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Unformatted text preview: Chapter 9 Cost Minimization Having formalized a firm’s production technology through the concept of a pro- duction set and its corresponding production function, we can now begin to an- swer specific questions firms face on a daily basis. The first, and most direct question we will address is “with a given set of input prices and a production technology, what is the cheapest way for a firm to produce q units of output?”. 9.1 Isocost Lines In order to answer this question, we need to formalize the concept of Cost. Con- sider a production process that uses x 1 units of Input Good 1 and x 2 units of Input Good 2. The price per unit of Input Good 1 is w 1 and the price per unit of Input Good 2 is w 2 . 1 Assuming Input Goods 1 and 2 are the only inputs necessary for production, the total cost, C, will satisfy the equation C = w 1 x 1 + w 2 x 2 (9.1) If we want to determine all the bundles of Input Goods that will cost exactly C , where C is a fixed cost level, we can rearrange Eqn. 9.1 to find x 2 = − w 1 w 2 x 1 + C w 2 (9.2) 1 We use the letter, w , to represent the prices of input goods necessary to produce a specific output good. We use p to represent the price for which the output good is sold to consumers (which naturally coincides with the price consumers have to pay for the good). 9.1 • ISOCOST LINES | 196 Thus, for every x 1 level, Eqn. 9.2 tells us exactly how much x 2 a firm can purchase to achieve a total cost equaling C . We define the line defined by Eqn. 9.2 as an isocost line . Definition 9.1 (Isocost Line): An Isocost Line is the set of all bundles that cause a firm to achieve a fixed cost level, C . It is worth noting the similarities between Eqn. 9.2 and the consumer’s Budget Line, Eqn. 5.2. In particular, the consumer’s Budget Line tells us all the combina- tions of x 1 and x 2 that cause an individual to spend exactly $ I . Equation 9.2 tells us all the combinations of x 1 and x 2 that cause the firm to spend exactly $ C . As the formulations of these lines are mathematically identical with the only differ- ence being a change in the variable names, it is not surprising that the properties of a budget line discussed in Chapter 5 have direct analogues to the properties of an isocost line. With x 1 on the x-axis and x 2 on the y-axis: • Eqn. 9.2 is in a “slope-intercept” form of a line, x 2 = mx 1 + b with m = − w 1 w 2 and b = C w 2 . • The y-axis intercept is equal to C w 2 . This should make intuitive sense, since the y-axis intercept of the isocost line tells us how many units of x 2 we can purchase if our total cost is C and we spend all our money on Input Good 2. Naturally, this will be the total cost divided by the price per unit of Input Good 2....
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- Fall '07
- Economics of production, Isocost Line, Input Goods