# How can I show convexity of this value function?

I have set up an optmization problem as follows:

$$V(A)=\max_{l, C} \quad u(C,l)$$

Where the only constraint is as follows:

$$C=f(l,A)$$

Here $$u$$ is the utility function which captures social welfare. $$f$$ is the production function that produces consumption $$C$$.

$$l$$ represents labour and so the utility function $$u$$ is decreasing in the $$l$$ input, while the production function $$f$$ is increasing in $$l$$. Clearly, $$u$$ is increasing in terms of consumption $$C$$. Finally, $$A$$ is a state variable, which is a negative input to the production function, i.e. more $$A$$ equals less production.

Here I assume the utility function captures the usual concavity/convexity properties, capturing decreasing marginal benefit of good things and increasing marginal cost of bad things. I also assume $$f$$ is a continuous function.

Now my question is: can we conclude that this value function is convex in terms of $$A$$?

My approach to this has been to use the following theorem:

Suppose $$V$$ is the upper envelope of convex functions i.e. $$V (a) = \max_{b} v(a, b)$$ where $$v(·, b)$$ is a convex function for each $$b$$, then $$V$$ is convex.

However, I am not entirely sure how to apply it here, any ideas?

## 1 Answer

Suppose that $$u(C,l)=\sqrt{C}-l^2$$ and $$f(l,A)=\big(l+g(A)\big)^2$$, where $$g$$ is any function of $$A$$ that is not convex.

Then $$u\big(f(l,A),l\big)=l+g(A)-l^2.$$ The optimal labor supply is given by $$1/2$$. So the value function is given by

$$V(a)=1/4+g(A).$$ Since $$g$$ is not convex, the value function is not convex either. Clearly, you need more assumptions on $$f$$.

• Thank you for you response. I understand what you are saying. So in order to have V(A) be convex in terms of A would the production function f need to be convex in terms of A? Or are there some other assumptions that tend to be made if one wishes to ensure this property? Commented Apr 19, 2022 at 14:04
• @L.Johnson I think something like that, but I'm not sure what exactly is needed. The point of the example is kinda that without any restriction on $f$, the assumption of decreasing marginal benefit in $C$ is meaningless. Commented Apr 19, 2022 at 15:24
• I see what you are saying. Would it make a difference if I assumed that the production function f was concave in the A input? Or would I then still have to make assumptions about u? Commented Apr 20, 2022 at 19:34
• @L.Johnson Concavity might be enough; I'm not sure about that. Commented Apr 20, 2022 at 20:41