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Unformatted text preview: Firm’s Problem Simon Board * This Version: September 20, 2009 First Version: December, 2009. In these notes we address the firm’s problem. We can break the firm’s problem into three questions. 1. Which combinations of inputs produce a given level of output? 2. Given input prices, what is the cheapest way to attain a certain output? 3. Given output prices, how much output should the firm produce? We study the firm’s technology in Sections 1–2, the cost minimisation problem in Section 3 and the profit maximisation problem in Section 4. 1 Technology 1.1 Model We model a firm as a production function that turns inputs into outputs. We assume: 1. The firm produces a single output q ∈ < + . One can generalise the model to allow for firms which make multiple products, but this is beyond this course. * Department of Economics, UCLA. http://www.econ.ucla.edu/sboard/. Please email suggestions and typos to sboard@econ.ucla.edu. 1 Eco11, Fall 2009 Simon Board 2. The firm has N possible inputs, { z 1 ,...,z N } , where z i ∈ < + for each i . We normally assume N = 2, but nothing depends on this. We can think of inputs as labour, capital or raw materials. 3. Inputs are mapped into output by a production function q = f ( z 1 ,z 2 ). This is normally assumed to be concave and monotone. We discuss these properties later. To illustrate the model, we can consider a farmer’s technology. In this case, the output is the farmer’s produce (e.g. corn) while the inputs are labour and capital (i.e. machinery). There is clearly a tradeoff between these two inputs: in the developing world, farmers use little capital, doing many tasks by hand; in the developed world, farmers use large machines to plant seeds and even pick fruit. In some examples inputs may be close substitutes. To illustrate, suppose two students are working on a homework. In this case the output equals the number of problems solved, while the inputs are the hours of the two students. The inputs are close substitutes if all that matters is the total number of hours worked (see Section 2.3). In other cases inputs may be complements. To illustrate, suppose an MBA and a computer engineer are setting up a company. Each worker has speed skills and neither can do the other’s job. In this case, output depends on which worker is doing the least work, and we say the inputs are perfect compliments (see Section 2.2). The marginal product of input z i is the output from one extra unit of good i . MP i ( z 1 ,z 2 ) = ∂f ( z 1 ,z 2 ) ∂z i , The average product of input i is AP i ( z 1 ,z 2 ) = f ( z 1 ,z 2 ) z i . 1.2 Isoquants An isoquant describes the combinations of inputs that produce a constant level of output....
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