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- *Find the elasticity of scale and the elasticity of substitution for the CES production function f(x1,x2) = (x{ + x%)¯,where 0 #p<1.For a specific production function, the associated Average Product function is (AP=) 12 L -0.2 L2 and the associated Marginal Product function is (MP=) 24 L-0.6 L². Calculate the "point" version of the elasticity of Output with respect. to labor (Eq,L) when labor is (L=) 35 units. Use the template provided to classify the value calculated.Consider the following production function: x = f0.K) = A e°k where xis the output e is the labour input, kis the capital input, and A, bare positive constants. (a) Set up the cost minimization problem and solve for the first order conditions using the Lagrange Method. Let wbe the wage rate and rthe rental rate of capital. (b) Using your answer in (a), find how much labour and capital would the firm use to produce x outputs, given wage rate wand rental rate of capital ? what are these functions called? (e) Using your answer in (b), find the minimum cost it takes to produce xoutputs, What is this function called?
- Given the following production function: q = 10KL. Assume that w = 25, r = 75 and C = 1200. (a) Mathematically find the minimum cost combination of capital and labour to produce a given level of output.b) Does the production function in part (a) show increasing returns to scale, decreasing returns to scale, or constant returns to scale? Explain. (c) Using isoquants and isocosts, graphically illustrate the effect of an increase in the wage rate, assuming the firm is producing at the same level of output.i) Derive the output elasticity of labour in the following alternative production function: Y = c + alnk + ßlnL ii) Suppose B = 0.5 and other factors are fixed, explain the unit change in output due to a percentage change in labour.b) Given the following Constant Elasticity of Substitution production function, show whether or not it is homogeneous, and the scale it exhibits. Q=[L² + K³]73
- Find the elasticity of scale and the elasticity of substitution for the CES production function: 1 1 f(x₁, x₂) = (x³ + x2)³. Solution: We first calculate the marginal products: 2 fx₁ = 3 + = x1fx1 -+ 2 2 10 - ² ( x² + x ) ( + x^²) = ( + + + + ² ) 15 ²0 x² -2/3 x₂ + x2fxz Elasticity of scale = _1₁_+_*__*(+)´<°¸«d«)*;»_ f(x1, f(x1, ‹2)² = -2/3 r2/3 = TRS = t (where TRS = =t). ⇒ r = t³/² ⇒ ln(r) = ln (t) and o = -2/3 dln (r) dln (t) To get elasticity of substitution, we first need TRS and denote r = 1 x3 X1 TMS - F (+4+2)0 = fx₁ fx₂ 2 1 + x²) x₂² MIN x2 3 -2/3 (x² -2/3 2 2/3 2/3 x1 -2/3 = r²/3 x¹/3 + 1/3 = 1.Maize is produced according to the production function q = 100(K°*L°) %3D Starting with a capital input of 4 and a labor input of 49, show that the marginal product of labor and the marginal product of capital are both decreasing. (i) (ii) Compute the marginal rate of technical substitution of labour for capital. (iii) Does this production function exhibit increasing, decreasing or constant returns to scale?The production function for soy beans is q = K 0.25 L 0.45. Dose this production function have constant, increasing or decreasing return to scale? A. may be increasing or decreasing B. Constant return to scale C. Decreasing return to scale D. Increasing return to scale
- in part b, how do we know that is the cost function? where are those variables coming from? thxDetermine the returns to scale for the following production functions. a) ? = K + 7L b) ? = KL3Both capital and labor are variable inputs. Suppose the wage rate of labor is 16 (per hour) and the rental price of capital is 20 (per hour). The downward sloping smooth curves that bend in towards the origin are a firm's isoquants. The downward sloping straight lines are isocost lines. If the ferm plans to produce 20 units of output lper hour), its minimized cost is Capital slope=-0.8 (6,5.5) (4,5) (2,3) q=30 q=20 q=10 Labor O 180 O 160 O 164 O 122