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This may be a stupid questions as I am just studying undergrad economics honours but how do we know that the solow model has diminishing returns to its input and harrod-domar has constant returns to its inputs from the definitions of the models? Like for example if in the harrod domar model, we take Y to be a function that has diminishing returns to scale then how can we say that it always has CRS?

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This is just assumption on the model.

There are some functions for which you can mathematically say they always have constant returns to scale.

For example, if $F(K,L)=AK^{0.5}L^{0.5}$ then we know it will always have constant returns to scale since when you increase both factors $K$ or $L$ by some scalar $z$ output also increases by $z$.

$$ F(zK,zL) = A(zK)^{0.5}(zK)^{0.5}=z AK^{0.5}L^{0.5} = zF(K,L)$$

So from the above we know that this (Cobb-Douglas) function always have constant returns to scale.

There is a large number of functions that have this property. Usually papers want to be general so they will just assume constant returns of scale without saying what the function is.

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