Cost Theory and Cost Minimization Notes

Distinguishing Between Accounting and Economic Cost

• General Perspectives:     - Accountant's View: Stresses out-of-pocket expenses, historical costs, depreciation, and other bookkeeping entries.     - Economist's View: Defines the cost of any input by the size of the payment necessary to keep the resource in its present employment. Alternatively, the economic cost is what that input would be paid in its next best use (opportunity cost).

• Labor Costs:     - Both economists and accountants view labor costs similarly.     - To accountants, expenditures on labor are current expenses and costs of production.     - For economists, labor is an explicit cost. Labor services (labor-hours) are contracted at an hourly wage rate ($w$), which is typically assumed to be equal to what the services would earn in their best alternative employment.

• Capital Costs:     - Concepts differ significantly between the two fields.     - Accountants: Use the historical price of the machine and apply arbitrary depreciation rules to determine charges for current costs.     - Economists: Regard the historical price as a sunk cost, which is irrelevant to output decisions. They view the implicit cost as the rental rate ($v$) for that machine in its best alternative use. By using the machine, the firm foregoes what others would pay to use it.

• Costs of Entrepreneurial Services:     - Accountant: The owner is a residual claimant; leftovers after paying input costs are called profits (positive or negative).     - Economist: Considers opportunity costs of the owner's time and funds. These services are treated as an input with an imputed cost.     - Example - Software Firm: A highly skilled computer programmer starts a firm. The programmer's time is an input. The cost imputed is the wage they would command working elsewhere.     - Economic vs. Accounting Profit: Economic profits are smaller than accounting profits because they subtract entrepreneurial opportunity costs. Economic profit might be negative if opportunity costs exceed accounting profits.

Economic Profits and Cost Minimization

• Two Simplifying Assumptions:     1. There are only two inputs: homogeneous labor ($L$, measured in labor-hours) and homogeneous capital ($K$, measured in machine-hours). Entrepreneurial costs are subsumed under capital costs.     2. Inputs are hired in perfectly competitive markets. Firms can buy/sell any amount of labor or capital at prevailing rental rates ($w$ and $v$). The supply curve for resources is horizontal; $w$ and $v$ are constants/parameters.

• Total Cost Equation:     - Total Costs ($TC$) during a period:     $TC = wl + vk$

• Economic Profit ($\pi$):     - Defined as the difference between total revenue ($P \times q$) and total economic costs:     $\pi = \text{Total Revenue} - \text{Total Cost} = Pq - wl - vk$     - In terms of the production function $q = f(K, L)$, the profit equation is:     $\pi = P \times f(k, l) - wl - vk$

• Objective of Cost Minimization:     - We assume the firm has decided to produce a particular output level ($q_0$) regardless of profit maximization goals.     - The task is to choose the input combination that produces $q_0$ at minimal costs.

Mathematical Analysis of Cost Minimization

• Constrained Minimization Problem:     - Minimize total costs given the constraint $q = f(k, l) = q_0$.     - Lagrangian Expression ($\mathcal{L}$):$\mathcal{L} = wl + vk + \lambda[q_0 - f(k, l)]$

• First-Order Conditions (FOCs):     1. $\frac{\partial \mathcal{L}}{\partial l} = w - \lambda \frac{\partial f}{\partial l} = 0$     2. $\frac{\partial \mathcal{L}}{\partial K} = v - \lambda \frac{\partial f}{\partial K} = 0$     3. $\frac{\partial \mathcal{L}}{\partial \lambda} = q_0 - f(k, l) = 0$

• Derivation of the Optimality Condition:     - From Eq 1.1: $\lambda \frac{\partial f}{\partial l} = w \implies \lambda = \frac{w}{\frac{\partial f}{\partial l}}$     - From Eq 1.2: $\lambda \frac{\partial f}{\partial k} = v \implies \lambda = \frac{v}{\frac{\partial f}{\partial k}}$     - Equating the two values of $\lambda$:     $\frac{w}{\frac{\partial f}{\partial l}} = \frac{v}{\frac{\partial f}{\partial k}} \implies \frac{w}{v} = \frac{\frac{\partial f}{\partial l}}{\frac{\partial f}{\partial k}} = \frac{MP_L}{MP_K} = RTS_{L,K}$

• Interpretations:     - Rate of Technical Substitution ($RTS$): The firm should equate the rate at which capital can be traded for labor in production ($RTS$) to the rate they are traded in the market ($w/v$).     - Marginal Productivity per Kwacha: Cross-multiplying yields $\frac{f_k}{v} = \frac{f_l}{w}$. For costs to be minimized, the marginal productivity per kwacha spent must be equal for all inputs. If one input gave a higher return per kwacha, the firm would hire more of it.     - Lagrangian Multiplier ($\lambda$) as Marginal Cost: The equation $\frac{w}{f_l} = \frac{v}{f_k} = \lambda$ shows that $\lambda$ represents the extra cost of obtaining an extra unit of output. It measures how much total costs would increase if the output constraint ($q_0$) were increased slightly.

