A firm has the production function f(x, y)= 20x^3/5 y^2/5.The slope of the firm's isoquant at the point (x, y)= (20, 40)is (pick the closest one) A)-3. B)-0.67. C)-1.50. D)-0.50. E)-0.25.

Question

Quizplus · Accepted Answer

The slope of an isoquant is given by the marginal rate of technical substitution (MRTS), which is the ratio of the marginal product of one input to the marginal product of the other input. For a Cobb-Douglas production function like $f(x, y) = x^{3/5}y^{2/5}$, the MRTS can be calculated by taking the partial derivatives of $f$ with respect to $x$ and $y$, and then finding their ratio at the given point (20, 40).The partial derivative of $f$ with respect to $x$ is $\frac{3}{5}x^{-2/5}y^{2/5}$ and with respect to $y$ is $\frac{2}{5}x^{3/5}y^{-3/5}$. At the point (20, 40), these derivatives are:$$\frac{\partial f}{\partial x} = \frac{3}{5} \cdot 20^{-2/5} \cdot 40^{2/5}$$$$\frac{\partial f}{\partial y} = \frac{2}{5} \cdot 20^{3/5} \cdot 40^{-3/5}$$The MRTS, which is the negative ratio of these derivatives ($-\frac{\partial f/\partial x}{\partial f/\partial y}$), at (20, 40) gives a value closest to -3, making choice A the correct answer. This calculation involves substituting the values of $x$ and $y$ into the expressions for the partial derivatives, calculating each, and then finding their ratio. The negative sign indicates the trade-off direction between $x$ and $y$.

A Firm Has the Production Function F(x, Y)= 20x3/5 Y2/5

A Firm Has the Production Function F(x, Y)= 20x^3/5 Y^2/5