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Machina Math Handout

Leads to no change in the value of the function

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Unformatted text preview: eneous of degree 0. Say that ƒ(x1,..., xn) is homogeneous of degree one, so that we have ƒ(λ⋅x1, ...,λ⋅xn) ≡ λ⋅ƒ(x1,..., xn). Differentiating this identity with respect to λ yields: n ∑ ƒ (λ ⋅x ,..., λ ⋅x ) ⋅ x i =1 i n 1 ≡ ƒ( x1 ,..., xn ) i for all x1 ,..., xn , all λ > 0 ≡ ƒ( x1 ,..., xn ) for all x1 ,..., xn and setting λ = 1 then gives: n ∑ ƒ (x ,..., x ) ⋅ x i =1 i 1 n i which is called Euler’s theorem, and which will turn out to have very important implications for the distribution of income among factors of production. Here’s another useful result: if a function is homogeneous of degree 1, then its partial derivatives are all homogeneous of degree 0. To see this, take the identity ƒ(λ⋅x1,...,λ⋅xn) ≡ λ⋅ƒ(x1,..., xn) and this time differentiate with respect to xi, to get: λ⋅ƒi(λ⋅x1,...,λ⋅xn) ≡ λ⋅ƒi(x1,..., xn) or equivalently: for all x1,..., xn and λ > 0 ƒi(λ⋅x1,...,λ⋅xn) ≡ ƒi(x1,..., xn) for all x1,..., xn and λ > 0 which establishes our result. In other words, if a production function exhibits constant returns to scale (i.e., is homogeneous of degree 1), the marginal products of all the factors will be scale invariant (i.e., homogeneous of degree 0). E. OPTIMIZATION #1: SOLVING OPTIMIZATION PROBLEMS The General Structure of Optimization Problems Economics is full of optimization (maximization or minimization) problems: the maximization of utility, the minimization of expenditure, the minimization of cost, the maximization of profits, etc. Understanding these is a lot easier if one knows what is systematic about such problems. Each optimization problem has an objective function ƒ(x1,..., xn;α1,...,αm) which we are trying to either maximize or minimize (in our examples, we’ll always be maximizing). This function depends upon both the control variables x1,..., xn which we (or the economic agent) are able to Econ 100A 6 Mathematical Handout set, as well as some parameters α1,...,αm, which are given as part of the problem. Thus a general unconstrained maximization problem takes the form: max ƒ( x1 ,..., xn ; α1 ,..., α m ) x1 ,..., xn Consider the following one-parameter maximization problem max ƒ( x1 ,..., xn ; α ) x1 ,..., xn (It’s only for simplicity that we assume just one parameter. All of our results will apply to the general case of many parameters α1,...,αm.) We represent the solutions to this problem, which obviously depend upon the values of the parameter(s), by the n solution functions: ∗ x1 ∗ x2 = = ∗ x1 (α ) ∗ x2 (α ) ∗ xn = ∗ xn (α ) It is often useful to ask “how well have we done?” or in other words, “how high can we get ƒ(x1,..., xn;α), given the value of the parameter α?” This is obviously determined by substituting in the optimal solutions back into the objective function, to obtain: ∗ ∗ ∗ ∗ φ (α ) ≡ ƒ( x1 ,..., xn ;α ) ≡ ƒ( x1 (α ),..., xn (α );α ) and φ(α) is called the...
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