Is there an economic interpretation of the envelope theorem?
First, let's consider a standard &quot;economic interpretation&quot; of the envelope theorem for unconstrained maximization:
if $x^*(b)$ solves $\max_x f(b,x)$, then $\frac{d f(b,x^*(b))}{d b_k} = \frac{\partial f(b,x^*(b))}{\partial bk},$ i.e., the total derivative of $f$ w.r.t. $b_k$ equals the partial derivative of $f$ w.r.t. $b_k$.
The economic interpretation (in the given problem): Let $\pi^*(p,w) = p f(x^∗) − w·x^*$ be the maximized value of profits given output price $p$ and input price vector $w$. Then the $i$'th input demand function is $x^*_i(·) = −\frac{\partial \pi^*(·,·)}{\partial w_i}$, known as Hotelling's Lemma, after Harold Hotelling, (1895–1973). This derivative is negative: if the price of the input increases, then the firm's maximal profit decreases.

Second, let's consider a standard &quot;economic interpretation&quot; of the envelope theorem for constrained maximization:
if $x^*(b), \lambda^*$ solves $\max_x f(b,x)$ s.t. $h^j(b,x) = 0$, $j= 1, ...m,$ then
$$\frac{d f(b,x^*(b))}{d b_k} = \frac{\partial f(b,x^*(b))}{\partial b_k} + \sum_{j=1}^m \lambda_j^*(b)\frac{\partial h^j(b,x^*(b))}{\partial b_k}, $$
i.e., the total derivative of $f$ w.r.t. $b_k$ equals the partial derivative of $f$ w.r.t. $b_k$, plus the $\lambda^*$-weighted sum of the partial derivatives of the $h^j$'s w.r.t. $b_k$.
The economic interpretation (in the given problem): Let $\hat c(\bar q, p, w) = w·\hat x$ be the minimized level of costs given prices $(p,w)$ and output level $\bar q$. Then the $i$'th conditional input demand function is $\hat x_i(·) = −\frac{\partial \hat c(·,·,.)}{\partial w_i}$,  known as Shepard's Lemma, after Ronald Shephard  (1912-1982). The partial derivatives of the expenditure function with respect to the prices of goods equal the Hicksian demand functions for the relevant goods.