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Production Functions, , [See Chap 9], 1, , 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 inputs., q = f(z1, … , zN), • Examples (with N=2):, – z1 = capital, z2 = labor., – z1 = skilled labor, z2 = unskilled labor, – z1 = capital, z2 = land., , 2, , Marginal Product, • The marginal product is the additional output, that can be produced by employing one more, unit of the input while holding other inputs, constant:, marginal product of z1 = MP1 =, , ∂q, = f1, ∂z1, , 3, , 1
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Isoquants, • Each isoquant represents a different level, of output., z2, , q = 30, q = 20, , z1, , 7, , Marginal Rate of Technical Substitution, • The slope of an isoquant shows the rate at, which z2 can be substituted for z1, • MRTS = number of z2 the firm gives up to get 1, unit of z1, if she wishes to hold output constant., z2, , z2*, , - slope = marginal rate of technical, substitution (MRTS), A, B, , z2, , In picture, MRTS is positive, and is diminishing for, increasing inputs of labor, q = 20, , Z1*, , z1, , z1, , 8, , MRTS, • The marginal rate of technical, substitution (MRTS) shows the rate at, which labor can be substituted for, capital while holding output constant, along an isoquant, MRTS = −, , dz 2, dz1, , q = q0, 9, , 3
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2. Quasi-Concavity, • Suppose z=(z1,z2) and z’=(z1’,z2’) are two, input bundles., • f(.) is quasi-concave in z if whenever, f(z)≥f(z’) then, f(tz+(1-t)z’)≥f(z’) 1>t>0., • Implications, – Isoquants are convex., – MRTS decreases in z1, as move along, isoquant., , 13, , 3. Concavity, • Suppose z=(z1,z2) and z’=(z1’,z2’) are two input, bundles., • f(.) is concave in z if, f(tz+(1-t)z’) ≥ tf(z) + (1-t)f(z’) 1>t>0., • Implies quasi-concavity (convex isoquants)., • Implies diminishing marginal productivity:, ∂MP1 ∂ 2 f, = 2 = f11 ≤ 0, ∂z1, ∂z1, , ∂MP2 ∂ 2 f, =, = f 22 ≤ 0, 2, ∂z 2, ∂z 2, , • Implies constant or decreasing returns to scale., 14, , 4. Returns to Scale, • How does output respond to increases, in all inputs together?, – suppose that all inputs are doubled, would, output double?, , • The effect of a proportional change in all, inputs on output is called the returns to, scale, 15, , 5
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Returns to Scale, • If the production function is given by q =, f(z1,z2) and all inputs are multiplied by the, same positive constant (t >1), then, Effect on Output Returns to Scale, f(tz1,t z2) = tf(z1,z2), , Constant, , f(tz1,t z2) < tf(z1,z2), , Decreasing, , f(tz1,t z2) > tf(z1,z2), , Increasing, 16, , Returns to Scale, • Why should there ever be DRS?, – If expand all inputs then shouldn’t output at, least double (just recreate what firm was, doing before)., – May be able to do better due to, specialization (leading to IRS)., , • DRS can be seen as coming from, omitted factor of production. For, example, limited management time., 17, , EXAMPLES, , 18, , 6
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Technical Progress, • Suppose that the production function is, q = A(t)f(z1, z2), , where A(t) represents all factors that, affect the production of q other than z1, and z2, – Changes in A over time represent technical, progress, – We would imagine that dA/dt > 0, 28, , 10
