Suppose 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.
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- Suppose the production function for a product is z = 65x2/5y3/5 where x is the capital expenditures and y is the number of work-hours. (a) Find the marginal productivity of x. (b) Find the marginal productivity of y.Suppose a firm uses capital K and labor L to produce output Q with the following production function: 1 Q=4KL + = 1² 3 (a) Find the marginal product of capital and the marginal product of labor. (b) Find the average production of capital and the average production of labor. (c) Does this production function have increasing, constant, or decreasing returns to scale? Explain. (d) Find the marginal rate of technical substitution (RTS). (e) Use your result in (d) to find the elasticity of substitution. (f) Find the contingent demand functions for capital and labor if w is the price of labor input and v is the price of capital input.The amount of fish caught per week an a trawler is a function of the crew size assigned to operate the boat. Based on past data, consider the following production function identifying the relationship between output and labor input. You may assume that capital is fixed at 10 units.Labour Input (No of. Workers) (L) Total Product TPL (Q). From the table below :-(a) Calculate APL and MPL. (b) Graph APL and MPL. Do they have the expected shape? (c) On your graph, identify the three stages of production Labour input (No of Workers) (L) Total Product TP L (Q) 0 0 1 80 2 170 3 300 4 450 5 540 6 620 7 700 8 770 9 830 10 880 11 920 12 950 13 970 14 980 15 985 16 988 17 988
- 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?(v) Find the ratio of the marginal productivities of two inputs, at L=2 and K =3 when the production function is given by Q= 8LK- L2 -K?Assuming a Cobb-Douglas production function with constant returns to scale, then, as L rises with K and A constant, it will be the case that: (a) Both the marginal product of labour and the marginal product of capital will fall(b) The marginal product of labour will fall and the marginal product of capitall will rise(c) Both the marginal product of labour and the marginal product of capital will rise(d) The marginal product of labour will rise and the marginal product of capital will fall
- 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?A firm has a production function of ?(?,?) = ??.???.? a) Explain the concept of returns to scale. Does the function provide increasing, decreasing, or constant returns to scale? b) Provide an example of a typical sector with increasing returns to scale. c) Explain the concept of MRTS and argue whether the MRTS for this production function is diminishing. Please also provide a graphical illustration using numbers.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.
- (1) Suppose the production function for widgets is given by q = kl – 0.8k? –0.272 where q is the annual quantity of widgets produced, k is the annual capital input and 1 is the annual labor input. (a) Suppose k-10; Find the average productivity of labor. At what level of labor input does the average productivity reach a maximum? (b) Suppose k-10; Find the marginal productivity of labor. At what level of labor input does the marginal productivity reach a maximum? Is the average product equal to the marginal product at the point where average product reaches its maximum? Please showL it.Consider the following production function:q = (KL)^α, where α > 0.Answer the following questions:(a) Under what conditions (i.e. values of α) will the production function exhibit decreasing returns to scale? Under what conditions will it exhibit constant returns to scale? Under what circumstances will it exhibit increasing returns to scale? (b) Confirm that the marginal physical product of capital is homogenous of degree zero in the case in which the production function exhibits constant returns to scale. (c) Derive an expression for the cost function of a firm using the productionfunction to produce output of a good. (d) Find the first and second partial derivatives of the cost function with respect to q. Interpret the second partial derivative and relate the sign of the derivative to the returns to scale.Consider the following production function: q = (KL)“, where a > 0. Answer the following questions: (a) Under what conditions (i.e. values of a) will the production function exhibit decreasing returns to scale? Under what conditions will it exhibit constant returns to scale? Under what circumstances will it exhibit increasing returns to scale? (b) Confirm that the marginal physical product of capital is homogenous of degree zero in the case in which the production function exhibits constant returns to scale. (c) Derive an expression for the cost function of a firm using the production function to produce output of a good. (d) Find the first and second partial derivatives of the cost function with respect to q. Interpret the second partial derivative and relate the sign of the derivative to the returns to scale.