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Cameron Churchill

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Principles of Economics: Short-run Cost We first examine the cost of firms in the period when capital is fixed to understand the importance of marginal cost in the determination of profit maximizing output. We do this by deriving cost functions from production functions. SHORT-RUN PRODUCTION AND COST FUNCTIONS Production Function Definition: The set of all maximum possible outputs from a given input or inputs that provide a technologically feasible way to produce a commodity or service. Q = f(Xo, X 1 X 2 X 3 X4, … X N where Xi represents the inputs into production We simplify this by dividing the inputs into Capital (K) and Labour (L) to get Q = f(L, K) Firm Definition: A business organization producing goods and services from resource inputs → ‘Business’ implies the assumption of profit maximization → The firm connects costs and output as the smallest unit of production. Industry Definition: All the firms that produce a given good or service Short-run Definition: The period in which one factor (input) is fixed Long-run Definition: The period in which all factors are variable Variable Inputs Definition: Inputs that vary in quantity with each additional unit of output - 1 - Principles of Economics: Short-run Cost →Variable inputs include Labour, Fuel, and Raw Materials but we use Labour (L) to represent them all Fixed Inputs Definition: Inputs that do not vary in quantity with an additional unit of output → Fixed inputs include equipment and machinery, buildings, and land but we use Capital (K) to represent them all SHORT-RUN PRODUCTION FUNCTIONS Total Product of Labour (TP ) L Definition: The total output produced from each total labour output given fixed capital →TP = L = f(LK) (Q = output of the industry and q = output of a firm so that Q = nq where n = #firms in the industry) Average Product of Labour (AP or siLply AP since we will assume that capital is fixed) Definition: Average output of each labour unit given fixed capital → AP = L/L Marginal Product of Labour (MP or simLly MP) Definition: The change in output due to a change in labour given fixed capital → MP = ΔqLΔL [(q – q )/1L – 0 ) f1r di0crete changes] (or = dq/dL for infinitesimal changes in Labour) NOTE: The Marginal Product of Labour is often expressed as the change in output due to an additional ‘unit’ of labour. - 2 - Principles of Economics: Short-run Cost Law of Eventually Diminishing Returns Definition: Marginal Product eventually decreases with increases in labour since the efficiency of additional labour eventually falls due to fixed capital e.g. The following table shows the Total Product, Average Product, and Marginal Products for units (days) of labour at cotton cloth manufacturer with a spinning, weaving, and dyeing machine. The equation q = - L + 12L describes the total product of labour. Labour Units Total Product Average Product Marginal Marginal Product (TP) (AP) Product MP = dTP/dL MP = L n+1- Ln 1 11 11 11 21 2 40 20 29 36 3 81 27 41 45 4 128 32 47 48 5 175 35 47 45 6 216 36 41 36 7 245 35 29 21 8 256 32 11 0 9 243 27 -13 -27 Marginal Product initially rises because additional workers can use the fixed capital more efficiently but eventually marginal product declines because the fixed capital limits the productivity of the workers. Marginal Product declines to below 0 when an additional worker results in a fall in Total Product, not just a decline in Marginal Product. Notice that Total Product shows us that output increases up to 8 workers and then decreases but does not show us the rate at which Total Product increases or decreases. Marginal Product is more informative since positive (negative) Marginal product tells us not only whether Total Product increases (decreases) but also the change in Total Product. - 3 - Principles of Economics: Short-run Cost The diagram below shows the output as a function of Labour in the upper diagram and output per worker as a function of Labour in the lower diagram. Total product is a function of total labour (e.g., the total product of six workers is 216) in the upper diagram. Average Product is a function of total labour but Marginal Product is a function of the last unit of labour (the discrete marginal product is 41 for the sixth worker) in the lower diagram. Output (Q) 3 2 300 TP = -L + 12L 250 200 TP 150 100 50 0 0 2 4 6 8 10 60 Output per L 40 20 AP 0 MP -20 -40 0 2 4 6 8 10 AP = - L + 12L and MP = - 3L + 24L 1 We will use the Marginal Product and Average Product diagrams to analyze costs since they are more informative than Total Product Note: MP = max MP The change in a variable (margin) has a specific mathematical relation with the average of a variable. 1 3 2 3 2 AP = TP/L = (-L + 12L )/L and MP = dTP/dL = d(-L + 12L )/dL - 4 - Principles of Economics: Short-run Cost Margin > Average => Increasing Average (the marginal pulls up the average) Margin < Average => Decreasing Average (the margin pulls down the average) Margin = Average => Maximum Average if the Margin increases then decreases => Minimum Average if the Margin decreased then increased Our diagram of Average and Marginal Product shows that increasing Marginal Product initially increases Average Product. Marginal Product reaches a maximum and falls but Average Product still rises so long as Marginal is greater than Average Product. Average Product falls when Marginal Product is below Average Product. Marginal Product equals Average Product, therefore, when Average Product is at a maximum. - 5 - Principles of Economics: Short-run Cost COST FUNCTIONS: (Assuming efficiency production, i.e., minimum cost for each output) Costs include: 1. Explicit Costs. These are the costs of purchasing or hiring inputs from firms or households. 2. Implicit Costs. These are the opportunity costs of inputs owned by the firm itself and thus not explicitly paid for. The most important implicit cost is the opportunity cost of employing owner capital in the firm (i.e., the equity of the firm). Accountants typically count explicit costs in the balance sheet to determine profit and then compare this profit and loss with the amount of equity on a separate page to determine the rate of return. Economists include capital as an opportunity cost in costs directly, usually as the foregone return (opportunity cost) on the capital. Fixed Costs Definition: Costs that do not change with every change in output (Cost of fixed inputs) Fixed costs are the opportunity cost of capital for us since we assume that fixed inputs are capital. This does not mean that fixed costs are the value of K, the total amount of capital, but rK, the average rate of return on capital ‘K’. Variable Costs: Definition: Costs that do change with every change in output (Cost of variable inputs) Variable Costs include the cost of Labour (wL) but also the cost of fuel and raw materials (which we include as Labour) - 6 - Principles of Economics: Short-run Cost Total Cost (TC) = Variable Costs + Fixed Costs = wL + rK Average Variable Cost (AVC) = Average variable cost of a unit of output = VC/q Average Fixed Cost (AFC) = Average fixed cost of a unit of output = FC/q Average Cost (AC) = Average cost of a unit of output = TC/q = (VC + FC)/Q = AVC + AFC Marginal Cost (MC) = Change in Total Cost for a change in output (= Change in Variable Cost for a change in output) = ΔTC/Δq (dTC/dq) = Δ(wL + rK)/Δq (d(wL + rK)/dq = Δ(wL)/Δq Since Δ(rK)/Δq = 0 (d(wL)/dq) (=>MC = dVC/dq since MC = d(wL)/dq) = wΔL/Δq wdL/dq => MC = w/MP Since ΔL/Δq = 1/(/Δq/ΔL) = 1/Marginal Product This derivation shows that Marginal Cost is simply the inverse of Marginal Product multiplied by the wage rate per unit of Labour. An increase (decrease) in Marginal Product implies a decrease (increase) in Marginal Cost. Since Marginal Product increases initially but eventually decreases, Marginal Cost decreases initially but eventually increases. Marginal Cost is - 7 - Principles of Economics: Short-run Cost at
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