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![Consider the following CES function
0-1
Y, = A( aK
+ (1 – a)L,º
||
As o → 0, prove the Leontief production function Y; = A min(Kt, Lt).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffec6819a-df1c-47e2-8279-27262374349f%2F33f4710b-26d9-42b7-883c-33ffc3889a84%2Fri8l8e_processed.png&w=3840&q=75)
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- How would you determine that a two-input Cobb-Douglas production function has decreasing returns to scale (DRS), increasing returns to scale (IRS) or constant returns to scale (CRS) depending on whether β is larger than, smaller than, or equal to one?For the production function Qs = K0.4L0.1 find the returns to scale, recall that a doubling of inputs that doulbes output is a CONSTANT returns to scale = 1.0 Please enter your response as a positive number with 1 decimal and 5/4 rounding (e.g. 1.15 = 1.2, 1.14 = 1.1).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
- For the production function Q = L2 +0.5K2, returns to scale: is constant Can be increasing, decreasing, or constant depending on the values of L and K. is increasing O is decreasingConsider the following production function: Y = F(K, L) A(2K + 3L) Does this production function exhibit constant returns to scale? =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.For the production function Q = 4LK, returns to scale: O is decreasing O Can be increasing, decreasing, or constant depending on the values of L and K. O is increasing O is constantAssume that the production function for a certain production process is given by the following Cobb-Douglas production function: Q = f(K,L) K²L² a) What is the expression for the marginal product of K (capital) at the point (K, L)? = b) For small increases in K, holding L constant, does the marginal product of K increase, decrease or stay constant? c) What is the marginal product of L (labour)? d) Using your own words, briefly explain the definition of marginal rate of technical substitution. Calculate the marginal rate of technical substitution between K and L. e) Does the production function exhibit increasing, decreasing or constant returns to scale? Explain your answer.
- Suppose that a rock star gives concerts. The production function of the concert is the following: Q(C,A,G) = (C – 10)'/4A/4G/4 where C represents hourly contribution of guitarist Chris; A represents hourly contribution of guitarist Ataman and G represents hourly contribution of guitarist George. a) If Chris has 26 units of hours, is there increasing returns to scale in production of concerts? b) Assume George has fixed amount of hours, G=16, and hourly wage of G is 5. Denote the wage of Chris= r and the hourly wage of Ataman =w. Derive Rock star's demand for inputs-C and A, respectively-as a function of output (Q). c) Assume that George has fixed amount of hours G=16 and hourly wage of G is 5. Additionally, you are given that the wage of C = r=100 and the unit wage of Ataman=w=100. Show that Rock star's short run cost function is given by TC(Q) = 1000 +50Q²-A production function is given by the following: F(L, K) = 5L2 + 4KiL %3D The production function exhibits: O Increasing returns to scale O Decreasing returns to scale O Constant returns to scale O Diminishing marginal product of laborFor the following production function: Y(K,L)= 25(KL)^(1/2) a) Compute the MRTS b) Define if it exhibits increasing, constant, or decreasing returns to scale c) Is the MRTS decreasing, increasing or constant as we increase the labor input? Provide numerical evidences and an economic interpretation of your answer d) Compute again the MRTS for this new production function: Y(K,L)= 2K+5L e) Compare now the MRTS of the two production functions and explain why the second case is a special case of the general result obtained at point a).
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