Consider a firm that has a Cobb-Douglas technology. The firm wishes to minimize the cost of producing y units of output and has access to perfectly competitive factor markets. The firm's cost minimization problem is given by: min wl +rk {k,l} s.t. k°18 = Y Let μ denote the Lagrange multiplier on the output constraint.
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Please answer in detail according to the picture.
Find µ*. What is its interpretation?
Find əc/əy and show that it is equal to µ*.
How does ə2c/əy2 depend on (α+β)?
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- Assume that tighter immigration laws cause a reduction in the supply of agriculturalworkers and this leads to higher wages in the sector. Dean’s Fruity Produce Ltd, afarm in Leicestershire, uses two inputs to harvest its output – machines (capital) andagricultural workers. Dean, the managing director of the company, has employedKarishma (a brilliant statistician) and she has estimated that the business’s long-runwage elasticity of labour demand is -1.7. Using isoquant/isocost analysis, explain the possible impact of the higher wages on the output level, total cost of production and input-factor mix of this business in boththe short and long-run and how would your answer change if Karishma had estimated that the long-run wageelasticity of labour demand was -0.4? (1000 words)Given the input-output matrix below, find the output matrix if final demand changes to 600 for water, 180 for electric power, and 700 for agriculture. Industry Electric Power Water 120 Agriculture Final Demand 240 320 160 Water 480 270 Electric Power Agriculture Other Industry: 60 120 170 400 180 240 240 360 80 The output matrix is X=. (Round to two decimal places as needed.)Find the elasticity of scale and the elasticity of substitution for the CES production function (x1; x2) = (x1rho + x2tho 1/P, where 0 * p > !!!
- 4. Find the cost function and the conditional demands for inputs associated to the CES production function f(r1, #2) = A(ar{ + (1 – a)a)/e, where A, 8 > 0, 0 < a < 1, and 0#p< 1.. An electricity producer has a constant marginal cost of production equal to $40 per megawatt. The residual demand for its electricity is given by P (q) = a−bq, where P is the price and q is the quantity of power generated by this producer. The producer knows the slope, b, but he vertical intercept of the residual demand curve, a is unknown. Assume A and B are greater than zero. If you get stuck, you may answer any of the following questions for special case where a = 80 And b = 0.5 for partial credit. (a) What is the marginal revenue, M R(q), for this producer? b) What is the optimal q for this producer? (c) What is the electricity producer’s optimal price? (d) What is the electricity producer’s optimal bid in a uniform price Auction? e) Suppose b is equal to zero. Would the producer have an incentive to submit a bid above its marginal cost? Explain.Derive formulae for the marginal products of the three inputs in the produc- tion function Q = 40K03L0.3R04. %D
- 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.A firm has a linear demand function for its product. When the price of the product isSh.220, the quantity demanded is 40 units. When the price increases to Sh.240, thequantity demanded becomes 30 units. In addition, the firm’s marginal cost function isgiven by:MC = 40q – 2q2 + 2Fixed cost = Sh.5 millionWhere q = quantity demanded, MC = marginal cost (Sh. million)Evaluate the level of output that maximizes profits.A firm is able to adjust both L and K and has a production function q = KL, where K is the amount of capital and L is the amount of labor it uses as inputs. The cost per unit of capital is r and the cost per unit of labor is w. The (conditional) demand for capital (also known as the optimal level of capital) is given by: O qwr O the square root of qr/w O qw/r O q/wr O the square root of qw/r
- Consider a firm that produces output (y) using only two inputs (A and B). The firm has a perfect substitutes production technology and a production function of the formf(A, B) = 5A + 8B. %3D Suppose the firm is currently using 200 units of input B. What is the marginal product of inputA when A 100? 500 What is the marginal product whenA 150? 750 Give your answers to two decimal places.Consider the following production function: y = lnx1 + lnx2, where x1>0, x2>0. Is it homogenous? Is the input requirement set monotonic and convex? Find its elasticity of substitution.Answer the Constrained Optimization: Cobb-Douglas Production Function:1. Based from the factor shares of the two inputs, what will happen to the number of output if it the firm decides to triple both the amount of labor and capital?