# 1. Interpretation of \( \gamma_C \) and \( \gamma_L \) -\( \gamma_C \):** Minimum threshold of consumption required before the individual derives positive utility. - **\( \gamma_L \):** Minimum threshold of leisure required before the individual derives positive utility. **If \( C < \gamma_C \) or \( L < \gamma_L \):** The utility function is undefined for these values (since it involves the square root of a negative number), so utility is not attainable if consumption or leisure falls below this threshold. # 2. Budget Constraint Let \( h \) be hours worked. Then: \[ L + h = T \implies h = T - L \] Total income is wage income plus non-labor income: \[ C = w h + V = w(T - L) + V \] **Budget constraint:** \[ C = w(T - L) + V \] # 3. Marginal Utilities and MRS **Utility function:** \[ U(C, L) = (C - \gamma_C)^{1/2}(L - \gamma_L)^{1/2} \] ## Marginal utilities: - With respect to \( C \): \[ U_C = \frac{\partial U}{\partial C} = \frac{1}{2}(C - \gamma_C)^{-1/2}(L - \gamma_L)^{1/2} \] - With respect to \( L \): \[ U_L = \frac{\partial U}{\partial L} = \frac{1}{2}(L - \gamma_L)^{-1/2}(C - \gamma_C)^{1/2} \] ## Marginal Rate of Substitution (MRS) of leisure for consumption: \[ MRS_{L, C} = \frac{U_L}{U_C} = \frac{(C - \gamma_C)}{(L - \gamma_L)} \] # 4. Optimality Condition At the optimum, the MRS equals the real wage: \[ MRS_{L, C} = w \] So, \[ \frac{C - \gamma_C}{L - \gamma_L} = w \] # 5. Optimal Choices \( (L^*, C^*) \) We have: 1. Budget constraint: \( C = w(T - L) + V \) 2. FOC/MRS: \( \frac{C - \gamma_C}{L - \gamma_L} = w \) **Step 1:** From the MRS condition: \[ C - \gamma_C = w(L - \gamma_L) \implies C = w(L - \gamma_L) + \gamma_C \] **Step 2:** Set equal to budget constraint: \[ w(L - \gamma_L) + \gamma_C = w(T - L) + V \] \[ wL - w\gamma_L + \gamma_C = wT - wL + V \] \[ wL + wL = wT + w\gamma_L - \gamma_C + V \] \[ 2wL = wT + w\gamma_L - \gamma_C + V \] \[ L^* = \frac{wT + w\gamma_L - \gamma_C + V}{2w} \] **Step 3:** Find \( C^* \) using the budget constraint: \[ C^* = w(T - L^*) + V \] Or, using the MRS solution: \[ C^* = w(L^* - \gamma_L) + \gamma_C \] Plug in \( L^* \) from above: \[ C^* = w \left( \frac{wT + w\gamma_L - \gamma_C + V}{2w} - \gamma_L \right) + \gamma_C \] \[ = w \left( \frac{wT + w\gamma_L - \gamma_C + V - 2w\gamma_L}{2w} \right) + \gamma_C \] \[ = w \left( \frac{wT - w\gamma_L - \gamma_C + V}{2w} \right) + \gamma_C \] \[ = \frac{wT - w\gamma_L - \gamma_C + V}{2} + \gamma_C \] \[ = \frac{wT - w\gamma_L + V + \gamma_C}{2} \] # 6. Are \( L \) and \( C \) Normal Goods? A good is **normal** if its demand increases with income (\( V \)). - \( \frac{\partial L^*}{\partial V} = \frac{1}{2w} > 0 \) - \( \frac{\partial C^*}{\partial V} = \frac{1}{2} > 0 \) **Conclusion:** Both leisure and consumption are normal goods. # 7. Reservation Wage \( \tilde{w} \) The reservation wage is the lowest wage at which the person is willing to work (i.e., for which \( L^* < T \)). Set \( L^* = T \): \[ T = \frac{wT + w\gamma_L - \gamma_C + V}{2w} \] Multiply both sides by \( 2w \): \[ 2wT = wT + w\gamma_L - \gamma_C + V \] \[ 2wT - wT - w\gamma_L = -\gamma_C + V \] \[ wT - w\gamma_L = V - \gamma_C \] \[ w(T - \gamma_L) = V - \gamma_C \] \[ \tilde{w} = \frac{V - \gamma_C}{T - \gamma_L} \] - **Dependence:** - Higher \( V \) increases \( \tilde{w} \) (more non-labor income, need higher wage to entice work). - Higher \( \gamma_C \) decreases \( \tilde{w} \) (need higher minimum consumption, so more likely to work for lower wage). - Higher \( \gamma_L \) increases \( \tilde{w} \) (value leisure more, need higher wage