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ECON 377/477,[object Object]
Topic 1.2,[object Object],Production functions,[object Object]
Outline,[object Object],Introduction,[object Object],Properties,[object Object],Quantities of interest,[object Object],An example,[object Object],Short-run production functions,[object Object],Transformation functions,[object Object],3,[object Object],ECON377/477 Topic 1.2,[object Object]
Introduction,[object Object],Consider a firm that uses amounts of N inputs to produce a single output,[object Object],The technological possibilities of such a firm can be summarised using the production function:,[object Object],q = f(x),[object Object],	where q is output and x = (x1, x2, …, xN)׳ is an Nx1 vector of inputs,[object Object],We assume these inputs are under the control of the decision maker,[object Object],4,[object Object],ECON377/477 Topic 1.2,[object Object]
Properties,[object Object],Non-negativity: the value of f(x) is a finite, non-negative, real number,[object Object],Weak essentiality: the production of positive output is impossible without the use of at least one input,[object Object],Non-decreasing in x (monotonicity): additional units of an input will not decrease output , i.e. if x0≥ x1, then f(x0)≥ f(x1),[object Object],ECON377/477 Topic 1.2,[object Object],5,[object Object]
Properties,[object Object],Concave in x: Any linear combination of the vectors x0 and x1 will produce an output that is no less than the same linear combination of f(x0) and f(x1),[object Object],	Formally, f(θx0) + (1 – θ)x1 ≥ θ f(x0) + (1 – θ)f(x1),[object Object],	If the production function is continuously differentiable, concavity implies that all marginal products (MPs) are non-increasing (the law of diminishing marginal productivity),[object Object],6,[object Object],ECON377/477 Topic 1.2,[object Object]
Properties,[object Object],The diagram on the next slide depicts a production function defined over a single input, x,[object Object],The values of q are all non-negative and finite real numbers for the values of x represented on the horizontal axis (non-negativity),[object Object],The function passes through the origin (weak essentiality),[object Object],The MP of x is positive at all points between the origin and point G (monotonicity) but monotonicity is violated on the curved segment GR,[object Object],7,[object Object],ECON377/477 Topic 1.2,[object Object]
q,[object Object],MP at G = 0,[object Object],G,[object Object],E,[object Object],AP at E is the slope of the ray through the origin and E,[object Object],R,[object Object],q = f(x),[object Object],Point of optimal scale,[object Object],D,[object Object],x,[object Object],0,[object Object],Concavity violated,[object Object],Feasible region,[object Object],Monotonicity violated,[object Object]
Properties,[object Object],As we move along the production function from the origin to point D, MPx increases,[object Object],Thus, the concavity property is violated at these points,[object Object],But concavity is satisfied at all points on the curve segment DR,[object Object],9,[object Object],ECON377/477 Topic 1.2,[object Object]
Properties,[object Object],Extending the graphical analysis to the multiple-input case is difficult,[object Object],In such cases it is common practice to plot the relationship between two of the variables while holding all others fixed,[object Object],We now consider a two-input production function and plot the relationship between the inputs x1 and x2 while holding output fixed at the values q0,[object Object],10,[object Object],ECON377/477 Topic 1.2,[object Object]
Properties,[object Object],We also plot the relationship between the two inputs when output is fixed at the values q1 and q2, where q2 > q1 > q0 (output isoquants), shown in the diagram on the next slide ,[object Object],If properties are satisfied, these isoquants are non-intersecting functions that are convex to the origin, as depicted ,[object Object],The slope of the isoquant is the marginal rate of technical substitution (MRTS) that measures the rate at which x1 must be substituted for x2 in order to keep output at its fixed level,[object Object],11,[object Object],ECON377/477 Topic 1.2,[object Object]
x2,[object Object],F,[object Object],F(x1,x2) = q2,[object Object],MRTS at F = slope of the isoquant at F,[object Object],F(x1,x2) = q1,[object Object],F(x1,x2) = q0,[object Object],x1,[object Object],0,[object Object],12,[object Object],ECON377/477 Topic 1.2,[object Object]
Properties,[object Object],An alternative representation of a two-input production function is provided in the diagram on the next slide,[object Object],The lowest of the four functions plots the relationship between q and x1:,[object Object],The value of x2 is held fixed,[object Object],The other functions plot the relationship between q and x1 when x2 is fixed at different values,[object Object],13,[object Object],ECON377/477 Topic 1.2,[object Object]
q,[object Object],0,[object Object],x1,[object Object],14,[object Object],ECON377/477 Topic 1.2,[object Object]
Quantities of interest,[object Object],If the production function is twice-continuously differentiable we can use calculus to define a number of economic quantities of interest,[object Object],For example, two quantities we have already encountered are the MP and the MRTS,[object Object],Related concepts that do not depend on units of measurement are the output elasticity and the direct elasticity of substitution, which is usually denoted as σin the two-input case,[object Object],15,[object Object],ECON377/477 Topic 1.2,[object Object]
