Machina Math Handout

X2 u0 and the level curves of a firms production

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Unformatted text preview: nts (defined by the equation ƒ(L, K) = Q0). The slope of a level curve is indicated by the notation: dx2 or dx1 ƒ( x , x ) = k 1 2 dx2 dx1 Δ ƒ =0 where the subscripted equations are used to remind us that x1 and x2 must vary in a manner which keeps us on the ƒ(x1, x2) = k level curve (i.e., so that Δƒ = 0). To calculate this slope, recall the vector of changes (Δx1,Δx2) will keep us on this level curve if and only if it satisfies the equation: 0 = Δƒ ≈ ƒ1(x1, x2)⋅Δx1 + ƒ2(x1, x2)⋅Δx2 which implies that Δx1 and Δx2 will accordingly satisfy: Δx2 Δx1 Econ 100A ≈− ƒ( x1 , x2 ) = k 4 ∂ ƒ( x1 , x2 ) ∂x1 ∂ ƒ( x1 , x2 ) ∂x2 Mathematical Handout so that in the limit we have: dx2 dx1 =− ƒ( x1 , x2 ) = k ∂ ƒ( x1 , x2 ) ∂x1 ∂ ƒ( x1 , x2 ) ∂x2 This slope gives the rate at which we can “trade off” or “substitute” x2 against x1 so as to leave the value of the function ƒ(x1, x2) unchanged. This concept is frequently used in economics. x2 slope = – ∂ƒ(x1,x2)/dx1 ∂ƒ(x1,x2)/dx2 ƒ(x1,x2) = k 0 x1 An application of this result is that the slope of the indifference curve at a given consumption bundle is given by the ratio of the marginal utilities of the two commodities at that bundle. Another application is that the slope of an isoquant at a given input bundle is the ratio of the marginal products of the two factors at that input bundle. In the case of a function ƒ(x1,..., xn) of several variables, we will have a level surface in n– dimensional space along which the function is constant, that is, defined by the equation ƒ(x1,..., xn) = k. In this case the level surface does not have a unique tangent line. However, we can still determine the rate at which we can trade off any pair of variables xi and xj so as to keep the value of the function constant. By exact analogy with the above derivation, this rate is given by: dxi dx j = ƒ( x1 ,..., xn ) = k dxi dx j =− Δ ƒ =0 ƒ j ( x1 ,..., xn ) ƒ i ( x1 ,..., xn ) Given any level curve (or level surface) corresponding to the value k, its better-than set is the set of all points at which the function yields a higher value than k, and its worse-than set is the set of all points at which the function yields a lower value than k. Econ 100A 5 Mathematical Handout D. SCALE PROPERTIES OF FUNCTIONS A function ƒ(x1,..., xn) is said to exhibit constant returns to scale if: ƒ(λ⋅x1,...,λ⋅xn) ≡ λ⋅ƒ(x1,..., xn) for all x1,..., xn and all λ > 0 That is, if multiplying all arguments by λ leads to the value of the function being multiplied by λ. Functions that exhibit constant returns to scale are also said to be homogeneous of degree 1. A function ƒ(x1,..., xn) is said to be scale invariant if: ƒ(λ⋅x1,...,λ⋅xn) ≡ ƒ(x1,..., xn) for all x1,..., xn and all λ > 0 In other words, if multiplying all the arguments by λ leads to no change in the value of the function. Functions that exhibit scale invariance are also said to be homog...
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