Diseconomies of Scale Long-Run with Economies and Diseconomies of Scale SC 1 SC 2 SC 3 Known: Three plant sizes with economies and diseconomies of scale. SMC 1 SMC 3 SMC 2 Long-Run with Economies and Diseconomies of Scale SC 1 SC 3 LC SC 2 Long-Run with Economies and Diseconomies of Scale SC 1 SC 3 LC SC 2 SMC 1 SMC 3 SMC 1 SMC 3 SMC 2 LMC SMC 2

12 Long-Run with Economies and Diseconomies of Scale $10 $8 SC 1 LMC SMC 1 B SMC 2 SMC 3 SC 3 LC SC 2 If the output is a manager would chose the small plant SC 1 and SC $8. Point B is on the LC because it is a least cost plant for a given output. What is the firms long-run cost curve? Firms can change scale to change output in the long-run. The long-run cost curve is the dark blue portion of the SC curve which represents the minimum cost for any level. Long-Run with Economies and Diseconomies of Scale Observations The LC does not include the minimum points of small and large size plants? Why not? LMC is not the envelope of the short-run marginal cost. Why not? Production with Two s-- Examples: Chicken farm--poultry and eggs utomobile company--cars and trucks University--Teaching and research Production with Two s-- Economies of scope exist when the joint output of a single firm is greater than the output that could be achieved by two different firms each producing a single output. What are the advantages of joint production? Consider an automobile company producing cars and tractors Production with Two s-- dvantages 1) Both use capital and labor. 2) The firms share management resources. 3) Both use the same labor skills and type of machinery.

13 Production with Two s-- Production: Firms must choose how much of each to produce. The alternative quantities can be illustrated using product transformation curves. Product Transformation Curve Number of tractors O 1 O 1 illustrates a low level. O 2 illustrates a higher level with two times as much labor and capital. O 2 Number of cars Production with Two s-- Observations Product transformation curves are negatively sloped Constant returns exist in this example Since the production transformation curve is concave is joint production desirable? Production with Two s-- Observations There is no direct relationship between economies of scope and economies of scale. May exience economies of scope and diseconomies of scale May have economies of scale and not have economies of scope Production with Two s-- The degree of economies of scope measures the savings in cost can be written: C( Q1) + C( Q2) C( Q1, Q2) SC = C( Q1, Q2) Production with Two s-- Interpretation: If SC > 0 -- Economies of scope If SC < 0 -- Diseconomies of scope C( ) is the cost of producing C(Q 2 ) is the cost of producing Q 2 C( Q 2 ) is the joint cost of producing both products

14 Example: in the Trucking Industry Issues Truckload versus less than truck load Direct versus indirect routing Length of haul Example: in the Trucking Industry Questions: Economies of Scale re large-scale, direct hauls chea and more profitable than individual hauls by small trucks? re there cost advantages from oating both direct and indirect hauls? Example: in the Trucking Industry Empirical Findings n analysis of 105 trucking firms examined four distinct outputs. Short hauls with partial loads Intermediate hauls with partial loads Long hauls with partial loads Hauls with total loads Example: in the Trucking Industry Empirical Findings Results SC = for reasonably large firm SC = for very large firms Interpretation Combining partial loads at an intermediate location lowers cost management difficulties with very large firms. Summary Managers, investors, and economists must take into account the opportunity cost associated with the use of the firm s resources. Firms are faced with both fixed and variable costs in the short-run. Summary When there is a single variable input, as in the short run, the presence of diminishing returns determines the shape of the cost curves. In the long run, all inputs to the production process are variable.

15 Summary The firm s expansion path describes how its cost-minimizing input choices vary as the scale or output of its oation increases. The long-run average cost curve is the envelope of the short-run average cost curves. Summary firm enjoys economies of scale when it can double its output at less than twice the cost. Economies of scope arise when the firm can produce any combination of the two outputs more cheaply than could two independent firms that each produced a single product. Summary firm s average cost of production can fall over time if the firm learns how to produce more effectively. functions relate the cost of production to the level of the firm.