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Production and Cost (intro)

Production and Cost (intro). Production . The ideas from Consumer theory, and in particular utility functions, are used in production theory. The bundle of inputs, (z 1 , z 2 ) is used to produce output, f(z 1 , z 2 ), just as a bundle of consumption goods was used to “produce utility.”

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Production and Cost (intro)

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  1. Production and Cost (intro)

  2. Production • The ideas from Consumer theory, and in particular utility functions, are used in production theory. The bundle of inputs, (z1, z2) is used to produce output, f(z1, z2), just as a bundle of consumption goods was used to “produce utility.” • The function f represents the maximum output possible using the inputs. The terminology is somewhat different: • marginal utilities become marginal products (the partial derivatives of the production function, f(z1, z2), with respect to the inputs), • indifference curves become isoquants (input bundles with equal outputs) • the marginal rate of substitution becomes the marginal rate of technical substitution. • The special utility functions we used while doing consumer theory (especially perfect substitutes, perfect complements, and Cobb-Douglas such as f(z1, z2) = z1z2) can be used in production theory.

  3. The main difference between utility and production functions is that utility is an ordinal concept (all that matters is the ordering based on “,” and negative numbers are fine) while production is a cardinal concept (the actual numbers matter, and negatives make no sense). • The three utility functions, x1+x2, (x1+x2)2, and (x1+x2)0.5, all represent the same preferences, but z1+z2, (z1+z2)2, and (z1+z2)0.5 represent very different production functions. • In all three cases the isoquants look the same, but the output levels assigned to them differ. • For example, the input bundle (2, 2) produces 4, 16, and 2 respectively in the three different production functions.

  4. The three functions, z1+z2, (z1+z2)2, and (z1+z2)0.5 provide a simple example of the different types of returns to scale which sometimes arise in production. • Returns to scale concerns the response of output to a proportional increase in all inputs. If for any starting bundle of inputs (with a strictly positive amount of each input) and any positive number, K, a 100K% increase in every input leads to a more than, equal to, or less than 100K% increase in output then the production function exhibits increasing returns to scale, constant returns to scale, or decreasing returns to scale, respectively. • Consider the second of the three production functions. Starting from input bundle (z1, z2), a 100K% increase in each input leads to input bundle ([1+K]z1, [1+K]z2), with corresponding output • ([1+K]z1 +[1+K]z2)2 = [1+K]2(z1 + z2)2, • which is a 100( [1+K]2 - 1)% = 100(2K + K2)% increase. • Since K is positive, this is more than a 100K% increase, and the second production function exhibits increasing returns to scale. • You should verify that z1+z2 exhibits constant returns to scale and (z1+z2)0.5 exhibits decreasing returns to scale. (food for thought)

  5. A second difference between utility and production functions is that we will also be interested in short-run production functions in which one of the inputs is fixed, and cannot be changed. • For example, in the short run a firm may have a contract with its workers that requires the firm to pay the workers even when there is little work to do (i.e., a “no-layoff’ clause in the contract). • In our case with two original variables, the short-run production function is effectively a function of just one variable.

  6. Cost • Assuming that the firm acts as a price-taker in the market for inputs (i.e., it thinks it can buy as much as it wants of any of the inputs at fixed per-unit prices) we can determine its cost function. For each output level, y, and set of input prices, w1 and w2, this is the minimum cost to produce the target output level given the input prices. • The cost of the inputs is just the sum of the price per unit of each input times the amount of the input used, summed over all the different inputs. • Note that we have not explained why the firm might be interested in producing output level y. That will be done in later examinations of competitive firms and monopoly firms. The same cost function analysis will be used in both cases.

  7. In mathematical terms, with production function f(z1, z2), the (long-run) cost minimization problem is to • minimize w1z1 +w2z2 • subject to f(z1, z2) = y. • This can be solved using the Lagrange multiplier method. • The optimal choices for z1 and z2 are called the conditional factor demands, since they represent the amount of the inputs (the factors of production) that would be demanded conditional on the desire to produce output y. They are written as z1 (y, w1 w2) and z2 (y, w1 , w2) since the optimal choices depend on the target output level, y, and the input prices, w1 and w2. • The (long-run) cost function is just the cost of using the optimal inputs, • C(y,w1,w2) = w1z1 (y,w1,w2) + w2z2(y,w1,w2). • It also depends on the target output level, y, and the input prices, w1 and w2.

  8. Example: What is the lowest cost to produce y = 10 with input prices w1 = 2 and w2 = 5 when the production function is f(z1, z2) = z1z2? • The Lagrange multiplier method may be used for minimization (as well as maximization) problems with an equality constraint. The Lagrangian is • L(z1, z2, λ) = 2z1 +5z2 + λ(z1z2 -- 10). • Taking partial derivatives of L with respect to z1, z2, and λ, setting the partial derivatives equal to zero, and solving as in our previous Lagrangian problems, the conditional factor demands are • z1(10, 2, 5) = 5 • z2(10, 2, 5) = 2, • with cost • C(10, 2,5) = (2)(5) + (5)(2) = 20.

  9. Compare this optimization problem to the utility maximization problem for the consumer. Graphically, the utility problem starts with a fixed, straight-line budget, and seeks the highest indifference curve that still touches the budget. • The cost problem starts with a fixed isoquant (the analog of the indifference curve) and seeks the lowest (since it is minimization) straight-line iso-cost curve (i.e., budget line of equal expenditure on inputs) that still touches the isoquant (graph it). • In the utility problem, if the optimal bundle has strictly positive amounts of each of the goods, and the indifference curve is smooth, then at the optimal bundle the budget holds and u1/u2 = MRS= p1/ p2(i.e., the ratio of marginal utilities equals the price ratio). • In the cost problem, if the optimal bundle has strictly positive amounts of each of the inputs, and the isoquant is smooth, then at the optimal input bundle the target output is produced and f1/f2 = MRTS = w1/w2(i.e., the ratio of marginal products equals the input price ratio).

  10. In the example we found the values of the conditional factor demands and the cost function for specific values of the output and the input prices. • In order to examine competitive firms and monopolies we must find the conditional factor demand functions and the cost function, at least as functions of the target output level.

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