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Part 3 PRODUCTION AND SUPPLY PRODUCTION FUNCTIONS COST FUNCTION PROFIT MAXINIZATION

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## Part 3 PRODUCTION AND SUPPLY PRODUCTION FUNCTIONS COST FUNCTION PROFIT MAXINIZATION

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**Part 3 PRODUCTION AND SUPPLY**• PRODUCTION FUNCTIONS • COST FUNCTION • PROFIT MAXINIZATION**Chapter 7**PRODUCTION FUNCTIONS**Contents**• Marginal productivity • Isoquant Maps and the rate of technical substitution • Returns to Scale • The elasticity of substitution • Four simple production function • Technical progress**Production Function**• The firm’s production function for a particular good (q) shows the maximum amount of the good that can be produced using alternative combinations of capital (k) and labor (l) q = f(k,l)**Average Physical Product**• Labor productivity is often measured by average productivity • Note that APl also depends on the amount of capital employed**Marginal Physical Product**• To study variation in a single input, we define marginal physical product as the additional output that can be produced by employing one more unit of that input while holding other inputs constant**Diminishing Marginal Productivity**• The marginal physical product of an input depends on how much of that input is used • In general, we assume diminishing marginal productivity**Diminishing Marginal Productivity**• Because of diminishing marginal productivity, 19th century economist Thomas Malthus worried about the effect of population growth on labor productivity • But changes in the marginal productivity of labor over time also depend on changes in other inputs such as capital • we need to consider flk which is often > 0**Isoquant Maps**• To illustrate the possible substitution of one input for another, we use an isoquant map • An isoquant shows those combinations of k and l that can produce a given level of output (q0) f(k,l) = q0**Each isoquant represents a different level of output**• output rises as we move northeast q = 30 q = 20 Isoquant Map k per period l per period**Isoquant Maps**Production Function : Multiple Inputs**The slope of an isoquant shows the rate at which l can be**substituted for k A kA B kB lA lB Marginal Rate of Technical Substitution (RTS) k per period - slope = marginal rate of technical substitution (RTS) RTS > 0 and is diminishing for increasing inputs of labor q = 20 l per period**Marginal Rate of Technical Substitution (RTS)**• The marginal rate of technical substitution (RTS) shows the rate at which labor can be substituted for capital while holding output constant along an isoquant**RTS and Marginal Productivities**• Take the total differential of the production function: • Along an isoquant dq = 0, so**RTS and Marginal Productivities**• Because MPl and MPk will both be nonnegative, RTS will be positive (or zero) • However, it is generally not possible to derive a diminishing RTSfrom the assumption of diminishing marginal productivity alone**RTS and Marginal Productivities**• To show that isoquants are convex, we would like to show that d(RTS)/dl < 0 • Since RTS = fl/fk**RTS and Marginal Productivities**• Using the fact that dk/dl = -fl/fk along an isoquant and Young’s theorem (fkl = flk) • Because we have assumed fk > 0, the denominator is positive • Because fll and fkk are both assumed to be negative, the ratio will be negative if fkl is positive quasi-concave function**RTS and Marginal Productivities**• Intuitively, it seems reasonable that fkl= flk should be positive • if workers have more capital, they will be more productive • But some production functions have fkl < 0 over some input ranges • when we assume diminishing RTS we are assuming that MPl and MPk diminish quickly enough to compensate for any possible negative cross-productivity effects**If the production function is given by q = f(k,l) and all**inputs are multiplied by the same positive constant (t >1), then Returns to Scale**Returns to Scale**• It is possible for a production function to exhibit constant returns to scale for some levels of input usage and increasing or decreasing returns for other levels • economists refer to the degree of returns to scale with the implicit notion that only a fairly narrow range of variation in input usage and the related level of output is being considered**Constant Returns to Scale**Labor(hours)**Constant Returns to Scale**• Constant returns-to-scale production functions are homogeneous of degree one in inputs f(tk,tl) = t1f(k,l) = tq • This implies that the marginal productivity functions are homogeneous of degree zero • if a function is homogeneous of degree k, its derivatives are homogeneous of degree k-1**Constant Returns to Scale**• The marginal productivity of any input depends on the ratio of capital and labor (not on the absolute levels of these inputs) • The RTS between k and l depends only on the ratio of k to l, not the scale of operation**The isoquants are equally**spaced as output expands q = 3 q = 2 q = 1 Constant Returns to Scale • The production function will be homothetic • Geometrically, all of the isoquants are radial expansions of one another • Along a ray from the origin (constant k/l), the RTS will be the same on all isoquants k per period l per period**homothetic production function**• The IRS and DRS also have a homothetic difference indifference curve , because the property of homotheticity is retained by any monotonic transformation of homogeneous function