To show that the Cobb-Douglas production function $F(K,L) = A K^a L^b$ is homogeneous of degree $a + b$, we need to prove the following:

A function $F(K,L)$ is said to be homogeneous of degree $d$ if for any scalar $t > 0$, the function satisfies:

$F(tK, tL) = t^d F(K,L)$

Step-by-step proof:

1. Start with the Cobb-Douglas function:

   $F(K,L) = A K^a L^b$

2. Apply a scalar $t$ to both inputs $K$ and $L$:

   $F(tK, tL) = A (tK)^a (tL)^b$

3. Simplify the terms:

   $F(tK, tL) = A \cdot (t^a K^a) \cdot (t^b L^b)$

4. Factor out the powers of $t$:

   $F(tK, tL) = A \cdot t^a \cdot t^b \cdot K^a \cdot L^b$

5. Combine the powers of $t$:

   $F(tK, tL) = t^{a+b} \cdot A K^a L^b$

6. Recognize that $A K^a L^b = F(K,L)$:

   $F(tK, tL) = t^{a+b} \cdot F(K,L)$

Conclusion:

Since $F(tK, tL) = t^{a+b} \cdot F(K,L)$, the Cobb-Douglas production function is homogeneous of degree $a + b$.

Would you like further details on homogeneity or other aspects of production functions? Here are five related questions you might be interested in:

1. How does homogeneity relate to returns to scale?
2. What is the economic interpretation of the parameters $a$ and $b$ in the Cobb-Douglas function?
3. Can we modify the Cobb-Douglas function to include technological progress?
4. What is Euler’s theorem on homogeneous functions, and how does it apply to production functions?
5. How does the elasticity of substitution behave in a Cobb-Douglas function?

Tip: Homogeneous functions are useful in economics to study how inputs scale together. Homogeneity of degree 1 (constant returns to scale) is especially significant for growth models.