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In the Solow-Swan Model, why is the (n+δ)k curve drawn as a line from the origin? What is the economic intuition underlying the same? Also, what is the economic intuition underlying the fundamental differential equation of the Solow-Swan model? [ k̇ = s.f(k) - (n+δ)k]

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In the Solow-Swan Model, why is the (n+δ)k curve drawn as a line from the origin?

In the graph output per capita ($y$) is graphed on $y$-axis and capital per capita $k$ on $x$-axis. Solow-Swan model typically uses Cobb-Douglas production function where (recall all is in per capita terms, your textbook might have full production function as $Y=AK^{\beta}L^{1-\beta}$, below we divide both sides by $L$ and $x=X/L$):

$$y= Ak^{\beta}$$

As a consequence of the above when $k=0$ both $y=0$ and $(n+\delta)k=0$ so you start at origin.

What is the economic intuition underlying the same?

The intuition is that since in the selected (Cobb-Douglas) production function to produce anything you need to have at least some capital, without any capital it is impossible to produce anything.

Also, what is the economic intuition underlying the fundamental differential equation of the Solow-Swan model? [ k̇ = sf(k) - (n+δ)k]

The $\dot{k} = sf(k) - (n+δ)k \quad$ in plain English says:

Change in per capita capital is equal to portion of saved per capita output $sf(k)=sy$ minus the depletion of capital stock per capita caused either by depreciation (damage to machines or buildings over time as they are used) or growth of population which depletes the per capita stock of capital since well its all in per capita terms so more people means less capital per person.

Its basically just common sense that change in per capita capital stock is equal to addition of new capital (through saving output) minus depreciation of capital or decrease of per capita capital stock.

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