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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?

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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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