# Proving that the utility is concave

Consider a household which solves the following problem: $$v(k,r,w)=\underset{c,l\in B{(k,r,ω)}}{\ {max}} \{u(c,l)\}$$

where $$u : R_+^2 \rightarrow R$$ is a strictly concave, twice continuously differentiable, strictly increasing function in its two arguments: consumption, $$c$$, and leisure, $$l$$. The constraints the household must obey in selecting $$c, l$$ are summarized by $$B$$: $$B(k, r, w) = {\{c, l : 0 ≤ c ≤ rk + w(1 − l), 0 ≤ l ≤ 1\}}$$

Here, $$k, r, w > 0$$ are numbers over which the household has no control. Prove that $$v$$ is concave in $$k$$ and that the derivative of $$v$$ with respect to $$k$$ exists for ‘interior $$k$$’. Display a formula for the derivative of $$v$$.

What I was thinking for solving this is by following Benveniste & Scheinkman theorem on differentiability $$ω : D → R$$ defined on the neighborhood $$D$$ of $$x_0$$, i.e. $$D ⊂ X$$ and $$x_0 ∈ int(D)$$ such that: $$ω(x) ≤ v(x)$$ and $$ω(x_0) = v(x_0)$$.

And $$ω(.)$$ is concave and differentiable, then $$v(.)$$ is differentiable at $$x_0$$ and $$v'(x_0) =ω'(x_0)$$.

I'm guessing we have to substitute c with $$rk + w(1 − l)$$ in the utility function, but I'm confused a bit with leisure. Because after that, I think we should just get the envelope theorem or not?

• Note that you could write $v(kr,w)$ instead of $v(k,r,w)$, this simplifies the expression of the derivatives of $v$ wrt $k$ and $r$. Dec 7 '20 at 11:38
• @Bertrand I'm sorry, I'm not following. Could you please elaborate more on that? Dec 7 '20 at 12:03
• If you define $z = rk$ and replace it in your problem, you end up with $v(z,w)$. This establishes a relationship between the marginal utility of $k$ and of $r$. Dec 7 '20 at 12:27
• You should probably use the Inverse Function Theorem to establish that solution exists and is differentiable. Jan 9 '21 at 17:18

Sharing my answer out there, correct me if I'm wrong. $$u(c,l)=u(rk+w(1-l), l)$$
$$U$$ is strictly concave and differentiable. Let $$max$$ u attained at $$(c^*, l^*)$$ i.e $$(k^*, l^*)$$.
Then $$v(k^*)=u(k^*)$$ and $$u(k) ≤ v(k)$$
Then by Theorem 4.10 (Benveniste & Scheinkman), $$v$$ is differentiable at $$k^*$$.
$$V_k^*=U_k(rk^* + w(1-l^*, l*) = U_c(rk^* + w(1-l^*, l*)$$
• Just few comments: 1). It is not (fully) clear to me how this is related to Benveniste and Scheinkman (who, in their 1979 paper, propose results that apply to dynamic problems). 2). Why do you write $k^*$? Is it optimally chosen? This fully change the statement of you question in which $k$ is exogenous. 3). You should avoid using the same notations for different functions $u(c,l)$ and then $u(k)$ Dec 7 '20 at 18:52