- Question 1 20 Marks Note That The Following Sub Questions Are Not Related To Each Other 1 Suppose That Y Is A Funct 1 (163.27 KiB) Viewed 23 times
not related to each other. (1) Suppose that y is a function of x
satisfying the equation 5x 2 + 10y + 3xy + 5y 2 + 6x = 0. Use the
implicit differentiation rule to find dy/dx (show the necessary
steps). (2) Assume that we have a Cobb-Douglas production function
for a certain industry: Q = cLαKβ , where c is a constant, and L
and K represent labor input and capital input, respectively. Assume
that the production is fixed at Q = Q0, and at this fixed level of
production, K is a function of L defined by Q0 = cLαKβ . Use the
implicit differentiation rule to find the “rate of substitution of
capital for labor”, which is measured by ∂K/∂L when the production
is fixed at Q = Q0 (show the necessary steps). (3) For x > 0 and
y > 0, the equation y 3 x 3 − (x + 2)2 (y + 3) = 0 defines y as
an implicit function of x. Find the elasticity of y with respect to
x at the point (x0, y0), where x0 and y0 are some known values.
Question 1 (20 marks) Note that the following sub-questions are not related to each other. (1) Suppose that y is a function of x satisfying the equation 5x2 + 10y + 3xy + 5y2 + 6x = 0. Use the implicit differentiation rule to find dy/dx (show the necessary steps). (2) Assume that we have a Cobb-Douglas production function for a certain industry: Q=cLºK, where c is a constant, and L and K represent labor input and capital input, respectively. Assume that the production is fixed at Q = Qo, and at this fixed level of production, K is a function of L defined by = Qo = clºK. Use the implicit differentiation rule to find the “rate of substitution of capital for labor”, which is measured by aK/aL when the production is fixed at Q = Qo (show the necessary steps). (3) For x > 0 and y > 0, the equation y3 - (x + 2)(y + 3) = 0 23 defines y as an implicit function of x. Find the elasticity of y with respect to x at the point (20, yo), where xo and yo are some known values.