# 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?

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 $$f$$.
• @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. Apr 19, 2022 at 15:24