please answer the both questions For Questions 31 and 32, use the following information:Consider a firm minimising the cost of production under the following Cobb-Douglasproduction technology constraint:q = KªL!-awhere q is the output level, K is the quantity of capital used and L is the quantity of labourused. Let w be the wage rate and r be the rental rate of capital.31. The Lagrangean for this optimisation problem is:a) wL + rK + µ[q = K«L²-«]b) KªL²-a – [wL +rK]c) wL + rK + µ[KªL!-«]d) wL + rK + µ[q – KªL!-«]32. The demand function for capital is:a w11-aa) K = ql,-ab) K = q[c) K = qd) K = ql1-a r1-a w1-aa ra r.1-a1-a e1-a1-a

Production function q=KαL1-α ............ (1) wage rate = w and rent on capital =r

Answered: For Questions 31 and 32, use the…

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter10: Cost Functions

Section: Chapter Questions

Problem 10.7P

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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?
Miguel and Jake run a paper company. Each week they need to produce 1,000 reams of paper to ship to their customers. The paper plant's long-run production function is: q= 4K¾L where q is the number of reams produced, K is the quantity of capital rented, and L is the quantity of labor hired. (K For this production, MP, L and MP = 3 K K The weekly cost function for the paper plant is C= 10K +2L where C is the total weekly cost What ratio of capital to labor minimizes Miguel and Jake's total costs? (Hint: Find the Marginal Rate of Technical Substitution (MRTS) for capital and labor.)
Consider a firm that utilizes Labor (L) and Capital (K) to produce a product Q, and the technology is given by the production function: Q = aL + bK. The wage rate of labor and the rental rate of capital are given by w and r respectively. (i) Draw the family of iso-quants and show that in a properly labelled diagram. (ii) Draw the iso-cost lines and show those in the same diagram. Note that the shape of these curves will depend on the parameter values of a, b, w and r. (iii) If the firm’s objective is to minimize cost of production, how should the two factors of production be combined in order to produce a given level of output in least costly manner? In other words, work out the cost minimizing input combinations. (iv) Derive the cost function, the total cost of production as a function of output and draw the cost function. (v) Can you find an interrelationship between the nature of the production function and that of the derived cost function?
Being driven by the high profit from digital asset trading, Doggycoin, Sulfikar has decided to set-up his own digital mining company. Let’s assume that two main essential inputs in digital asset production be the high-quality graphic card (C) and electricity (E). After observing the business for few months, he finds the production function as Cobb-Douglas technology as seen in image: Where q is continuous, strictly increasing, and strictly quasi-concave and q(0) = 0. Answer the following questions: a. If the prices of graphic card and electricity denoted by c and e, respectively, and we assume the cost-minimizing behavior. Solve for C and E in their prices and output level, and what is the total cost function C(e, c, q)? b. What are the conditional demands for input C and E?
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
Coca cola uses labour(L) and capital (k) in its production process. It estimates that the production function facing it is given by Q= K0.5 L0.5. The company can sell a bottle of coke at GH4. The cost of a machine is GH3 and that of Man- hour is GH5. The firm's total cost if production at the end of production is expected to be GH 3000. Required: Determine The optimal level of capital and labour usage The firm’s firm’s profit at the optimal levels of capital and labour usage The firm’s elasticity of output with respect to labour and capital and interpret it
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A farmer uses K units of machinery and L hours of labor to produce C tons of corn, with the production function, C = 2 L0.5 K. The farmer currently uses L = 20 and K = 10. Suppose the input price ratio is w/r = 1. What would you tell the farmer to do in order to minimize his cost of production given the current level of output?
10. Extension of the Cobb-Douglas Production Function-The Cobb-Douglas produc- tion function (Equation 7.16) can be shown to be a special case of a larger class of linear homogeneous production functions having the following mathematical form:" Q = y[@K* + (1 – a)L°]W/P where y is an efficiency parameter that shows the output resulting from given quantities of inputs; a is a distribution parameter (0 sas 1) that indicates the division of factor income between capital and labor; p is a substitution parameter that is a measure of substitutability of capital for labor (or vice versa) in the pro- duction process; and v is a scale parameter (v > 0) that indicates the type of re- turns to scale (increasing, constant, or decreasing). Show that when v = 1, this function exhibits constant returns to scale. [Hint: Increase capital K and labor L each by a factor of A, or K* = (A)K and L* = (1)L, and show that output Q also increases by a factor of A, or Q* = (A)(Q).]

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