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A company production function Is Q = 20K^3.2 L ^m + K^8 + 7L^m, where Q is the quantity produced, L is the number of units of labour and K is the number of units of capital
(A) Find the first-order partial derivatives of Q
(B) Calculate the value of A so that L ∂Q/∂L + K ∂Q/∂K= AQ
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- 100VKL, where q is 4. The production function for puffed rice is given by q = the number of boxes produced per hour, K is the number of puffing guns used each hour, and L is the number of workers hired each hour. (a) Calculate the q = 1,000 isoquant for this production function and show it on a graph. (b) If K = 10, how many workers are required to produce q =1,000? What is the average productivity of puffed-rice workers? (c) Suppose technical progress shifts the production function to q = 200 VKL. Answer parts (a) and (b) for this new situation.Question 1 (a) A company production function is Q = 15K¹.7 L¹.3 + K³ + 5L³, where Q is the quantity produced, L is the number of units of labour and K is the number of units of capital. (1) Find the first order partial derivatives of Q (II) Calculate the value of A so that: L + K aq aQ ƏL ак = AQx+2 (1) Find the rate of change of the function f(x) (ii) The number of units Q of a particular commodity that will be produced with K with respect to x when x = 1. 1- 8x thousand dollars of capital expenditure is modeled by Q(K) = 500 Kố. Suppose that capital expenditure varies with time in such a way that t months from now there will be K(t) thousand dollars of capital expenditure, where 2t* + 3t + 149 K(t) t+2 (a) What will be the capital expenditure 3 months from now? How many units will be produced at this time? (b) At what rate will production be changing with respect to time 5 months from now? Will production be increasing or decreasing at this time?
- Suppose the production function for widgets is given by KL – 0.5K2 – 0.1 L2 , where q represents the annual quantity of widgets produced, K represents annual capital input, and L represents annual labor input. (a). Suppose K=5; what is the average productivity of labor (Average product of Labor, MPL) (b). Suppose K=10; at what level of labor input does the total output reach the maximum?Suppose that a furniture manufacturer produces tables. The manufacturer's production function is Q(KL)- 2K0.8 0.2 where Kis the number of wood saws and L is the number of labor hours used to produce tables. The prevailing wage rate is $10 per hour and the rental rate associated with wood saws is $50. The manufacturer has a goal to produce 1,000 units. What Lagrangian equation can be used to solve the manufacturer's cost minimization problem? Oa. C = 2K12 s.t. 1,000 = 50K + 10L b. = Oc L = Od. L = 10K + 50L s. t. 1,000 = 2K08102 50K + 10L s.t. 1,000= 2KL02 2K10 s. t. 1,000= 10K + 50LSuppose the production function for good q is given by q = (k+1)¹/2 where k and I are capital and labor inputs: (a) The function exhibits diminishing marginal productivities in both inputs. (b) The function exhibits diminishing marginal productivities in capital but not in labor. (c) The function exhibits diminishing marginal productivities in labor but not in capital. (d) The function exhibits increasing marginal productivities in both inputs.
- The total cost of producing 1 unit of a product is ху C(x, y) = 28 + 4x + 6y + 56 dollars where x is the cost per pound of raw materials and y is the cost per hour of labor. (a) If labor costs are held constant, find the function that describes the rate at which total cost increases for each increase of $1 per pound in material cost. (b) If material costs are held constant, find the function that describes the rate at which total cost increases for each $1 per hour increase in labor costs.Let C(T) be a function that models the dependence of the cost (C) in thousands of dollars on the amount of ore to extract from a copper mine measured in tons (T): 1) If you computed the average rate of change of cost with respect to tons for production levels between T = 20000 and T = 40000, give the units of your answer (no calculations - describe the units of the rate of change). 2) If you had a function for C(T) and were able to calculate the answer to part 1, explain why you would not expect your answer to be negative (explanation should be in terms of cost, tons of ore to extract, and rates of change).A production process using two inputs, labor and capital, can be written as: Q = 3LK where Q represents output per day (tons). Find the MRTS for this production. It is
- Does the output maximization level of input use exist for the following function? If yes, then what is the exact input-use level? y = x1 + 0.1x12 - 0.05x13 + x2 + 0.1x22 - 0.05x23The output (Q) of a production process is a function of two inputs (L and K) and K is given by the following relationship: Q = 0.50LK− 0.10L2− 0.05K2 The per-unit prices of inputs L and K are $20 and $25, respectively. The firm is interested in maximizing output subject to a cost constraint of $500. Use the lagrangian optimization techniques to find the following: How many units of L and K should be used by the firm? What is the total output of this combination? What is the marginal rate of substitution between L and K?3) If we duplicate all the production factors of the production function f (L,K) = L 2/3. K1/3, where L are working hours, k the capital then the amount of product will be multiplied by: Choose one: A) 4 B) 2 C) 1 D) 8 E) 0.5