The AI for Students.

This question gets at a really important distinction between two major types of production functions. The Cobb-Douglas function from part (a) is a workhorse, but this Constant Elasticity of Substitution (CES) function offers a lot more flexibility and realism.

Part 1: Show Constant Returns to Scale

A function has constant returns to scale (CRS) if, when you multiply all inputs by some positive constant $\lambda$, output is multiplied by that same constant.

• Let's start with the production function: $Y_t=[\alpha K_t^{\gamma }+(1-\alpha )L_t^{\gamma }{]}^{1/\gamma }$
• Substitute $\lambda K_t$ and $\lambda L_t$ for $K_t$ and $L_t$:$F(\lambda K_t,\lambda L_t)=[\alpha (\lambda K_t{)}^{\gamma }+(1-\alpha )(\lambda L_t{)}^{\gamma }{]}^{1/\gamma }$
• Distribute the exponent $\gamma$ and factor out the common term, ${\lambda }^{\gamma }$:$=[{\lambda }^{\gamma }(\alpha K_t^{\gamma }+(1-\alpha )L_t^{\gamma }){]}^{1/\gamma }$
• Now, pull the ${\lambda }^{\gamma }$ term out of the main brackets:$=({\lambda }^{\gamma }{)}^{1/\gamma }[\alpha K_t^{\gamma }+(1-\alpha )L_t^{\gamma }{]}^{1/\gamma }$
• Simplifying the exponent on $\lambda$:$={\lambda }^1\cdot Y_t=\lambda Y_t$

Since $F(\lambda K_t,\lambda L_t)=\lambda Y_t$, the function exhibits constant returns to scale.

Part 2: Show Income Shares Depend on $K_t$ and $L_t$

In a competitive market, profit-maximizing firms pay inputs their marginal product. So, the wage rate ($w_t$) is the marginal product of labor (MPL), and the rental rate of capital ($q_t$) is the marginal product of capital (MPK).

• Find the Marginal Product of Labor (MPL):$w_t=\frac{\partial Y_t}{\partial L_t}=(1-\alpha )L_t^{\gamma -1}[\alpha K_t^{\gamma }+(1-\alpha )L_t^{\gamma }{]}^{(1/\gamma )-1}$
• Find the Labor Share of Income: $\frac{w_t{L}_t}{Y_t}$$\frac{w_t{L}_t}{Y_t}=\frac{(1-\alpha )L_t^{\gamma }[\alpha K_t^{\gamma }+(1-\alpha )L_t^{\gamma }{]}^{(1/\gamma )-1}}{[\alpha K_t^{\gamma }+(1-\alpha )L_t^{\gamma }{]}^{1/\gamma }}=\frac{(1-\alpha )L_t^{\gamma }}{\alpha K_t^{\gamma }+(1-\alpha )L_t^{\gamma }}$
• Find the Marginal Product of Capital (MPK):$q_t=\frac{\partial Y_t}{\partial K_t}=\alpha K_t^{\gamma -1}[\alpha K_t^{\gamma }+(1-\alpha )L_t^{\gamma }{]}^{(1/\gamma )-1}$
• Find the Capital Share of Income: $\frac{q_t{K}_t}{Y_t}$$\frac{q_t{K}_t}{Y_t}=\frac{\alpha K_t^{\gamma }[\alpha K_t^{\gamma }+(1-\alpha )L_t^{\gamma }{]}^{(1/\gamma )-1}}{[\alpha K_t^{\gamma }+(1-\alpha )L_t^{\gamma }{]}^{1/\gamma }}=\frac{\alpha K_t^{\gamma }}{\alpha K_t^{\gamma }+(1-\alpha )L_t^{\gamma }}$

As you can see, both the labor share and the capital share are functions of the ratio of capital and labor in the economy.

A common misconception is that "constant returns to scale" automatically means that the shares of income going to capital and labor are also constant. This problem elegantly proves that is not the case. Think of it this way: A production process is like a recipe.

• A Cobb-Douglas function is a strict recipe for a cake. You must always use 70% of your budget on flour and 30% on sugar, regardless of price.
• A CES function is a more flexible recipe, like for a fruit smoothie. You can easily substitute strawberries for raspberries. If strawberries become very cheap and abundant, you'll use more of them.

This matters for debates on inequality, as it suggests that if technology makes capital much more productive (e.g., AI and robotics), the share of national income going to owners of capital could rise significantly.