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||'''Unit'''|| ECC5650 ||
||'''Topics'''|| Gambles, Returns to scale ||

$$\sigma_{ij} = \frac{d\ln(x_j/x_i)}{d \ln MRTS_{ij}(x)} = \frac{d\ln (x_j/x_i)}{d \ln(f_i(x)/f_j(x))}$$
$$R_a(w) = \frac{-u''(w)}{u'(w)}$$

 1. [JR 2.23] Consider the utility function, $$U(w) = a + bw + cw^2$$
  1. What restrictions must be placed on parameters, ''a'', ''b'', and ''c'' for this function to display risk aversion?
  1. Over what domain of wealth can a quadratic VNM utility function be defined?
  1. Given the gamble $$g = ((1/2)\cdot (w+h),(1/2)\cdot(w-h))$$ show that $CE < E(g)$ and that $P > 0$.
  1. Show that this function, satisfying the restrictions in part (a), ''cannot'' represent preferences that display ''decreasing'' absolute risk aversion.
 1. Provide a proof that the CES utility function $$y=(x_1^{\rho}+x_2^{\rho})^{\frac{1}{\rho}}$$ actually exhibits ''constant elasticity of substitution'' and thus deserves its name.
 1. [JR p.125] Consider the production function $$y = k(1+x_1^{-\alpha}x_2^{-\beta})^{-1}$$
  1. Show that this function exhibts ''variable'' returns to scale;
  1. Find the output level $\bar{y}$ for which this function exhibits ''constant returns to scale''.