# What would happen if we had decreasing returns to scale in the solow-swan growth model? Would there still be a steady state and perfect competition?

I understand with IRS we have explosive growth and a single firm would eventually dominate the market since MC < AC, and hence P < AC, so firms would leave the market. I am struggling to find what would happen if we had the opposite case - DRS. Would it be similarly problematic?

• "Would there still be [...] perfect competition?" Perfect competition is a modelling assumption, not a result, so I am not sure what you are asking here? Commented Apr 22 at 8:04

If you recall the simple intensive form (capital-labor ratio) of the Solow model: $$y_t = k_t^\alpha,$$ with $$0 < \alpha < 1$$, you would realize that it exhibits decreasing returns to scale (to capital-labor ratio). When the competitive firm maximizes its profit, it chooses the optimal capital-labor ratio.

In Varian's words, the decreasing-returns-to-scale is just constant-returns-to-scale but with the presence of a fixed factor (such as land). Say you have a production function with labor $$L$$ and land $$T$$ $$Y_t = L_t^{\alpha} T_t^{1-\alpha}$$ This is CRTS. But by fixing the land factor: $$T_t = \bar{T} = 1 \ \forall t$$, you have a DRTS technology: $$Y_t = L_t^\alpha$$ You can solve this model (profit maximization) in 2 ways.

(2) The first way is to solve it as a DRTS model $$\max L^{\alpha} - wL$$

(2) The second way is solving it as a CRTS model $$\max L^{\alpha} T^{1-\alpha} - wL - qT$$ and then imposing the land market clearing condition in equilibrium $$T_t = \bar{T} = 1 \ \forall t$$.

The results are the same regardless of how you solve it as a DRTS or CTRS-in-disguise. For more exposition, please read Hal Varian's Microeconomic Analysis (p.352).

• Hi teddi! Having read your answer, I don't understand what question "The short answer is no." applies to? Could you please elaborate? Commented Apr 23 at 7:30
• Thank you that's extremely helpful! Commented Apr 23 at 9:37
• @Giskard Hi. I think what I meant was the production function can have CRRS on the scale level, but when you consider the intensive form, it is actually DRRS. So it's not problematic, just 2 faces of the same coin. Also, without DRRS to the capital-labor ratio, there will be no unique steady state. Commented Apr 23 at 10:45