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Consider a production function $$y = F(A,K,L)$$ where $A$ represent "technology" (in the broad sense including for example also organizational technology), $K$ is capital and $L$ is labor (again both broadly defined) "Economies of scale" (a looser synonym of increasing returns to scale), is, in economics, always defined with respect to all arguments in ...


5

You want to find a relation between $tF(z)$ and $F(tz)$ for all $t>1$ (or $0$ for CRS). So since $2t=tF(z)>F(tz)=2$ for all $t>1$, we see decreasing returns to scale.


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The problem is to distinguish between changes along the production function, from $f(K_1, L_1)$ to $f(K_2, L_2)$, and changes of the production function, from $f(K_1, L_1)$ to $g(K_1, L_1)$. A very simple example/model. Imagine that the only input is labor and that this input doubles between t and t+1, but output more than doubles : $L_{t+1} = 2 L_t$, $Y_{t+...


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In production with more than one input, "diminishing returns" refers to what happens when we increase one input while keeping all the rest constant. "Economies of scale" is a more informal term for "increasing returns to scale" and so relates to what happens when we increase all inputs by the same proportion (while, and this is important, we keep the ...


3

Quoting from the question: $F(aK,aL)$ : simultaneous and proportional inputs increase $aF(K,L)$ : output increase But $F(aK,aL)$ is an output increase, that is why it starts with $F$. It is the output increase due to the simultaneous and proportional increase of inputs. At the same time, $aF(K,L)$ is a hypothetical output, the output were this if ...


3

What you have proven is that $$f(tx_1, tx_2) > tg(x_1,x_2)$$ which translates "scaled $f$ is higher than some other, homogeneous function $g$ scaled by the same factor". This does not prove anything about the returns to scale related to $f$, although I can see why it may appear otherwise. You can go in reverse, starting from $$tf(x_1,x_2) = t^{0.5}t^{...


3

Second derivative of cost function is actually the first derivative of marginal cost function. i.e. $$ \frac{\partial^2C(q)}{\partial q^2} =\frac{\partial}{\partial q}\frac{\partial C(q)}{\partial q}=\frac{\partial}{\partial q}MC(q) $$ Now if $\frac{\partial^2C(q)}{\partial q^2}<0$, this means that marginal cost is decreasing in output. If marginal cost ...


3

Under standard assumptions (some of which you state in your question: no externalities, etc.), no. This follows from the First Welfare Theorem. Perhaps there are departures from standard models that would support something resembling your conclusion, but my guess is that most economists would view any such departure as the absence of “perfect competition”. ...


2

When you lend your money to a bank (by depositing it), they don't just make money off of lending out your money. They also make money out of charging you for banking services. If the bank makes more money off of offering you services then they can afford to pay you a higher rate of interest (all other things equal). Check out: How do banks make money? The ...


2

In the US at least, purchases and wages cost more than rent for a fast-food type business: Purchases include some other hidden labor costs (drivers for transportation etc.) I'm not sure what's in the "other" category, perhaps insurance, lawyers etc. And I'm guessing in Europe (where you are) "value menus" might not be as common as in the US. These are loss ...


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Returns to scale is directly related to homogeneous functions: For a homogeneous function $F(x,y)$ given $\theta>0$, for simplicity, $$F(\theta x,\theta y) = \theta^r F(x,y)$$ where would we refer to $F$ being a homogeneous function of degree $r$. Here it is much clearer to see that if $r>1$, we have increasing returns to scale because for a given $\...


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If you are looking for term for the reason why the efficiencies occur it would be the increasing returns to scale.


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The above chart is from Agroecosystems, nitrogen-use efficiency, and nitrogen management by Cassman KG, Dobermann A, Walters DT, and it deals with this sort of question quite extensively for those interested in all the gory details.


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I would give you a simple answer: when the marginal cost is equal to the marginal income. This would be the equilibrium and beyond that the company would lose money.


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I found book "Intermediate Microeconomics" by John Hey, it seems to support my conclusion. And there is nothing radical. I will quote some places from the chapter 29. "One very obvious reason why a single large firm might be more appropriate in some industry is simply that a single large firm might have access to a more efficient technology than lots of ...


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Trying to avoid posting further comments above. Ok, as you suggest, let's assume that demand curve does not shift, because P=ATC condition moves the price along the deman curve. The only situation where surplus is higher under monopoly is when ATC curve shifts downward so that at the new scale of operations where P=ATC, ATC is lower than that observed in ...


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I would say it exhibits no returns to scale. What follows is a counter argument to the idea that it can exhibit returns to scale. A 'return to scale' means that production will change in response to a change in the input. A 'constant return to scale' is a straight-line function (or a portion thereof) including the origin (0, 0), with the input on the ...


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