Quantities of interest,[object Object],In the next slide, an infinitesimal movement from one side of point A to the other results in an infinitesimal change in the input ratio but an infinitely large change in the MRTS, implying that σ = 0,[object Object],Thus, in the case of a right-angled isoquant, an efficient firm must use its inputs in fixed proportions,[object Object],That is, no input substitution is possible,[object Object],ECON377/477 Topic 1.2,[object Object],16,[object Object]
x2,[object Object],σ = 0,[object Object],A,[object Object],x1,[object Object],0,[object Object],17,[object Object],ECON377/477 Topic 1.2,[object Object]
Quantities of interest,[object Object],In the next slide, a movement from D to E results in a large percentage change in the input ratio but leaves the MRTS unchanged,[object Object],This result implies that the isoquant is a straight line and inputs are perfect substitutes,[object Object],18,[object Object],ECON377/477 Topic 1.2,[object Object]
x2,[object Object],D,[object Object],σ = infinity,[object Object],E,[object Object],x1,[object Object],0,[object Object],19,[object Object],ECON377/477 Topic 1.2,[object Object]
Quantities of interest,[object Object],In the next slide, an intermediate (and more common) case is depicted where σ lies somewhere between zero and infinity,[object Object],20,[object Object],ECON377/477 Topic 1.2,[object Object]
x2,[object Object],σ lies between zero and infinity,[object Object],x1,[object Object],0,[object Object],21,[object Object],ECON377/477 Topic 1.2,[object Object]
Quantities of interest,[object Object],In the multiple-input case it is possible to define at least two other elasticities of substitution: the Allen partial elasticity of substitution (AES) and the Morishima elasticity of substitution (MES),[object Object],The DES is sometimes regarded as a short-run elasticity because it measures substitutability between xn and xm while holding all other inputs fixed,[object Object],22,[object Object],ECON377/477 Topic 1.2,[object Object]
Quantities of interest,[object Object],Economists use the term, short-run, to refer to time horizons so short that at least one input is fixed,[object Object],The AES and MES are long-run elasticities because they allow all inputs to vary,[object Object],When there are only two inputs, DES = AES,[object Object],23,[object Object],ECON377/477 Topic 1.2,[object Object]
Quantities of interest,[object Object],The MP measures the output response when one input is varied and all other inputs are held fixed,[object Object],But we are often interested in measuring output response when all inputs are varied simultaneously,[object Object],If a proportionate increase in all inputs results in a less than proportionate increase in output, then we say the production function exhibits decreasing returns to scale (DRS),[object Object],24,[object Object],ECON377/477 Topic 1.2,[object Object]
Quantities of interest,[object Object],If a proportionate increase in inputs results in the same proportionate increase in output, the production function is said to exhibit constant returns to scale (CRS),[object Object],If a proportionate increase inputs leads to a more than proportionate increase in output the production function exhibits increasing returns to scale (IRS),[object Object],There are many reasons why firms may experience different returns to scale,[object Object],25,[object Object],ECON377/477 Topic 1.2,[object Object]
Quantities of interest,[object Object],A widely used measure of returns to scale is the elasticity of scale or total elasticity of production,[object Object],The production function exhibits locally DRS, CRS or IRS as the elasticity of scale is less than, equal to or greater than 1,[object Object],For the Cobb-Douglas production function defined over N inputs:,[object Object],q = ax1β1x2β2 … xNβN,[object Object],The output elasticities are En = Σβnfor n = 1, …, N,[object Object],26,[object Object],ECON377/477 Topic 1.2,[object Object]
An example,[object Object],Refer to pages 18-19 for an example illustrating the computation of MPs and elasticities in a two-input Cobb-Douglas production function,[object Object],27,[object Object],ECON377/477 Topic 1.2,[object Object]
Short-run production functions,[object Object],Short-run production functions are obtained by holding one or more inputs fixed,[object Object],Consider equation (2.10) on page 18 of CROB:,[object Object],q = 2x10.5x20.4,[object Object],If x2 were fixed at 100 in the short run, the short-run production function would be:,[object Object],q = 2x10.51000.4, = 12.619x10.5,[object Object],This function is depicted in the diagram on the next slide, along with another function based on the assumption that x2 is fixed at 150, when:,[object Object],q = 14.841x10.5,[object Object],28,[object Object],ECON377/477 Topic 1.2,[object Object]
Short-run production functions,[object Object],q = 2x10.51500.4, = 14.841x10.5,[object Object],q = 2x10.51000.4, = 12.619x10.5,[object Object],29,[object Object],ECON377/477 Topic 1.2,[object Object]
Short-run production functions,[object Object],A family of short-run production functions could be constructed in this way, each of which satisfies the four properties outlined above,[object Object],As a group, this family could be viewed as a long-run production function because it depicts the production possibilities of the firm when both inputs vary,[object Object],30,[object Object],ECON377/477 Topic 1.2,[object Object]
Transformation functions,[object Object],The production function concept can be generalised to more than one output,[object Object],The technological possibilities of a firm that uses N inputs to produce M outputs can be summarised by the transformation function:,[object Object],T(x,q) = 0,[object Object],	where q = (q1, q2, …, qM)׳ is an Mx1 vector of outputs,[object Object],Transformation functions are special cases of distance functions, which are discussed in detail later,[object Object],31,[object Object],ECON377/477 Topic 1.2,[object Object]

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