Question

A single-input production function is given by

• A. The cost function is concave and the production function is convex.
• B. The cost function is convex and the production function is concave.
• C. The cost function and the production function are both differentiable and convex.
• D. The cost function and the production function are both differentiable and concave.

Hint:

If \(Q=L^a\) with \(0<a<1\), the production function is concave. The corresponding cost function often becomes convex because input requirement rises more than proportionately with output.

Answer 1

The Correct Option is B

Step 1: Write the production function.
\[ Q=L^{0.25} \] This can also be written as
\[ Q=L^{\frac{1}{4}} \]

Step 2: Check the nature of production function.
Since the exponent is between \(0\) and \(1\), the production function is concave.
\[ 0<\frac{1}{4}<1 \]
Thus, production increases at a decreasing rate.

Step 3: Express input \(L\) in terms of output \(Q\).
\[ Q=L^{\frac{1}{4}} \] Raise both sides to the power \(4\):
\[ L=Q^4 \]

Step 4: Write the cost function.
Total cost is fixed cost plus input cost.
\[ C(Q)=C_1+wL \]
Substituting \(L=Q^4\),
\[ C(Q)=C_1+wQ^4 \]

Step 5: Check the nature of cost function.
Differentiate cost function twice with respect to \(Q\).
\[ C'(Q)=4wQ^3 \] \[ C''(Q)=12wQ^2 \]
Since
\[ w>0,\quad Q>0, \] we have
\[ C''(Q)>0 \]
Therefore, the cost function is convex.

Step 6: Compare with given options.
We found that:
Production function is concave and
Cost function is convex

Step 7: Final conclusion.
Hence, the correct statement is
\[ \boxed{\text{The cost function is convex and the production function is concave.}} \]
Therefore, the correct option is (B).