Is there an economic interpretation of the envelope theorem?

Question

I wonder if envelope theorem has also some hidden economic interpretation. For example, Lagrangian multiplier in economics can have interpretation of 'shadow price' which is useful economic concept.

The envelope theorem seems like something that should have some deeper economic interpretation as it is about examining how function adjusts when parameters change given that we are at some optimum. The envelope theorem says that only the direct effects of a change in an exogenous variable need be considered, even though the exogenous variable may enter the maximum value function indirectly as part of the solution to the endogenous choice variables. But I am not sure if this makes sense. Is there any economic concept attached to the envelope theorem?

Answer 1

First, let's consider a standard "economic interpretation" of the envelope theorem for unconstrained maximization:

if $x^*(b)$ solves $\max_x f(b,x)$, then $\frac{d f(b,x^*(b))}{d b_k} = \frac{\partial f(b,x^*(b))}{\partial b_k},$ i.e., the total derivative of $f$ w.r.t. $b_k$ equals the partial derivative of $f$ w.r.t. $b_k$.

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.

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Second, let's consider a standard "economic interpretation" 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.