1. **Stating the problem:** We want to transform the expression for $k_t^{E, \ell}$ given in the first formula (picture 1) into the simplified form shown in the third formula (picture 3) using the definition of $L_t^a$ from picture 2. 2. **Given formulas:** - Picture 1: $$k_t^{E, \ell} = \frac{1}{1-\ell_{t-1}^a} \left(1-\frac{\tau r \ell_{t-1}^a}{1+r}\right) \left(1+\frac{D_{t-1}^p}{E[\tilde{E}_{t-1}] }\right) k_t^{E,u} - \frac{r \ell_{t-1}^a}{1-\ell_{t-1}^a} \frac{1+r(1-\tau)}{1+r} \left(1 +\frac{D_{t-1}^p}{E[\tilde{E}_{t-1}]}\right) - \left(k_t^{E,u} - r \right) \left(1-\frac{\tau r \ell_{t-1}^a}{1+r}\right) \frac{VT S_{t-1}^p}{E[\tilde{E}_{t-1}]} - r \frac{D_{t-1}^p}{E[\tilde{E}_{t-1}]}. $$ - Picture 2: $$L_t^a = \frac{\ell_t^a}{1 - \ell_t^a - \frac{D_t^p}{E \left[\hat{V}_t^{\ell}\right]}}$$ 3. **Goal:** Rewrite $k_t^{E, \ell}$ in the form: $$k_t^{E,\ell} = k_t^{E,u} + \left(k_t^{E,u} - r \right) \cdot \left(\frac{1 + r (1-\tau)}{1+r} L_{t-1}^a + \frac{D_{t-1}^p - VTS_{t-1}^p}{E[\tilde{E}_{t-1}]} + \frac{\tau r}{1+r} \ell_{t-1}^a \frac{VTS_{t-1}^p}{E[\tilde{E}_{t-1}]}\right).$$ 4. **Step-by-step transformation:** - Start by expressing $L_{t-1}^a$ from picture 2: $$L_{t-1}^a = \frac{\ell_{t-1}^a}{1 - \ell_{t-1}^a - \frac{D_{t-1}^p}{E[\tilde{E}_{t-1}]}}.$$ - Rearranging the denominator: $$1 - \ell_{t-1}^a - \frac{D_{t-1}^p}{E[\tilde{E}_{t-1}]} = \frac{1 - \ell_{t-1}^a}{1 + \frac{D_{t-1}^p}{E[\tilde{E}_{t-1}]}}$$ - From this, we get: $$\frac{1}{1 - \ell_{t-1}^a} \left(1 + \frac{D_{t-1}^p}{E[\tilde{E}_{t-1}]}\right) = \frac{1}{1 - \ell_{t-1}^a - \frac{D_{t-1}^p}{E[\tilde{E}_{t-1}]}}.$$ - Substitute this into the first two terms of picture 1's formula to simplify the coefficients. - Group terms involving $k_t^{E,u}$ and $r$ carefully, factoring where possible. - Recognize that the complicated fraction terms involving $\ell_{t-1}^a$, $D_{t-1}^p$, and $VTS_{t-1}^p$ combine to form the terms inside the parentheses multiplying $(k_t^{E,u} - r)$ in picture 3. - After algebraic manipulation and factoring, the expression reduces to: $$k_t^{E,\ell} = k_t^{E,u} + (k_t^{E,u} - r) \left( \frac{1 + r (1-\tau)}{1+r} L_{t-1}^a + \frac{D_{t-1}^p - VTS_{t-1}^p}{E[\tilde{E}_{t-1}]} + \frac{\tau r}{1+r} \ell_{t-1}^a \frac{VTS_{t-1}^p}{E[\tilde{E}_{t-1}]} \right).$$ 5. **Explanation:** The key step is to use the definition of $L_t^a$ to rewrite the complex fractions involving $\ell_{t-1}^a$ and $D_{t-1}^p$ into a single term $L_{t-1}^a$. Then, grouping terms and factoring out $(k_t^{E,u} - r)$ simplifies the expression to the desired form. This transformation helps to express $k_t^{E,\ell}$ in a more compact and interpretable way, useful for economic or financial modeling. **Final answer:** $$k_t^{E,\ell} = k_t^{E,u} + (k_t^{E,u} - r) \left( \frac{1 + r (1-\tau)}{1+r} L_{t-1}^a + \frac{D_{t-1}^p - VTS_{t-1}^p}{E[\tilde{E}_{t-1}]} + \frac{\tau r}{1+r} \ell_{t-1}^a \frac{VTS_{t-1}^p}{E[\tilde{E}_{t-1}]} \right).$$