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||'''Unit'''|| ECC5650 ||
||'''Topics'''|| Identities and the Expenditure Function ||

A useful identity: '''Roy's Identity''' $$x^*_i = -\frac{\delta v}{\delta p_i}\bigg / \frac{\delta v}{\delta y}$$


 1. For the general consumer's utility maximization problem, prove the following identities for  ''y=1'':
  1. Inverse demand function: $$p_i(\mathbf{x}^*) = \frac{\delta u}{\delta x_i}(\mathbf{x}^*) \bigg / \sum^m_{j=1}\left ( \frac{\delta u}{\delta x_j} x_j\right )$$
  1. Direct demand function: $$x_i(\mathbf{p},1) = \frac{\delta v}{\delta p_i} \bigg / \sum^m_{j=1}\left ( \frac{\delta v}{\delta p_j} p_j\right )$$
 1. Prove Shephard's Lemma (for the consumer). That is, that if $e(\mathbf{p},u)$ is differentiable in $\mathbf{p}$ at $(\mathbf{p}^0,u^0)$ with $\mathbf{p}^0 > 0$ then, $$x^h_i(\mathbf{p}^0,u^0) = \frac{\delta e(\mathbf{p}^0,u^0)}{\delta p_i}\quad , \quad i = 1,\dots,m$$
 1. For the budget minimization problem in $\mathbf{R^2}$ with utility given by the CES utility function $$u(x_1,x_2) = (x_1^{\rho}+x_2^{\rho})^{\frac{1}{\rho}}$$ ($\rho\neq 0, \rho < 1$), show that the ''Hicksian demand functions'' are given by $$x^h_i(\bar{u},\mathbf{p}) = \bar{u}(p_i^r + p_j^r)^{\frac{1}{r}-1}p_i^{r-1}\quad\quad i,j\in\{1,2\}$$ where $r = \frac{\rho}{\rho-1}$ and the corresponding ''expenditure function'' is given by $$e(\mathbf{p},\bar{u}) = \bar{u}(p_1^r + p_2^r)^{\frac{1}{r}}$$
 1. Verify your answer above by using the fact that $v(\mathbf{p},y) = y(p_1^r + p_2^r)^{-\frac{1}{r}}$ and $v(\mathbf{p},e(\mathbf{p},u)) = \bar{u}$.