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Essential Microeconomics -1 Essential Microeconomics -1- 1.1 SUPPORTING PRICES Production plan and production set 2 Efficient plan 6 Supporting hyperplane theorem 13 Supporting prices 22 Linear production set 20 Linear Programming problem 40 © John Riley 3 October 2012 Essential Microeconomics -2- SUPPORTING PRICES Key ideas: convex and non-convex production sets, price based incentives, Supporting Hyperplane Theorem When do prices provide incentives to “support” desired outcomes? Firm or plant: produces n different outputs qq= (1 ,..., qn )using m different inputs zz= (1 ,..., zm ). mn+ A particular production plan is then the input-output vector (,zq )∈ + Production set A firm has available some collection of alternative feasible production plans (production vectors) Let Y be the set of feasible production plans for this plant. This is the plant’s “production set.” © John Riley 3 October 2012 Essential Microeconomics -3- Production function and production set Consider a firm producing q units of a single output. Let fz() be the maximum output that a firm can produce using input vector z. Therefore output must satisfy the constraint q≤ fz(). The production set Y is the set of feasible plans so Y={(,)|(,) zq zq ≥≤ 0, q f ()} z . Example: Y={(,)|(,) zq zq ≥≤ 0, q 2 z1/2 }, equivalently, Y={( zq , ) | z ≥ 0, 4 z −≥ q2 0} © John Riley 3 October 2012 Essential Microeconomics -4- New notation: treat inputs as negative numbers 1 production vector y=( y1 ,..., ynm+ ) =−− ( z 11 ,..., zqm , ,..., q n ) Any negative component of y is an input. Any positive component is an output. The production set Y is the set of feasible production vectors. Fig.1.1-1a: Production set * © John Riley 3 October 2012 Essential Microeconomics -5- New notation: treat inputs as negative numbers 1 production vector y=( y1 ,..., ynm+ ) =−− ( z 11 ,..., zqm , ,..., q n ) Any negative component of y is an input. Any positive component is an output. The production set Y is the set of feasible production vectors. Example: Fig.1.1-1a: Production set Y={( zq , ) | z ≥ 0, 4 z −≥ q2 0}. Using the new notation, a production vector is (y12 , y )= ( − zq ,). The production set is 2 Y ={(yy12 , ) | y 1 ≤−− 0, 4 y 1 y 2 ≥ 0} This set is depicted in Fig 1.1-11b. Fig.1.1-1b: Production set __________________ 1 For a vector, our convention is that if yyjj≥ for all j we write yy≥ . If, in addition, the inequality is strict for some j we write yy> and if the inequality is strict for all j we write yy>> . © John Riley 3 October 2012 Essential Microeconomics -6- Efficient plan: A production plan y is wasteful if there is another plan in the production set for which outputs are larger and inputs are smaller. Non-wasteful plans are said to be production efficient. Formally the plan y is production efficient if there is no y ∈Y such that yy> . * © John Riley 3 October 2012 Essential Microeconomics -7- Efficient plan: A production plan y is wasteful if there is another plan in the production set for which outputs are larger and inputs are smaller. Non-wasteful plans are said to be production efficient. Formally the plan y is production efficient if there is no y ∈Y such that yy> . Profit maximization Let pp= (1 ,..., pmn+ ) be the vector of m input and n output prices. mn+ total revenue TR = ∑ pyii im= +1 m m total cost TC = ∑ pzii=∑ pyii() − . i=1 i=1 mn+ m profit π =∑∑pyi i −=⋅ pz ii p y. im=+=11i A production vector is profit maximizing if it solves Max{| p⋅∈ y y Y } y © John Riley 3 October 2012 Essential Microeconomics -8- Supporting prices When do prices provide an incentive for a firm to produce any efficient production plan? Mathematically, we seek prices that “support” an efficient production plan. Example: a production plant that uses a single input to produce a single output. The input can be purchased at a price p1. Suppose, furthermore, that the plant is part of a large firm. 0 Objective: Produce y2 units of output efficiently. For delivery to another “down-stream” plant within the firm. 0 From the figure, zy11= − units of the input is efficient. Fig.1.1-2a: Production set * © John Riley 3 October 2012 Essential Microeconomics -9- Supporting prices When do prices provide an incentive for a firm to produce any efficient production plan? Mathematically, we seek prices that “support” an efficient production plan. Example: a production plant that uses a single input to produce a single output. The input can be purchased at a price