Question

Find all the first and second-partial derivatives for the utility function, U -50x 5y 0.2 (a) Give a verbal description of each derivative. (b) Are the marginal functions increasing or decreasing. Use the derivatives to justify your answer b. Given the function Q- Al'KB, explain the terms: constant returns to scale; increasing returns to scale; decreasing returns to scale. Show that the production function, Q -100L0.3K0.5, exhibits decreasing returns to scale and diminishing returns to labour Show that the production function, Q -100L 0.5 K0.5, exhibits constant returns to scale and diminishing returns to labour C.

Answer 1

(a) U = 50 x^0.5 y^0.2 First order derivative partial differential U/partial differential x = f_x = 50(0.55) x^-0.5 y^0.2 = 25 x^-0.5 y^0.2 It shows that If x increased by 1 unit then utility will increase by 25 x^-0.5 y^0.2 unit keeping y constant (i.e., It shoes marginal utility from good x) partial differential U/partial differential y = f_y = 50 x^0.5 (0.2) y^-0.8 = 10 x^0.5 y^-0.8 (It shows marginal utility from consumption of additional unit of good y keeping good x constant) second order derivative f_xx = 25(-0.5) x^-1.5 y^0.2 = -12.5 x^-1.5 y^0.2 \r\n It shows that marginal utility tends to Decline as we consume move & move of good i.e., (Diminishing marginal utility) partial differential^2 U/partial differential y^2 = f_yy = 10(-0.8) x^0.5 y^-1.8 f_yy = -8x^0.5 y^-1.8 It also shows that marginal utility tends to decline as we consume move & move of good-y (i.e., diminishing marginal productivity for good-y) (b)Q = A L^alpha K^beta constant Return to scale (CRS)