Question

Explain with the help of a diagram the relationship between Total Cost, Average Cost and Marginal Cost.

Hint:

A helpful analogy for the AC/MC relationship is your grade point average (GPA). If your grade in the next (marginal) subject is lower than your current average, your average will fall. If your grade in the next subject is higher than your average, your average will rise. The marginal always pulls the average in its direction.

Answer 1

Step 1: Definitions of Cost Concepts
\begin{itemize} \item Total Cost (TC): It is the total expenditure incurred by a firm to produce a given level of output. It is the sum of Total Fixed Cost (TFC) and Total Variable Cost (TVC). \(TC = TFC + TVC\). \item Average Cost (AC): It is the cost per unit of output. It is calculated by dividing the total cost by the quantity of output produced. \(AC = TC/Q\). \item Marginal Cost (MC): It is the addition made to the total cost by producing one more unit of output. \(MC = \Delta TC / \Delta Q = TC_n - TC_{n-1}\). Mathematically, MC is the slope of the TC curve. \end{itemize}

Step 2: Diagrammatic Representation
The relationship between TC, AC, and MC is shown in the two-panel diagram below. The shapes of these curves are determined by the Law of Variable Proportions. \begin{center} \begin{tikzpicture} % Upper Panel for Total Costs \begin{axis}[ name=plot1, axis lines=left, xlabel={Output (Q)}, ylabel={Total Cost}, xmin=0, xmax=10, ymin=0, ymax=100, xtick=\empty, ytick=\empty, legend style={at={(0.05,0.95)},anchor=north west}, height=7cm, width=10cm ] \addplot[smooth, thick, blue, domain=0:10] {10 + 0.2*x^3 - 2*x^2 + 10*x}; \addlegendentry{TC} \addplot[smooth, thick, green, domain=0:10] {0.2*x^3 - 2*x^2 + 10*x}; \addlegendentry{TVC} \addplot[dashed, thick, red, domain=0:10] {10}; \addlegendentry{TFC} \coordinate (inflection) at (axis cs:3.33, 24.8); \end{axis} % Lower Panel for Average and Marginal Costs \begin{axis}[ name=plot2, at=(plot1.below), anchor=above, axis lines=left, xlabel={Output (Q)}, ylabel={Per-Unit Cost}, xmin=0, xmax=10, ymin=0, ymax=20, xtick=\empty, ytick=\empty, legend style={at={(0.6,0.95)},anchor=north west}, height=7cm, width=10cm ] \addplot[smooth, thick, blue, domain=1:10] {10/x + 0.2*x^2 - 2*x + 10}; \addlegendentry{AC} \addplot[smooth, thick, magenta, domain=0:10] {0.6*x^2 - 4*x + 10}; \addlegendentry{MC} \coordinate (mc_min) at (axis cs:3.33, 3.33); \coordinate (ac_min) at (axis cs:5, 5); \end{axis} % Dashed line connecting inflection point and MC minimum \draw[dashed, gray] (inflection) -- (mc_min); \draw[dashed, gray] (ac_min) -- (axis cs:5, 0) node[below] {$Q^*$}; % Annotation for MC=AC \node[above right, text width=2cm] at (ac_min) {MC intersects AC at its minimum}; \end{tikzpicture} \end{center}

Step 3: Explanation of the Relationships
The relationships between the cost curves can be summarized as follows: \begin{enumerate} \item Relationship between TC and MC: \begin{itemize} \item Marginal Cost (MC) is the slope of the Total Cost (TC) curve. \item The TC curve initially increases at a decreasing rate (becomes flatter) as long as MC is falling. This corresponds to increasing returns to the variable factor. \item The TC curve has its point of inflection (where the slope is minimum) at the level of output where the MC curve reaches its minimum point. \item After the inflection point, the TC curve starts increasing at an increasing rate (becomes steeper) as the MC curve rises. This corresponds to diminishing returns. \end{itemize} \item Relationship between AC and MC: This is the most important relationship. \begin{itemize} \item Both AC and MC curves are U-shaped due to the law of variable proportions. They initially fall, reach a minimum, and then rise. \item When MC \(<\) AC, the cost of the additional unit is less than the average cost so far. This pulls the average cost down. Therefore, the AC curve is falling. \item When MC \(>\) AC, the cost of the additional unit is more than the average cost. This pulls the average cost up. Therefore, the AC curve is rising. \item When MC = AC, the AC curve is neither falling nor rising; it is at its minimum point. This is why the MC curve cuts the AC curve at the lowest point of the AC curve. \end{itemize} \item Shape of the Curves: \begin{itemize} \item The Total Cost (TC) curve starts from the level of Total Fixed Cost (TFC) on the Y-axis because there is a fixed cost even at zero output. It is always upward sloping. \item The Average Cost (AC) and Marginal Cost (MC) curves are derived from the TC curve and are typically U-shaped in the short run. \end{itemize} \end{enumerate}