John Riley 8 February 7 CHAPTER 3: OPTIMIZATION 3. TWO VARIABLES 8 Second Order Conditions Implicit Function Theorem 3. UNCONSTRAINED OPTIMIZATION 4 Necessary and Sufficient Conditions 3.3 CONSTRAINED OPTIMIZATION- an intuitive approach 4 FOC for a Constrained Maimum Sufficient Conditions 3.4 THE ENVELOPE THEOREM 8 4 pages
John Riley March 7 3. TWO VARIABLES Rather than leap directly to the analysis of multi-variable optimization problems we begin by eamining the two variable case. Maimization Suppose that the function f takes on its maimum over R at = (, ). Taking one variable at a time, we know that each of the first partial derivatives must be zero and each of the second partial derivatives must be negative. To obtain a further necessary condition we consider the change in f as changes from.we do this by considering the weighted average λ of λ = ( λ) + λ = + λ( ), in the direction of some other vector and, that is and define g( λ) = f ( + λ( ). This function is depicted below. f g( λ ) O λ ( b, b ) Fig. 3.-: Cross-section of f The mapping, g( λ ), depicted in the cross-section is a function from R into R. Then we can appeal to the necessary conditions for one-variable maimization. In particular, for a maimum at derivative of g is, the second derivative of g( λ) must be negative at λ =. The first Section 3. page
John Riley March 7 g ( λ) = ( ) z + ( ) z, where λ λ z. For a maimum this must be zero for all z, hence the partial derivatives must both be zero. Differentiating again, f λ f λ f λ g ( λ) = z ( ) + z z ( ) + z ( ) = [ z z ] f f z z f f Note that the right hand side of the last equation is a quadratic form. Appealing to Proposition.- we have the following result. Proposition 3.-: Second Order Conditions for a Maimum If f (, ) takes on its maimum over R at, then f f f f ( i) ( ), i =, and ( ii) ( ) ( ) ( ( )). i Implicit Function Rule Consider an economy where inputs can be used to produce commodity (guns) or commodity (roses). For any level of gun production, the maimum output of roses is ( ) 4 = f =. This is depicted below. Consider the point = (,3). Differentiating the function f ( ), we can determine the opportunity cost of additional gun production, in terms of roses foregone. = = at (,3). Section 3. page
John Riley March 7 roses = (,3) Fig. 3.-: Guns-roses trade-off guns Note also that the higher the production of guns, the greater the opportunity cost in terms of roses. constraint More generally, we can represent the feasible outputs of guns and roses with the g(, ) b where the derivatives of g are strictly positive. For any, let = h( ) be the maimum output of roses. That is, g(, h( )) = b. Differentiating by, Therefore dg g g = + = g dh = =. g. This is known as the Implicit Function Rule. For eample, suppose 4 3 g(, ) = + + + and b = 33. Since g (, 3) = 33, the point (,3) lies on the boundary curve. Both partial derivatives are Section 3. page 3
John Riley March 7 positive for all so the constraint g( ) the Implicit Function Rule, g 3 4 + 4 = = g + 3. = b implicitly defines a function h( ). By Thus the opportunity cost at (,3) is /7. Note that even without solving eplicitly, we can learn a lot about the function from its slope. As increases, declines. Thus the numerator increases and the denominator declines. Thus as increases the slope becomes more negative and so the opportunity cost rises. Qualitatively, the guns-roses trade-off is again as depicted in Fig. 3.-. Rule applies. The following proposition provides conditions under which the Implicit Function Proposition 3.-: Implicit Function Theorem If the partial derivatives of g are continuous in some neighborhood of g ( ) then the slope of the function = h( ) implied by the equation g(, ) g g = b is = /. = (, ) and Constrained Maimization As an application of the Implicit Function Rule consider the following maimization problem. Ma{ f ( ) g( ) b, } We will assume that the partial derivatives of f and g are both strictly positive. Since the sum of conve functions is conve, g( ) is conve, thus the function f ( ) = b g( ) is concave. It follows that the upper contour set f ( ) is conve as depicted. Section 3. page 4
