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: $$y=f\left( x_1 \right)=200\sqrt{x_1}$$

• 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 $$t>1$$
• $$y=f(x_1,x_2)=100\sqrt{x_1x_2}$$
• $$f\left( tx_1,tx_2 \right)=100\sqrt{tx_1⋅tx_2}=100\sqrt{t^2x_1x_2}=100t\sqrt{x_1x_2}=tf(x_1,x_2)$$