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Multivariable Optimization

Lecture 5. Multivariable Optimization. 1 . Necessary Conditions. Optimization problems: max x1,x2 y = f(x 1 ,x 2 ) min x1,x2 y = f(x 1 ,x 2 ). E.g., profit maximization, cost minimization. When x 2 is kept constant, f(x 1 , c) is a function of x 1 only.

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Multivariable Optimization

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  1. Lecture 5 Multivariable Optimization ECON 1150, Spring 2013

  2. 1. Necessary Conditions Optimization problems: maxx1,x2 y = f(x1,x2) minx1,x2 y = f(x1,x2) E.g., profit maximization, cost minimization ECON 1150, Spring 2013

  3. When x2 is kept constant, f(x1, c) is a function of x1 only. When x1 is kept constant, f(c, x2) is a function of x2 only First-order conditions for a stationary point ECON 1150, Spring 2013

  4. Example 5.1: Find stationary values of the following functions: (a) y = 2x1²+x2²; (b) y = 4x1²-x1x2+x2²-x1³. ECON 1150, Spring 2013

  5. Possible cases of stationary points ECON 1150, Spring 2013

  6. 2. Sufficient Conditions Second-order conditions for a local maximum at a point (SOC max) f11< 0; f11·f22 – (f12)2> 0 Second-order conditions for a local minimum at a point (SOC min) f11> 0; f11·f22 – (f12)2> 0. ECON 1150, Spring 2013

  7. Classifying a stationary point • If f1(x1*,x2*) = f2(x1*,x2*) = 0, then • a. SOC max  local maximum • SOC min  local minimum • f11f22 – (f12)2 = 0  no conclusion • f11f22 – (f12)2 < 0  saddle point ECON 1150, Spring 2013

  8. Example 5.2: Identify the nature of the stationary points of the following function, y=f(x1,x2)=x1³+5x1x2–x2². ECON 1150, Spring 2013

  9. Example 5.3: Identify the nature of the stationary points of the following functions: • f(x1,x2) = x12 + x24; • f(x1,x2) = x12 – x24; • f(x1,x2) = x12 + x23; • f(x1,x2) = x14 + x24; ECON 1150, Spring 2013

  10. If for all (x1,x2), f11 0 and f11f22 – (f12)2  0. then f(x1,x2) is a concave function. • Example 5.4: Concave functions • y = x1 + x2; • y = x10.4x20.6 for positive x1 and x2; • y = lnx1 + lnx2. ECON 1150, Spring 2013

  11. If for all (x1,x2), f11 0 and f11f22 – (f12)2  0. then f(x1,x2) is a convex function. • Example 5.5: Convex functions • y = x1 + x2; • y = 3x12 + 3x1x2 + x22. ECON 1150, Spring 2013

  12. For a stationary point (x10, x20), concave function  global maximum convex function  global minimum Example 5.6: Show that the function f(x1,x2) = -2x12 – 2x1x2 – 2x22 + 36x1 + 42x2 – 158 is a concave function. Find the global maximum point. ECON 1150, Spring 2013

  13. Remarks: a. 2nd order conditions hold at a point  Local extremum b. 2nd order conditions hold for all points  Global extremum c. SOC are sufficient, but not necessary ECON 1150, Spring 2013

  14. 3. Economic Applications A firm produces 2 different kinds A and B of a commodity. The daily cost of producing x units of A and y units of B is C(x,y) = 0.04x2 + 0.01xy + 0.01y2 + 4x + 2y + 500. Suppose that that firm sells all its output at a price per unit of 15 for A and 9 for B. Find the daily production levels x* and y* that maximize profit per day. ECON 1150, Spring 2013

  15. Long-run profit maximization of a competitive firm Output price: 200 Inputs: K with a price of 42 L with a price of 5 Production function: 3.1K0.3L0.25. ECON 1150, Spring 2013

  16. General profit maximization problem maxK,L(K,L) = pf(K,L) – wKK – wLL FOC: K(K,L) = pMPK – wK = 0 L(K,L) = pMPL – wL = 0 • pMPn = wn (n = K,L) • wK / MPK = wL / MPL = p = MC • wK/wL = MPK/MPL = MRTS ECON 1150, Spring 2013

  17. Profit maximization of a 2-product firm A two-product firm faces the demand and cost functions below: Q₁ = 40-2P₁-P₂ Q₂ = 35-P₁-P₂ TC = Q₁²+2Q₂²+10 Find the profit-maximizing output levels and the maximum profit. ECON 1150, Spring 2013

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