p1. Suppose, furthermore, that the plant is part of a large firm. 0 Objective: Produce y2 units of output efficiently. For delivery to another “down-stream” plant within the firm. 0 From the figure, zy11= − units of the input is efficient. Fig.1.1-2a: Production set Transfer price A price p2 , paid for each unit delivered to the downstream division. The plant manager’s bonus will be based on the profit π ()y= py11 + py 2 2. © John Riley 3 October 2012 Essential Microeconomics -10- Two contour sets (or “iso-profit” lines) of π ()y= py11 + py 2 2 are depicted in Fig. 1.1-2a. The steepness of any such line is the input-output price ratio pp12/ . As shown, the ratio is too low since profit is maximized at a point to the North-West of y0 . Fig.1.1-2a Transfer price too high However, with the transfer price lowered appropriately, as in Fig. 1.1-2b, the optimal production plan is achieved. The correct transfer price thus provides the manager with the appropriate incentive. Fig.1.1-2b: Optimal transfer price © John Riley 3 October 2012 Essential Microeconomics -11- Unfortunately, this approach does not always work. Consider Fig. 1.1-3. 0 Suppose, once again that the output target is y2 units. While y0 is locally profit maximizing, the profit 000 p⋅= y py11 + py 2 2 is negative. Profit is maximized by producing nothing. Fig.1.1-3: No optimal transfer price * © John Riley 3 October 2012 Essential Microeconomics -12- Unfortunately, this approach does not always work. Consider Fig. 1.1-3. 0 Suppose, once again that the output target is y2 units. 0 While y is locally profit maximizing, the profit 000 p⋅= y py11 + py 2 2 is negative. Profit is maximized by producing nothing. Fig.1.1-3: No optimal transfer price Convex production sets Y is convex if for any yy01, ∈Y every convex combination yλ ≡−(1λλ ) yy01 + ∈Y As we shall see, prices can be used to support all efficient production plans if the production set is convex. © John Riley 3 October 2012 Essential Microeconomics -13- Proposition 1.1-1: Supporting Hyperplane Theorem Suppose Y ⊂ n is non-empty and convex and y0 lies on the boundary of Y . Then there exists 0 0 p ≠ 0 such that (i) for all y ∈Y , py⋅≤⋅ pyand (ii) for all y ∈intY , py⋅<⋅ py. For a general proof see EM Appendix C. Here we consider the special case in which the convex set Y is an upper contour set of the function h∈1 , that is Y ={yhy | ( ) ≥ 0}. Since y0 is on the boundary of Y, hy()00 = . As long as the gradient vector is non-zero at y0 , the linear approximation of h at y0 is ∂h hy()= hy ()0 + ()( y 00 ⋅− y y ). ∂y Note that hy() and hy() have the same value and gradient at y0 © John Riley 3 October 2012 Essential Microeconomics -14- In two dimensions, the contour set of the linear approximation is the tangent plane as depicted in Fig. 1.1-4. ≥ 0 If the upper contour set Y={yhy | ( ) hy ( )} is convex as depicted, all the points in Y lie in the upper contour set of h (i.e. the lightly and heavily shaded areas.) In mathematical terms, 0∂h 00 hy()≥ hy () ⇒ ()( y ⋅− y y )0 ≥. Fig.1.1-4: Supporting hyperplane ∂y * © John Riley 3 October 2012 Essential Microeconomics -15- In two dimensions, the contour set of the linear approximation is the tangent plane as depicted in Fig. 1.1-4. 0 If the upper contour set Y={yhy | ( )≥ hy ( )} is convex as depicted, all the points in Y lie in the upper contour set of h (i.e. the lightly and heavily shaded areas.) In mathematical terms, 0∂h 00 hy()≥ hy () ⇒ ()( y ⋅− y y )0 ≥. Fig.1.1-4: Supporting hyperplane ∂y Formally, we have the following Lemma. ∂h Lemma 1.1-2: If Y ={yhy | ( ) ≥ hy (0 )} is convex then yY∈⇒(y00 )( ⋅ yy − ) ≥ 0. ∂y ∂h To prove Proposition 1.1-1, choose py= − ()0 . Appealing to the lemma, ∂y yY∈ ⇒−pyy ⋅( −0 )0 ≥ that is y∈⇒⋅≤⋅Y py py0 . © John Riley 3 October 2012 Essential Microeconomics -16- ∂h Lemma 1.1-2: If Y ={yhy | ( ) ≥ hy (0 )} is convex then yY∈⇒(y00 )( ⋅ yy − ) ≥ 0. ∂y Proof: Pick any y in Y . Since Y is convex, all convex combinations of y0 and y lie in Y . That is, for all λ ∈(0,1) , hy()−≥⇒−≥ hy ()000 hy (λ ) hy ()0 where yλ =−+(1λλ ) yy0 . ** © John Riley 3 October 2012 Essential Microeconomics -17- ∂h Lemma 1.1-2: If Y ={yhy | ( ) ≥ hy (0 )} is convex then yY∈⇒(y00 )( ⋅ yy − ) ≥ 0. ∂y Proof: Pick any y in Y . Since Y is convex, all convex combinations of y0 and y lie in Y . That is, for all λ ∈(0,1) , hy()−≥⇒−≥ hy ()000 hy (λ ) hy ()0 where yλ =−+(1λλ ) yy0 . Define g()()((1)λ≡ hyλ =− h λλ y0 += y )( hy 00 +− λ ( y y )).
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