Final answer:
The production function Y=F(K,L)=K¹⁄₃L²⁄₃ does indeed have constant returns to scale, as output proportionately increases with an increase in inputs. The LRAC curve's flat portion around Q3 indicates this phase where average costs remain stable with changes in scale.
Step-by-step explanation:
The student's question refers to whether the production function Y=F(K,L)=K¹⁄₃L²⁄₃ has constant returns to scale. To determine this, we can examine what happens to output when all inputs are proportionately increased. We apply a test where we scale both capital (K) and labor (L) by a factor 't': Y = F(tK, tL). According to the given production function:
Y = F(tK, tL) = (tK)¹⁄₃ (tL)²⁄₃ = t¹⁄₃K¹⁄₃ t²⁄₃L²⁄₃ = tK¹⁄₃L²⁄₃ = tY
Since the output Y scales up exactly by the factor 't', this indicates that the production function has constant returns to scale. This means that if we double both inputs (capital and labor), output will also double. In contrast, economies of scale refer to a situation where the percentage increase in output is greater than the percentage increase in inputs. Conversely, diseconomies of scale occur when the percentage increase in output is less than the percentage increase in inputs. The long-run average cost (LRAC) curve reflects these concepts, where the flat portion around Q3 indicates constant returns to scale, meaning average costs remain stable as production scales.