PLEASE CHECK AND CORRECT MY ANSWERS!!!!
MRTS = marginal rate of technical substitution
RTS = returns to scale
MP = marginal product
Suppose the production function is Cobb-Douglas and f(x1, x2) = [x1^(1/2)][x2^(3/2)]. (Note: x1 and x2 are variables).
a) Select an expression for MP of factor 1 at point (x1, x2).
b) MP of factor 1 ___ for small increases in x1, holding x2 fixed.
c) MP of factor 2 is _____ and it ____ for small increases in x2.
d) An increase in the amount of x2 _____ the MP of factor 1.
e) The technical rate of substitution between x2 and x1 is _____.
f) Does this technology have a diminishing technical rate of substitution?
g) This technology demonstrates ____ returns to scale.
My answers:
a) [x1^(-1/2)][x2^(3/2)]
b) increases
c) [3/2][x1^(1/2)][x2^(1/2)], decreases
d) increases
e) -(x2)/(3x1)
f) yes
g) decreasing
a) It should be (1/2)[x1^(-1/2)][x2^)3/2)
b) I think decreases. Plug in an example. If x1 grows x1^(-1/2) gets smaller.
c) the MP is correct. I think increases. Again, plug in an example. if x2 grows x2^(1/2) gets bigger.
d) I agree
e) i agree
f) I agree
g) I disagree. As the sum of the two exponents is > 1, this implies an increasing return to scale. Plug in an example. Increase x1 and x2 by some z percentage. Does output grow by more than z percent
a) To find the marginal product (MP) of factor 1 at a specific point (x1, x2), we need to take the partial derivative of the production function with respect to x1. In this case, the production function f(x1, x2) = [x1^(1/2)][x2^(3/2)]. Taking the partial derivative with respect to x1, we obtain MP of factor 1 = [1/2][x1^(-1/2)][x2^(3/2)].
b) When determining the MP of factor 1 for small increases in x1, holding x2 fixed, we need to evaluate the partial derivative of the production function f(x1, x2) = [x1^(1/2)][x2^(3/2)] with respect to x1, again. Since the exponent of x1 is positive (1/2), the MP of factor 1 will increase for small increases in x1, assuming x2 is held constant.
c) Similarly, to find the MP of factor 2 for small increases in x2, we need to take the partial derivative of the production function with respect to x2. In this case, the partial derivative of f(x1, x2) = [x1^(1/2)][x2^(3/2)] with respect to x2 is MP of factor 2 = [3/2][x1^(1/2)][x2^(1/2)]. Since the exponent of x2 is (1/2), the MP of factor 2 decreases for small increases in x2.
d) An increase in the amount of x2 will not directly affect the MP of factor 1. The MP of factor 1 is influenced only by changes in x1.
e) The technical rate of substitution (TRS) between x2 and x1 can be calculated by taking the ratio of the partial derivatives of the production function with respect to x2 and x1. In this case, TRS = -(partial derivative of f(x1, x2) with respect to x2) / (partial derivative of f(x1, x2) with respect to x1) = -(x2)/(3x1).
f) Yes, this technology exhibits a diminishing technical rate of substitution since the TRS value is negative. As more of factor 2 (x2) is substituted for factor 1 (x1), the decrease in x2 yields smaller increases in production.
g) To determine the returns to scale (RTS), we need to analyze how changes in inputs affect the overall output. In a Cobb-Douglas production function, the RTS can be determined by assessing the sum of the exponents of the production function. In this case, f(x1, x2) = [x1^(1/2)][x2^(3/2)], so the sum of the exponents is 1/2 + 3/2 = 2. Since the sum of the exponents is less than 3, this technology demonstrates decreasing returns to scale.