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||'''Unit'''|| ECC5650 ||
||'''Topics'''|| Slutsky Equation, Shepherd's Lemma, Elasticities ||

A useful identity, ''Shephard's Lemma'' $$x_i^h(\mathbf{p},\bar{u}) = \frac{e(\mathbf{p},\bar{u})}{\delta p_i}$$

 1. Provide a proof for the Slutsky Equation, $$\frac{\delta x_i(\mathbf{p},y)}{\delta p_j} =\frac{\delta x_i^h(\mathbf{p},\bar{u})}{\delta p_j} - \frac{\delta x_i(\mathbf{p},y)}{\delta y}x_j(\mathbf{p},y)$$ recalling that, $x^h(\mathbf{p},\bar{u}) = x(\mathbf{p},e(\mathbf{p},\bar{u}))$.
 2. Graphically illustrate and explain the resulting change in $x_i$ for a given change in $p_j$ by appealing to the components of the Slutsky Equation.
 3. Given that the following elasticity relations hold for the Marshallian Demand Functions,$$\eta_i=\frac{\delta x_i(\mathbf{p},y)}{\delta y}\frac{y}{x_i(\mathbf{p},y)} \quad \quad \quad \epsilon_{ij} = \frac{\delta x_i(\mathbf{p},y)}{\delta p_j}\frac{p_j}{x_i(\mathbf{p},y)}\quad\quad\quad s_i = \frac{p_i x_i(\mathbf{p},y)}{y}$$ show that:
  1. ''Engel aggregation'' is given by: $$\sum_{i=1}^m s_i\eta_i = 1$$
  2. ''Cournot aggregation'' is given by: $$\sum_{i=1}^m s_i \epsilon_{ij} = -s_j \quad\quad, \forall \, j = 1,\dots,m$$
  ''(Hint: make use of Walras' Law, $y = \mathbf{p}\cdot x(\mathbf{p},y)$.)''