Final answer:
The parameter alpha must take the value of 2/3 for the production function to exhibit constant returns to scale, considering the exponent on labor is 1/3. This aligns with the concept of a constant-cost industry, where the average cost of production remains the same as firms scale their operations.
Step-by-step explanation:
The subject of this question is the study of production functions in an economic context, specifically examining a Cobb-Douglas production function and the concept of constant returns to scale. When there are constant returns to scale in an economy with a Cobb-Douglas production function, this means that the sum of the exponents on the production function's inputs must equal one. As the question states that the exponent on labor (represented by L) is 1/3, the exponent on capital (represented by K) must be 2/3 to ensure that the production function exhibits constant returns to scale (since 1/3 + 2/3 = 1).
Under constant returns to scale, scaling up all inputs in the same proportion results in a proportional increase in output, and this property is crucial in defining a constant-cost industry. In such an industry, as demand increases and the firm scales up its production, the average cost of production remains constant, and thus, the supply curve is very elastic. This scenario is prevalent in industries such as agriculture, where the supply of inputs like labor and raw materials can be increased without significant changes in their prices.