to give it up). - **Interpretation if \( \gamma_C > V \):** The reservation wage is negative. This means the person cannot attain their minimum consumption without working, so they will work at any positive wage. # 8. Numerical Application Given: \( \gamma_C = 12 \), \( \gamma_L = 8 \), \( T = 24 \), \( V = 20 \), \( w = 4 \) ## Step 1: Calculate \( L^* \) \[ L^* = \frac{wT + w\gamma_L - \gamma_C + V}{2w} \] \[ = \frac{4 \times 24 + 4 \times 8 - 12 + 20}{2 \times 4} \] \[ = \frac{96 + 32 - 12 + 20}{8} \] \[ = \frac{136}{8} \] \[ = 17 \] ## Step 2: Calculate \( C^* \) \[ C^* = \frac{wT - w\gamma_L + V + \gamma_C}{2} \] \[ = \frac{4 \times 24 - 4 \times 8 + 20 + 12}{2} \] \[ = \frac{96 - 32 + 20 + 12}{2} \] \[ = \frac{96}{2} \] \[ = 48 \] ## Step 3: Utility at these choices \[ U^* = (C^* - \gamma_C)^{1/2} (L^* - \gamma_L)^{1/2} \] \[ = (48 - 12)^{1/2} (17 - 8)^{1/2} \] \[ = (36)^{1/2} (9)^{1/2} \] \[ = 6 \times 3 = 18 \] **Summary:** - \( L^* = 17 \) hours leisure - \( C^* = \$48 \) consumption - \( U^* = 18 \) # 9. Wage Change and Decomposition Suppose wage drops from \( w=4 \) to \( w=2 \). ## Step 1: New \( L^* \) and \( C^* \) at \( w=2 \) \[ L^* = \frac{wT + w\gamma_L - \gamma_C + V}{2w} \] \[ = \frac{2 \times 24 + 2 \times 8 - 12 + 20}{2 \times 2} \] \[ = \frac{48 + 16 - 12 + 20}{4} \] \[ = \frac{72}{4} \] \[ = 18 \] \[ C^* = \frac{wT - w\gamma_L + V + \gamma_C}{2} \] \[ = \frac{2 \times 24 - 2 \times 8 + 20 + 12}{2} \] \[ = \frac{48 - 16 + 20 + 12}{2} \] \[ = \frac{64}{2} \] \[ = 32 \] ## Step 2: Total Effect on Leisure \[ \Delta L = L^*_{w=2} - L^*_{w=4} = 18 - 17 = 1 \] **Leisure increases by 1 hour.** ## Step 3: Income and Substitution Effects - **Substitution effect:** Lower wage makes leisure cheaper (relative to consumption), so should increase leisure. - **Income effect:** Lower wage lowers income for any \( L < T \), which (since leisure is normal) decreases leisure. ### To decompose: - **Hypothetical ("compensated") wage:** Find leisure that keeps utility constant at the old utility \( U^* = 18 \), but at new wage and some hypothetical non-labor income. #### 1. Find income (\( V' \)) at \( w=2 \) such that \( U^* = 18 \) Let \( L = L^{c} \) (compensated leisure), \( C = w(T - L^{c}) + V' \): \[ U = (C - \gamma_C)^{1/2}(L^{c} - \gamma_L)^{1/2} = 18 \] But from the optimality condition at \( w=2 \): \[ \frac{C - \gamma_C}{L^{c} - \gamma_L} = 2 \implies C - \gamma_C = 2(L^{c} - \gamma_L) \] \[ C = 2(L^{c} - 8) + 12 = 2L^{c} - 16 + 12 = 2L^{c} - 4 \] Plug into utility: \[ U = (C - \gamma_C)^{1/2}(L^{c} - \gamma_L)^{1/2} = [2(L^{c} - 8)]^{1/2}(L^{c} - 8)^{1/2} \] \[ = \sqrt{2}(L^{c} - 8) \] Set equal to 18: \[ \sqrt{2}(L^{c} - 8) = 18 \implies L^{c} - 8 = \frac{18}{\sqrt{2}} = 9\sqrt{2} \] \[ L^{c} = 8 + 9\sqrt{2} \approx 8 + 12.728 = 20.728 \] #### 2. Substitution Effect \[ SE = L^{c} - L^*_{w=4} \approx 20.728 - 17 = 3.728 \] #### 3. Income Effect \[ IE = L^*_{w=2} - L^{c} = 18 - 20.728 = -2.728 \] **Interpretation:** - Substitution effect increases leisure by 3.73 hours. - Income effect decreases leisure by 2.73 hours. - The **substitution effect is stronger**. --- # **Summary Table** | Quantity | \( w = 4 \) | \( w = 2 \) | Change | |----------------------|-------------|-------------|-------------| | Leisure (\(L^*\)) | 17 | 18 | +1 | | Consumption (\(C^*\))| 48 | 32 | -16 | | Utility (\(U^*\)) | 18 | 12 | -6 | - Total effect on leisure: **+1 hour** - Substitution effect: **+3.73 hours** - Income effect: **–2.73 hours** (reducing leisure due to lower real income) - **Substitution effect dominates.**