L E A R N I N G - B Y- D O I N G E X E R C I S E 6.
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Learning-By-Doing Exercise 6.1 shows how to derive such an equation. For a production function like the ones we have been considering, where quantity of output Q depends on two inputs (quantity of labor L and quantity of capital K ), the equation of an isoquant would express K in terms of L. An isoquant can also be represented algebraically, in the form of an equation, as well as graphically (like the isoquants in Figure 6.8). Hence, isoquants Q2 and Q3, to the northeast of Q1 in Figure 6.8, correspond to larger and larger quantities of output. When both inputs have positive marginal products, using more of each input increases the amount of output attainable. Notice that points B and D along this isoquant correspond to the highlighted input combinations in Table 6.4. In Figure 6.8, isoquant Q1 corresponds to 25 units of output.
Any production function has an infinite number of isoquants, each one corresponding to a particular level of output. Such substitution is always possible whenever both labor and capital (e.g., robots) have positive marginal products. If we apply this idea to a semiconductor firm, it tells us that the firm could produce a given quantity of semiconductors using lots of workers and a small number of robots or using fewer workers and more robots. The fact that the isoquants are downward sloping in Figure 6.8 illustrates an important economic trade-off: A firm can substitute capital for labor and keep its output unchanged. Figure 6.8 shows isoquants for the production function in Table 6.4 and Figure 6.6.
The total product hill in Figure 6.6 is analogous to the three-dimensional map of Mount Hood in panel (a) of Figure 6.7, and the isoquants of the total product hill (see Figure 6.8) are analogous to the lines on the topographical map of Mount Hood in panel (b) of Figure 6.7. Topographical map shows points in geographic space at which the elevation of the land is constant. As we move to the northeast, the isoquants correspond to progressively higher outputs. Isoquants for the Production Function in Table 6.4 and Figure 6.6 Every input combination of labor and capital along the Q1 25 isoquant (in particular, combinations B and D) produces the same output, 25,000 semiconductor chips per day. ,QFOXGHV ,QWHUDFWLYH 7H[WERRN 5HVRXUFHV «VXSSRUWVLQVWUXFWRUVZLWK UHOLDEOHUHVRXUFHVWKDW UHLQIRUFH?RXUVHJRDOV LQVLGH=QGRXWVLGHRI WKH?ODVVURRP
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