# Solow model - concavity vs strict concavity

I'm studying the Solow growth model from the Acemoglu's book.

Consider the following standard assumptions:

The fact that $$F()$$ exhibits constant returns to scale means that F() is linear homogenous. Why the book says that $$F()$$ is concave? Why it is not strictly concave?

• I'm not certain that I've understood your question fully. If the book assumes that $F$ is concave, this is a weaker condition than assuming it is strictly concave. Therefore, the analysis conducted in the book, which holds for concave functions, will also hold for strictly concave functions.
– Amit
Commented Apr 1, 2023 at 0:49
• Thanks for your answer. The book says : "Prove that Assumption 1 implies that F(A,K, L) is concave in K and L but not strictly so." Commented Apr 1, 2023 at 13:35
• @JohnM. Are you asking if there any concave functions that are not strictly concave, or what exactly is your question here? Commented Apr 1, 2023 at 18:45
• @JohnM. Let me ask you a follow-up question. Can you think of any production function $F$ that satisfy assumption $1$?
– Amit
Commented Apr 1, 2023 at 21:52
• Thanks for your replies. Assumption 1, in my understanding implies that $F$ is strictly concave. Strict concavity implies concavity. Why the book says that $F$ is concave only (and not strictly concave as well)? Commented Apr 2, 2023 at 8:38

If you consider the production function $$f(K, L) = F(K, L, A = 1)= K^{\frac{1}{2}}L^{\frac{1}{2}}$$, this function meets all the conditions specified in assumption 1 within the interior, but not on the axes due to its lack of differentiability on the axes. Typically, this detail is not explicitly mentioned. Check that $$f(K, L)= K^{\frac{1}{2}}L^{\frac{1}{2}}$$ is concave and not strictly concave. To show that this is not strictly concave, consider $$(K_1, L_1)=(0,0)$$, $$(K_2, L_2)=(2,2)$$ and $$\lambda = \frac{1}{2}$$,
$$f(\lambda (K_1, L_1) +(1-\lambda) (K_2, L_2)) = f(1,1) =1 =\frac{1}{2}(0) + \frac{1}{2}(2) = \lambda f(K_1, L_1) + (1-\lambda) f(K_2, L_2)$$
• I understand the definition of concave function as you wrote above, and makes sense. But I also know that if the second (partial) derivatives are strictly negative, then $F$ is strictly concave. Why this is not the case? Commented Apr 2, 2023 at 10:46
• @JohnM. The example example is a counter-example to your claim "if the second (partial) derivatives are strictly negative, then $F$ is strictly concave". Your claim is wrong which is why you have a counter-example.