MediumCalculusca-169

Envelope Theorem Derivative

Problem

Maximize f(x, theta) = a*x - 0.5*b*x^2 + c*theta*x over x. The optimizer is x*(theta) = (a + c*theta) / b and the value function is V(theta) = f(x*(theta), theta).

By the envelope theorem V'(theta) equals the partial derivative of f with respect to theta at the optimum, which is c*x*. Given a, b != 0, c, and theta, return V'(theta).

Hint

Differentiate f with respect to theta, then substitute the optimal x.

Test cases

case 1

in: [1,2,3,0]

expected: 1.5

case 2

in: [0,1,2,5]

expected: 20

case 3

in: [2,4,-1,1]

expected: -0.25

case 4

in: [5,1,0,3]

expected: 0

Starter codePython

def envelope_derivative(a, b, c, theta):
    # Your code here
    pass

4 test cases · run your code against them in the editor