John Riley March 7 A special case is the standard commodity consumer choice problem. The consumer has a utility function f ( ) and must pick a point satisfying the budget constraint, p + p I. The feasible set is depicted below, along with the sets of points satisfying f ( ) = f ( ) and f ( ) = f ( ). f ( ) f ( ) g( ) b f ( ) = f ( ) As depicted, is optimal since for all other feasible points, f ( ) < f ( ). Let h( ) be the maimum feasible, for any given. This is depicted above along with level curves for f. Suppose that Fig. 3.-3: Constrained maimum, the maimizing value of, is not at a corner. The set of points satisfying, f ( ) = f ( ) also implicitly defines a function l( ). Graphically, at the maimum, the slopes of the two implicit functions must be equal. Appealing to the Implicit Function Theorem, we have the following first order condition. g dl dh = ( ) = ( ) = g (3.-) Formally, since f is increasing, for any it is optimal to choose the maimum feasible, that is, = h( ). Substituting for in f, we can rewrite the maimization problem as follows. Section 3. page 5
John Riley March 7 Ma{ f (, h( ))}. Differentiating the maimand by, df f dh = +. Appealing to the Implicit Function Theorem, g g df = = ( ) g f g Assuming that the maimizing value of is not at a corner, the FOC, be satisfied. Hence condition (3.-) must hold. g( ) df ( ) =, must In the special case of consumer maimization subect to a budget constraint, = p so the first order condition becomes p =. p If the consumer purchases one more unit of commodity he increases spending by p. He must then reduce his consumption of commodity by p / p units. This is the market trade-off. Also and is the utility he gains if he consumes one more of commodity is the utility he loses if he consumes one less unit of commodity. Thus the ratio is the number of units of commodity he is willing to sacrifice in order to consume one additional unit of commodity. At the maimum this willingness to substitute or marginal rate of substitution must be equal to the market trade-off. Alternatively, the first order condition can be rewritten as follows. = p p Section 3. page 6
John Riley March 7 The term ( ) p is the number of additional units that can be purchased if an additional dollar is spent on commodity and is the marginal benefit of an additional unit. Thus the consumer equates the marginal utility per dollar on each commodity that he consumes. Eercise 3.-: Consumer Choice Bev faces prices p and p and her income is I. Her utility function U (, ) satisfies U ( ) >, =,, R +. (a) Etend the argument in this section to obtain necessary conditions for a corner solution. (b) Suppose that U lim ( ) =, =,, that is, the marginal utility of each commodity increases without bound as consumption of the commodity declines to zero. Show that the necessary conditions for a maimum cannot be satisfied at a corner. (c) If U ( ) = α ln, where α >, =,..., n, show that the solution cannot be at a corner. = (d) Hence solve for Bev s optimal choice. (e) What if instead U ( ) =? α α Eercise 3.-: Firm with interdependent demands A firm sells two products. Demand prices for these products are as follows. p = 8 q q, p = 8 q q The cost of production is C( q) = q + αq q + q. (a) If α =, show that the profit function is concave and solve for the outputs that satisfy the first order conditions. Section 3. page 7
John Riley March 7 (b) Show that the first order conditions for profit maimization are satisfied at q = (, ) if α = 4. (c) Are the second order (necessary) conditions satisfied at q = (, )? (f) Show that there are two corner solutions to the FOC. Eplain why profit must be maimized at these outputs. (e) Show that, fiing q, profit is a concave function of q and hence solve for the profit maimizing output of commodity. (d) Substitute for q and hence epress profit as a function of q, q π q q q π ( ) = ( ( ), ). Solve for the profit maimizing output a neat figure. q and hence q. Depict the function q ( q ), q π ( ) in Eercise 3.-3: Quality and quantity choice A firm that produces q units of quality z commands a demand price p( q, z) = 8z q. The cost of production is C( q, z) = z α q, where z. (a) If α =, fi q and write down the first order condition for a profit maimum and solve for the profit maimizing quality as a function of q hence solve for the profit maimizing output and quality. (b) Confirm that the first order conditions hold at is concave in some neighborhood of this point. Thus maimum. ( q, z ) = (4,4) and show that profit ( q, z ) = (4,4) is a local (c) Show that, for any output, it is optimal to choose a quality level of 4. Hence or otherwise, eplain why the local maimum must also be the (global) maimum. (d) Re-eamine the problem if α =. Hint: If you like spread-sheet analysis you could first plot the graph of the profit function. Section 3. page 8