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