Discretization, Integration and Interpolation

Expected Utility of Consumption

Assume consumption is given by \(C(\epsilon)=(\exp({\epsilon}))\) with \(\epsilon\) is a random variable with standard deviation \({\sigma}_{\epsilon}=0.05\).

Set utility of consumption to be: \(U(x)=\frac{x^{-\gamma}}{-\gamma}\).

Goal is to compare several ways to compute \(E_{\epsilon}(U(C(\epsilon)))\) .

We’ll use the calibration \(\gamma=40\) and \({\sigma}_{\epsilon}=0.05\).

(optional) Compute the result in closed form

Hint: \[\int_{-\infty}^{\infty}e^{-(a x^2+bx +c)} dx = \sqrt{\frac{\pi}{a}} e^{\frac{b^2}{4a}+c}\]

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Compute the result using Monte-Carlo simulations. Evaluate the variance of the estimation.

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Compute the result using Gauss-Hermite quadrature

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Interpolation Methods

Take

\[ f(x)=\frac{1}{1+25x^2}, \quad x\in[-1,1] \]

and compare several interpolation methods.

1. Implement piecewise linear interpolation on an equally spaced grid.

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2. Compute interpolation errors (max error and RMSE) on a fine grid.

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3. Plot

4. Compare with:

• nearest-neighbor interpolation on the same nodes
• natural cubic spline interpolation on the same nodes
• global polynomial interpolation on Chebyshev nodes

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5. Repeat for \(N=9\) and \(N=21\).

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6. Comments

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