A firm has the production function f(X, Y)= X^3/4 Y^1/4, where X is the amount of factor x used and Y is the amount of factor y used.On a diagram we put X on the horizontal axis and Y on the vertical axis.We draw some isoquants.Now we draw a straight line on the graph and we notice that wherever this line meets an isoquant, the isoquant has a slope of -9.The straight line we drew A)is a ray through the origin with slope 3. B)is a ray through the origin with slope 4. C)is vertical. D)is horizontal. E)has a negative slope.

Question

Quizplus · Accepted Answer

If the slope of the isoquant is F1F1F1S1F1F1F109, then it must be tangent to a ray through the origin with slope 3. This is because the marginal rate of technical substitution (MRTS) at any point on an isoquant is equal to the slope of the isoquant at that point, and the MRTS is given by:
MRTS = (Marginal Product of X) / (Marginal Product of Y)
Taking the partial derivative of the production function with respect to X and Y gives:
MRTS = -(F1F1F1S1/Y) / -(F1F1F109S1/X)
Simplifying this expression gives:
MRTS = F1F1F1S1*X / F1F1F109S1*Y
The MRTS is constant along an isoquant, so we can set it equal to F1F1F1S1F1F1F109 and solve for Y in terms of X to get the equation of the isoquant. Doing this gives:
Y = (F1F1F1S1*X) / (F1F1F109S1*(F1F1F1S1F1F1F109))
This equation corresponds to a straight line passing through the origin with slope 3. Therefore, the answer is (A) a ray through the origin with slope 3.

A Firm Has the Production Function F(X, Y)= X3/4 Y1/4

A Firm Has the Production Function F(X, Y)= X^3/4 Y^1/4