Question

Explain diagrammatically the law of Constant Return to Scale.

Hint:

To remember the three returns to scale visually: \begin{itemize} \item Increasing: Isoquants get closer together. \item Constant: Isoquants are equally spaced. \item Decreasing: Isoquants get farther apart. \end{itemize}

Answer 1

Step 1: Understanding the Concept:
The law of returns to scale describes the relationship between inputs and output in the long run when all inputs are variable. Constant Returns to Scale (CRS) is a situation where if a firm increases all its inputs (e.g., labor and capital) by a certain percentage, its output increases by the exact same percentage. For example, if all inputs are doubled, the output also doubles.

Step 2: Diagrammatic Explanation:
Constant returns to scale can be illustrated using an isoquant map. An isoquant represents all combinations of two inputs (like labor and capital) that produce a given level of output. \begin{center} \begin{tikzpicture} \begin{axis}[ xlabel={Labor (L)}, ylabel={Capital (K)}, xmin=0, xmax=12, ymin=0, ymax=12, axis lines=left, ticks=none, clip=false, width=0.7\textwidth, height=0.6\textwidth, ] % Isoquants \addplot[smooth, thick, blue, domain=1:8] {8/x} node[pos=0.9, right] {IQ$_1$ (100 units)}; \addplot[smooth, thick, blue, domain=2:10] {16/x} node[pos=0.9, right] {IQ$_2$ (200 units)}; \addplot[smooth, thick, blue, domain=3:12] {24/x} node[pos=0.9, right] {IQ$_3$ (300 units)}; % Ray from origin \draw[->, thick, red] (axis cs:0,0) -- (axis cs:10,10) node[above] {Scale Line}; % Points on the ray \node[circle, fill, inner sep=1.5pt, label=below left:A] at (axis cs:2,2) {}; \node[circle, fill, inner sep=1.5pt, label=below left:B] at (axis cs:4,4) {}; \node[circle, fill, inner sep=1.5pt, label=below left:C] at (axis cs:6,6) {}; % Dashed lines to axes \draw[dashed, gray] (axis cs:2,2) -- (axis cs:2,0); \node at (axis cs:2,-0.5) {L$_1$}; \draw[dashed, gray] (axis cs:2,2) -- (axis cs:0,2); \node at (axis cs:-0.5,2) {K$_1$}; \draw[dashed, gray] (axis cs:4,4) -- (axis cs:4,0); \node at (axis cs:4,-0.5) {L$_2$=2L$_1$}; \draw[dashed, gray] (axis cs:4,4) -- (axis cs:0,4); \node at (axis cs:-0.5,4) {K$_2$=2K$_1$}; \end{axis} \end{tikzpicture} \end{center} \begin{itemize} \item The diagram shows an isoquant map with three isoquants: \(IQ_1, IQ_2,\) and \(IQ_3\), representing output levels of 100, 200, and 300 units, respectively. \item The ray from the origin (0) is the "scale line," which shows that the ratio of inputs (K/L) remains constant as the firm expands its scale of production. \item At point A, the firm uses \(K_1\) units of capital and \(L_1\) units of labor to produce 100 units of output. \item When the firm doubles its inputs to \(K_2 = 2K_1\) and \(L_2 = 2L_1\), it moves to point B. As shown, point B lies on the isoquant \(IQ_2\), which represents 200 units of output. \item The key feature of CRS is that the distance between consecutive isoquants along the scale line is equal (OA = AB = BC). This equal spacing indicates that a proportional increase in inputs leads to the same proportional increase in output. \end{itemize}

Step 3: Final Answer:
In Constant Returns to Scale, output increases in the same proportion as the increase in all inputs. This is represented by an isoquant map where isoquants corresponding to equal increments in output are equally spaced.