1. 1 Ch. 12 Optimization with Equality Constraints 12.1Effects of a Constraint 12.2Finding the Stationary Values 12.3Second-Order Conditions 12.4Quasi-concavity and Quasi-convexity 12.5Utility Maximization and Consumer Demand 12.6Homogeneous Functions 12.7Least-Cost Combination of Inputs 12.8Some concluding remarks

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6. 6 12.2-2 Total-differential approach dL = f x dx + f y dy = 0differential of L=f(x,y) dg = g x dx + g y dy = 0 differential of g=g(x,y) dx & dy dependent on each other dy/dx = -f x / f y slope of isoquant curve dy/dx = -g x /g y slope of the constraint line -g x /g y = -f x / f y equal at the tangent f x / g x = f y /g y = equi-marginal principle

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14. 14 12.2 Finding the Stationary Values 12.2-1 Lagrange-multiplier method 12.2-2 Total-differential approach 12.2-3 An interpretation of the Lagrange multiplier 12.2-4 n-variable and multi-constraint case

15. 15 12.2-1 Lagrange-multiplier method

16. 16 12.2-2 Total-differential approach dL = f x dx + f y dy = 0differential of L=f(x,y) dg = g x dx + g y dy = 0 differential of g=g(x,y) dx & dy dependent on each other dy/dx = -f x / f y slope of isoquant curve dy/dx = -g x /g y slope of the constraint line -g x /g y = -f x / f y equal at the tangent f x / g x = f y /g y = equi-marginal principle

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19. 19 12.3 Second-Order Conditions 12.3-1 Second-order total differential 12.3-2 Second-order conditions 12.3-3 The bordered Hessian 12.3-4 n-variable case 12.3-5 Multi-constraint case

20. 20 11.4 n-variable soc principal minors test for unconstrained max or min

21. 21 12.3-1 Second-order total differential has no effect on the value of Z * because the constraint equals zero but … A new set of second-order conditions are needed The constraint changes the criterion for a relative max. or min.

22. 22 12.3-1 Second-order total differential

23. 23 12.3-1 Second-order total differential

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27. 27 12.4 Quasi-concavity and Quasi- convexity 12.4-1 Geometric characterization 12.4-2 Algebraic definition 12.4-3 Differentiable functions 12.4-4 A further look at the bordered Hessian 12.4-5 Absolute vs. relative extrema

28. 28 12.5 Utility Maximization and Consumer Demand 12.5-1 First-order condition 12.5-2 Second-order condition 12.5-3 Comparative-static analysis 12.5-4 Proportionate changes in prices and income

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39. 39 Graph: Substitution and Income Effects B A C U1U1 U0U0 Quantity Q 1 Quantity Q 2 P0P0 P0P0 P1P1 P1P1 P0P0 If the price of Q 1 increases, then the change in demand equals the substitution effect (AB) and the income effect (BC).

40. 40 B A C U1U1 U0U0 Quantity X Quantity Y -P x1 /P y0 If the price of Q 1 increases, then the change in ordinary demand equals the sum of the substitution effect (AB) and the income effect (BC). X 1 X 1 ' X 0 Price X P1P1 P0P0 Ordinary demand Compensated demand Quantity X -P x1 /P y0 -P x0 /P y0 Y1'Y0Y1Y1'Y0Y1 Graph: Substitution and Income Effects

41. 41 12.5-4 Proportionate changes in prices and income

42. 42 12.6 Homogeneous Functions 12.6-1 Linear homogeneity 12.6-2 Cobb-Douglas production function 12.6-3 Extension of the results

43. 43 12.6-1 Linear homogeneity A function f(x 1,..., x n ) is homogeneous of degree r if multiplication of each of its independent variables by a constant j will alter the value of the function by the proportion j r, that is; if f (jx 1,..., jx n ) = j r f(x 1,... x n ) for all f (jx 1,... jx n ) in the domain of f If r = 0, j 0 = 1,, the function is homogeneous of degree zero (e.g., utility function subject to a budget constraint) If r = 1, j 1 = j, the function is homogeneous of degree one, (e.g., production function with constant returns to scale)

44. 44 12.6-1 Linear homogeneity Given the linearly homogeneous production function Q = f(K, L), The average physical product of labor (APP L ) and of capital (APP K ) can be expressed as the capital- labor ratio, k  K/L, alone. Let j = 1/L

45. 45 12.6-1 Linear homogeneity Given the linearly homogeneous production function Q = f(K, L), The marginal physical products MPP L and MPP K can be expressed as functions of k alone

46. 46 12.6-1 Linear homogeneity Given the linearly homogeneous production function Q = f(K, L), if each input is paid the amount of its marginal product the total product will be exactly exhausted by the distributive shares for all the inputs, i.e., no residual. Euler’s theorem (No fixed factors?)

47. 47 12.6-2 Cobb-Douglas production function Q = AK  L  is homogeneous of degree j  + 

48. 48 12.6-2 Cobb-Douglas production function Q = AK  L  1)Function is homogeneous of degree j  +   +  > 1 increasing returns (paid < share)  +  share) 2) If  +  =1, function is linearly homogeneous 3) Isoquants are negatively sloped and strictly convex (K, L > 0) 4) Function is quasi-concave (K, L > 0)

49. 49 12.6-2 Cobb-Douglas production function

50. 50 12.6-2 Cobb-Douglas production function

51. 51 12.6-2 Cobb-Douglas production function

52. 52 12.7 Least-Cost Combination of Inputs 12.7-1 First-order condition 12.7-2 Second-order condition 12.7-3 The expansion path 12.7-4 Homothetic functions 12.7-5 Elasticity of substitution 12.7-6 CES production function 12.7-7 Cobb-Douglas function as a special case of the CES function

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58. 58 12.7-2 Second-order condition =  (Q aa Q b 2 -2Q ab Q a Q b +Q bb Q a 2 )<0

59. 59 12.7-1 First-order condition  b QoQo a

60. 60 12.7-3 The expansion path Points of tangency that describes the least cost combinations required to produce varying levels of Q o All points on the expansion path must show the same fixed input ratio, i.e., the expansion path must be a straight line emanating from the origin.

61. 61 12.7-4 Homothetic functions Suppose H is meat, Q is feed grain, and a, & b are fertilizer and land respectively Although a homothetic function is derived from a homogeneous function, the function H (a,b) itself is not necessarily homogeneous in the variables a & b. None the less, the expansion paths of H (a,b) are linear like those of Q (a,b)

62. 62 12.7-4 Homothetic functions Let H = Q 2 Q isoquants share the same slopes so therefore the expansion path of H (a,b) like those of Q (a,b) are linear

63. 63 12.7-5 Elasticity of substitution

64. 64 12.7-6 CES production function

65. 65 12.7-6 CES production function

66. 66 12.7-6 CES production function

67. 67 12.7-6 CES production function

68. 68 12.7-6 CES production function

69. 69 12.7-6 CES production function

70. 70 12.7-6 CES production function

71. 71 12.7-6 CES production function