Final answer:
The given production function experiences diminishing returns to scale as indicated by the exponent rho, which is less than 1. This means that as labor and capital are increased, the additional output gained from each new unit of input decreases, signifying a decreasing efficiency as scale increases.
Step-by-step explanation:
When analyzing the following production function given by f (L,K)= L + K+ ψ(LK)^p rho, where ψ is a constant and rho, < 1, we're looking at its behavior when inputs are scaled up. The behavior of this function will tell us if it experiences increasing returns to scale, constant returns to scale, or decreasing returns to scale. To determine this, we would usually increase both labor (L) and capital (K) by a certain factor to see how output (Q) changes. Given that rho is less than 1, the function shows diminishing returns for the multiplicative part (ψ(LK)^p), meaning that as more units of L and K are used, the additional output produced from each additional unit decreases, indicating diminishing returns to scale.
The presence of a coefficient rho less than 1 means that when we scale labor and capital, this part of the production function increases less than proportionately. When all parts of the production function are considered, if the sum of the output elasticities of labor and capital is less than 1, which is implied by rho being less than 1 for the multiplicative part, the production function experiences diminishing returns to scale. This contrasts with constant returns to scale, where output would increase proportionally with inputs, and increasing returns to scale, where output would increase more than proportionally.