Long run, short run and return to scale

Summary

Long run and short run

• Short run: At least one factor of production is fixed
• Long run: All factors of production can be varied
• Example
• Production function: $y=f\left( x_1,x_2 \right)=100\sqrt{x_1x_2}$
• Short run: $x_2$ is fixed at $x_2={\bar{x}}_2=4$
• Short run: $y=f\left( x_1 \right)=200\sqrt{x_1}$

Return to scale

• Constant return to scale: $f\left( tx_1,tx_2 \right)=tf\left( x_1,x_2 \right)$ for all $t$
• Increasing return to scale: $f\left( tx_1,tx_2 \right)>tf\left( x_1,x_2 \right)$ for all $t>1$
• Decreasing return to scale: $f\left( tx_1,tx_2 \right)<tf\left( x_1,x_2 \right)$ for all $t>1$
• Example:
• $y=f(x_1,x_2)=100\sqrt{x_1x_2}$
• $f\left( tx_1,tx_2 \right)=100\sqrt{tx_1⋅tx_2}=100\sqrt{t^2x_1x_2}=100t\sqrt{x_1x_2}=tf(x_1,x_2)$
• Constant return to scale