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||'''Unit'''|| ECC2810 ||
||'''Topics'''|| Indirect Utility Functions ||

 1. Suppose a consumer faces the following utility maximization problem:
 $$\max_{x} u(x) = (1+x_1)(1+x_2) \quad : \quad p\cdot x \leq y$$
 Construct the consumer's ''indirect utility function v(p,y)'' and demand functions $x_i(p,y)$ (i=1,2).
 2. Prove that the solution to a consumer's constrained utility maximization problem in $\mathbf{R^2_+}$ will occur when the slope of the indifference curve equals the slope of the budget line. That is, prove that:
 $$\frac{\delta u}{\delta x_1}\bigg/\frac{\delta u}{\delta x_2} = \frac{p_1}{p_2}$$
 3. Consider the general ''indirect utility function'', $v(p,y) = \max_{x}\{u(x) | p \cdot x \leq y, x\geq 0\}$ and a useful identity known as ''Roy's Identity'', namely that $x^*_i = -\frac{\delta v}{\delta p_i}\bigg/\frac{\delta v}{\delta y}$.
 To show that in general,
  1. $\frac{\delta v}{\delta y} = \lambda^*$;
  2. $\frac{\delta v}{\delta p_i} = -\lambda^* x_i^*$;
  3. And in particular, for the Cobb-Douglas Utility function, $u(x) = \prod_{i=1}^m x_i^{\alpha_i}$, that, $x_i^* = \frac{\alpha_i y}{p_i}$ (assuming that $\sum_{i=1}^m \alpha_i = 1$).
 4. For the standard budget minimization problem, where the ''expenditure function'', $e(p,\bar{u})$ is given by: $$e(p,\bar{u}) = \{min_x p\cdot x | u(x)\geq\bar{u},x\geq 0\}$$ show that a good's price effect on the consumer's expenditure is given by the optimized demand for that good. That is,
 $$\frac{\delta e}{\delta p_i}(p,\bar{u}) = x^*_i(p,\bar{u})$$
 5. (Riley, Challenge) Suppose that a consumer has the utility function, $$u(x) = \sum^2_{i=1}-\alpha_i e^{-Ax_i}$$ and faces a budget constraint ''y'' and price vector ''p''.
  1. Show that the consumer's optimal consumption bundle can be represented by, $$x^*_2 - x^*_1 = a + b \ln \frac{p_1}{p_2}$$ where ''a'' and ''b'' are constants.
  1. (harder) Using the above result, show that the Marginal Rate of Substitution has the form, $$MRS = B\left(\frac{p_1}{p_2}\right )^c$$ where ''B'' and ''c'' are constants.