Cost Minimization Example 1: Cobb-Douglas Production Function

• Production Function: $q_0 = f(k, l) = k^{\alpha} l^{\beta}$

• Lagrangian: $\mathcal{L} = vk + wl + \lambda(q_0 - k^{\alpha} l^{\beta})$

• FOCs:     1. $\frac{\partial \mathcal{L}}{\partial l} = w - \lambda \beta k^{\alpha} l^{\beta-1} = 0$     2. $\frac{\partial \mathcal{L}}{\partial k} = v - \lambda \alpha k^{\alpha-1} l^{\beta} = 0$     3. $\frac{\partial \mathcal{L}}{\partial \lambda} = q_0 - k^{\alpha} l^{\beta} = 0$

• Optimal Input Ratio:$\frac{w}{v} = \frac{\beta k^{\alpha} l^{\beta-1}}{\alpha k^{\alpha-1} l^{\beta}} = \frac{\beta}{\alpha} \frac{k}{l}$

• Numerical Illustration:     - Parameters: $\alpha = 0.5$, $\beta = 0.5$, $w = 12$, $v = 3$, and $q_0 = 40$.     - Using the ratio: $v \beta k = w \alpha l \implies 3(0.5)k = 12(0.5)l \implies 1.5k = 6l \implies k = 4l$.     - Substituting into the production function:     $40 = (4l)^{0.5} (l)^{0.5} = 2l \implies l = 20$     - Finding capital: $k = 4(20) = 80$.     - Minimum Total Cost:$wl + vk = (12)(20) + (3)(80) = 240 + 240 = 480$

• Comparison with other points producing $q=40$:     - If $K=40, l=40$: $TC = (12)(40) + (3)(40) = 600$.     - If $K=10, l=160$: $TC = (12)(160) + (3)(10) = 1950$.     - If $K=160, l=10$: $TC = (12)(10) + (3)(160) = 600$.     - Any other combination results in a cost higher than 480.

Graphical Analysis and the Expansion Path

• Cost Minimization Graph (Fig 1):     - The least costly point for output $q_0$ is the point of tangency between the isoquant and the lowest possible isocost line ($C_1$).     - This combination is denoted as $(l^*, k^*)$.     - A true minimum requires the isoquant to be convex (diminishing $MRTS$).

• The Firm's Expansion Path (Fig 2):     - The locus of cost-minimizing points (tangencies) for successively higher levels of output ($q_0, q_1, q_2$) while holding factor prices ($w, v$) constant.     - For constant returns to scale (homothetic functions), the expansion path is a straight line through the origin because the $MRTS$ depends only on the ratio of $k$ to $l$.

• Input Inferiority (Fig 3):     - The expansion path is not always a straight line.     - An inferior input is one where the quantity used decreases as output expands beyond a certain point.     - Example: The use of shovels may decline while the use of a backhoe increases as production of building foundations grows.

Cost Functions: Total, Average, and Marginal

• Total Cost Function:     - Shows the minimum total cost for any set of input prices and output levels: $C = C(v, w, q)$.     - Total costs increase as output $q$ increases.

• Unit Cost Measures:     - Average Cost ($AC$): Cost per unit of output.     $AC(v, w, q) = \frac{C(v, w, q)}{q}$     - Marginal Cost ($MC$): Cost of one additional unit of output.     $MC(v, w, q) = \frac{\partial C(v, w, q)}{\partial q}$

• Constant Returns to Scale Case (Fig 4):     - Total cost is proportional to output ($C = aq$, where $a$ is cost of 1 unit).     - $AC$ and $MC$ are constant and equal ($AC = MC = a$). Graphically, this is a horizontal line.

• Cubic Total Cost Curve Case (Fig 5):     - The total cost curve is initially concave (costs rise rapidly then slow down) and later becomes convex (costs rise progressively rapidly).     - Reasons for shape: A fixed third factor (e.g., entrepreneurial services). Concavity represents increasingly optimal usage of the entrepreneur; convexity represents the entrepreneur becoming overworked (diminishing returns).     - Relationship between MC and AC:         - $MC$ is the slope of the total cost curve; it is U-shaped.         - If $MC < AC$, $AC$ is falling.         - If $MC > AC$, $AC$ is rising.         - $MC = AC$ at the low point of the $AC$ curve ($q^*$).         - The point of minimum average cost is the Minimum Efficient Scale (MES).

Short Run vs. Long Run Costs

• Definitions:     - Short Run: A period where economic actors have limited flexibility. Specifically, one input (capital $K$) is held fixed at a level $k_1$.     - Long Run: A longer period providing greater degrees of freedom; all inputs are variable.

• Short Run Total Costs ($SC$):     - Defined as: $SC = vk_1 + wl$.     - Short Run Fixed Costs: The term $vk_1$. These costs do not change with output level in the short run.     - Short Run Variable Costs: The term $wl$. These change as labor is varied to change output.

• Non-Optimality of Short Run Costs (Fig 7):     - In the short run, the firm is forced to use non-optimal input combinations because it cannot adjust capital.     - The $MRTS$ will typically not equal the input price ratio ($w/v$).     - Case 1 (Point a): Producing $q_0$ with fixed capital $k_1$ uses "too much" capital relative to the optimal level $k_0$.     - Case 2 (Point c): Producing $q_2$ with fixed capital $k_1$ uses "too little" capital relative to the optimal level $k_2$.     - Optimal Point (Point b): The short run cost matches the minimal long-run cost only at the one level of output ($q_1$) where capital usage $k_1$ happens to be the cost-minimizing amount. All other points in the short run cost more than in the long run.

• Transition: A firm can only reach cost-minimizing combinations for all output levels in the long run by adjusting capital usage.