A short run equilibriam production function is given as Q= X½ were Q is output and X is input.
a) is the production function concave? Show your working
Yes, the production function is concave. To show this, we can take the second derivative of the production function:
d2Q/dX2 = -1/2X-3/2
Since this is always negative, the production function is concave.
To determine if the production function Q = X^(1/2) is concave, we need to check the second derivative of the function. If the second derivative is negative, then the function is concave.
First, let's find the first derivative of the production function:
dQ/dX = (1/2) * X^(-1/2)
Next, we can find the second derivative by taking the derivative of the first derivative:
d^2Q/dX^2 = (-1/4) * X^(-3/2)
Now, we need to check if the second derivative is negative for all values of X. Since the second derivative has a negative constant factor (-1/4), and X raised to a negative power, we can conclude that the second derivative is indeed negative for all X.
Therefore, the production function Q = X^(1/2) is concave.
To determine whether the production function is concave or not, we need to examine the second derivative of the function.
First, let's find the first derivative of the production function, Q, with respect to X:
dQ/dX = (1/2)X^(-1/2)
Now, let's find the second derivative of Q with respect to X:
d^2Q/dX^2 = (d/dX) ((1/2)X^(-1/2))
Using the power rule for differentiation, we can find the second derivative:
d^2Q/dX^2 = (1/2) * (-1/2) * X^(-3/2)
Simplifying the expression for the second derivative:
d^2Q/dX^2 = -1/4 * X^(-3/2)
Since the second derivative is negative, we can conclude that the production function is concave.