<p>You only provide partial information. E.g., this production function is unusual; is anything else unusual? Is depreciation still linear in <span class="math-container">$k$</span>? Is the rate of population growth constant? etc.</p>
<p>If nothing else is unusual: you can repeat the usual steps for finding a steady state, find if a <span class="math-container">$k$</span> exists for which per capita savings/investment equal per capita depreciation:
<span class="math-container">$$
s \cdot e^k = \delta \cdot k \tag{1}
$$</span>
Note that <span class="math-container">$k = 0$</span> does not work as you have
<span class="math-container">$$s \cdot e^0 = s \neq 0 = \delta \cdot 0.$$</span></p>
<p>Note that convexity only guarantees that any interior equilibrium you may find would be <em>unstable</em>; increasing (decreasing) <span class="math-container">$k$</span> will increase (decrease) the LHS of (1) more than the RHS of (1), thus a positive (negative) shock will be further reinforced by increased (decreased) <span class="math-container">$k$</span> levels.</p>

CC BY-SA 4.0
2022-03-14T12:03:21.420Z
2022-03-14T12:03:21.420Z
Giskard
<p>Suppose an economy is producing <span class="math-container">$e^k$</span> amount of output per capita if it uses <span class="math-container">$k$</span> amount of capital per capita.</p>
<p>As the production function is strictly convex I am thinking the only steady state is at <span class="math-container">$k=0$</span>;
hence the answer should be that 'this model may not have a steady state equilibrium'.</p>
<p>Can someone please let me know if my thinking is correct?</p>

<solow><steady-state>
Steady state equilibrium in Solow model with a convex production function
petearGriffin
# Steady state equilibrium in Solow model with a convex production function

(Asked by: username_0 on 2022-03-14)

Suppose an economy is producing $e^k$ amount of output per capita if it uses $k$ amount of capital per capita.

As the production function is strictly convex I am thinking the only steady state is at $k=0$;
hence the answer should be that 'this model may not have a steady state equilibrium'.

Can someone please let me know if my thinking is correct?

**Tags:** solow, steady-state

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## 1 answers

### Answer by username_1 on 2022-03-14. Score: 3 | Accepted answer

You only provide partial information. E.g., this production function is unusual; is anything else unusual? Is depreciation still linear in $k$? Is the rate of population growth constant? etc.

If nothing else is unusual: you can repeat the usual steps for finding a steady state, find if a $k$ exists for which per capita savings/investment equal per capita depreciation:
$$
s \cdot e^k = \delta \cdot k \tag{1}
$$
Note that $k = 0$ does not work as you have
$$s \cdot e^0 = s \neq 0 = \delta \cdot 0.$$

Note that convexity only guarantees that any interior equilibrium you may find would be *unstable*; increasing (decreasing) $k$ will increase (decrease) the LHS of (1) more than the RHS of (1), thus a positive (negative) shock will be further reinforced by increased (decreased) $k$ levels.


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