Let's solve each part step-by-step. --- # **1. Production Function: \( F(K, L) = K^\alpha L^\beta \)** ## **(a) Returns to Scale** - **Returns to scale:** Examine \( FK, t) = (K)^\ (tL^\beta = t^{\alphabeta} Kalpha L^\beta \) - **If \( \alpha+\beta = 1 \):** Constant returns to scale - **If \( \alpha+\beta > 1 \):** Increasing returns to scale - **If \( \alpha+\beta < 1 \):** Decreasing returns to scale --- ## **(b) Marginal Products and Diminishing Marginal Product** - **Marginal Product of Capital (MPK):** \[ MP_K = \frac{\partial F}{\partial K} = \alpha K^{\alpha-1} L^\beta \] - **Marginal Product of Labor (MPL):** \[ MP_L = \frac{\partial F}{\partial L} = \beta K^\alpha L^{\beta-1} \] - **Diminishing marginal product:** The second derivative w.r.t. each input should be negative. - For capital: \[ \frac{\partial^2 F}{\partial K^2} = \alpha(\alpha-1)K^{\alpha-2}L^\beta \] Diminishing if \( \alpha < 1 \). - For labor: \[ \frac{\partial^2 F}{\partial L^2} = \beta(\beta-1)K^\alpha L^{\beta-2} \] Diminishing if \( \beta < 1 \). --- ## **(c) Examples** - **Diminishing marginal products, increasing returns to scale:** \( \alpha < 1, \beta < 1, \alpha+\beta > 1 \) - **Diminishing marginal products, constant returns:** \( \alpha < 1, \beta < 1, \alpha+\beta = 1 \) - **Diminishing marginal products, decreasing returns:** \( \alpha < 1, \beta < 1, \alpha+\beta < 1 \) --- ## **(d) Contingent Demands (Cost Minimization)** Minimize cost \( C = wL + rK \) s.t. \( Q = K^\alpha L^\beta \): Set up the Lagrangian: \[ \mathcal{L} = wL + rK + \lambda(Q - K^\alpha L^\beta) \] First-order conditions: - \( \frac{\partial \mathcal{L}}{\partial K} = r - \lambda \alpha K^{\alpha-1} L^\beta = \) - \( \frac{\partial \mathcal{L}}{\partial L} = w - \lambda \beta K^\alpha L^{\beta-1} = \) - \( \frac{\partial \mathcal{L}}{\partial \lambda} = Q - K^\alpha L^\beta = \) Solve for \( K \) and \( L \): Divide FOCs: \[ \frac{r}{w} = \frac{\alpha}{\beta} \cdot \frac{L}{K} \implies \frac{L}{K} = \frac{r\beta}{w\alpha} \implies L = K \frac{r\beta}{w\alpha} \] Substitute into constraint: \[ Q = K^\alpha \left(K \frac{r\beta}{w\alpha}\right)^\beta = K^{\alpha+\beta} \left(\frac{r\beta}{w\alpha}\right)^\beta \] \[ K = Q^{\frac{1}{\alpha+\beta}} \left(\frac{w\alpha}{r\beta}\right)^{\frac{\beta}{\alpha+\beta}} \] \[ L = Q^{\frac{1}{\alpha+\beta}} \left(\frac{w\alpha}{r\beta}\right)^{\frac{\beta}{\alpha+\beta}} \frac{r\beta}{w\alpha} = Q^{\frac{1}{\alpha+\beta}} \left(\frac{w\alpha}{r\beta}\right)^{\frac{\beta-1}{\alpha+\beta}} \left(\frac{r\beta}{w\alpha}\right) \] --- ## **(e) Effect of \( \alpha \) and \( \beta \) on Demand for Inputs** - If \( \frac{\alpha}{\beta} \) increases (i.e., alpha increases relative to beta), from above: \[ \frac{L}{K} = \frac{r\beta}{w\alpha} \] - As \( \frac{\alpha}{\beta} \) increases, \( \frac{L}{K} \) **decreases**. So, **demand for labor relative to capital falls**. --- # **2. Production Function: \( F(K, L) = K^{1/2} + L^{1/2} \)** ## **(a) Cost Minimization Problem** - **Objective:** Minimize \( C = wL + rK \) - **Constraint:** \( Q = K^{1/2} + L^{1/2} \) Lagrangian: \[ \mathcal{L} = wL + rK + \lambda(Q - K^{1/2} - L^{1/2}) \] --- ## **(b) Contingent Demands** FOCs: - \( \frac{\partial \mathcal{L}}{\partial K} = r - \lambda \frac{1}{2}K^{-1/2} = \implies \lambda = 2rK^{1/2} \) - \( \frac{\partial \mathcal{L}}{\partial L} = w - \lambda \frac{1}{2}L^{-1/2} = \implies \lambda = 2wL^{1/2} \) Set equal: \[ 2rK^{1/2} = 2wL^{1/2} \implies rK^{1/2} = wL^{1/2} \implies \frac{K^{1/2}}{L^{1/2}} = \frac{w}{r} \implies K = \left(\frac{w}{r}\right)^2 L \] Substitute into constraint: \[ Q = \left(\frac{w}{r}\right)L^{1/2} + L^{1/2} = L^{1/2}\left(1 + \frac{w}{r}\right) \implies L^{1/2} = \frac{Q}{1 + \frac{w}{r}} \implies L = \left(\frac{Q}{1+\frac{w}{r}}\right)^2 \] \[ K = \left(\frac{w}{r}\right)^2 L = \left(\frac{w}{r}\right)^2 \left(\frac{Q}{1+\frac{w}{r}}\right)^2 \] --- ## **(c) Total Cost Function** \[ C(w, r, Q) = wL + rK \] \[ = w \left(\frac{Q}{1+\frac{w}{r}}\right)^2 + r \left(\frac{w}{r}\right)^2 \left(\frac{Q}{1+\frac{w}{r}}\right)^2 \] Factor out \(\left(\frac{Q}{1+\frac{w}{r}}\right)^2\): \[ = \left[\ w + r \left(\frac{w}{r}\right)^2 \right] \left(\frac{Q}{1+\frac{w}{r}}\right)^2 = \left[w + \frac{w^2}{r}\right]\left(\frac{Q}{1+\frac{w}{r}}\right)^2 = \frac{w r + w^2}{r} \left(\frac{Q}{1+\frac{w}{r}}\right)^2 \] \[ 1 + \frac{w}{r} = \frac{r + w}{r} \] So, \[ C(w, r, Q) = \frac{w r + w^2}{r} \left(\frac{Q r}{r + w}\right)^2 = \frac{w r + w^2}{r} \frac{Q^2 r^2}{(r + w)^2} = (w + w^2/r) \frac{Q^2 r^2}{(r + w)^2} \] Alternatively, you can leave it in the more compact form above. --- ## **(d) Average Cost Function** \[ AC = \frac{C(w, r, Q)}{Q} \] --- ## **(e) Marginal Cost Function** \[ MC = \frac{\partial C(w, r, Q)}{\partial Q} \] --- # **Summary Table** | Part | Final Answer (Summary) | |------|------------------------| | 1(a) | Returns to scale depend on \( \alpha+\beta \): \(=1\) (constant), \(>1\) (increasing), \(<1\) (decreasing) | | 1(b) | Diminishing marginal product if \( \alpha < 1, \beta < 1 \) | | 1(c) | Give parameter examples as per above | | 1(d) | See contingent demand formulas above | | 1(e) | As \( \frac{\alpha}{\beta} \) increases, \( \frac{L}{K} \) decreases | | 2(a)-(e) | See cost minimization, contingent demands, total/average/marginal cost above | --- Let me know if you need any step explained in more detail!

# **Production Functions Explanation** ## **1. Production Function: \( F(K, L) = K^\alpha L^\beta \)** ### **Definition** This function models how inputs (capital \( K \) and labor \( L \)) combine to produce output. The parameters \( \alpha \) and \( \beta \) represent the output elasticities, indicating the percentage change in output resulting from a 1% change in inputs. ### **Returns to Scale** The sum \( \alpha + \beta \) determines returns to scale: - **Constant Returns:** \( \alpha + \beta = 1 \) - **Increasing Returns:** \( \alpha + \beta > 1 \) - **Decreasing Returns:** \( \alpha + \beta < 1 \) ### **Marginal Products** Marginal products measure the additional output from an extra unit of input: - **Capital:** \( MP_K = \alpha K^{\alpha-1} L^\beta \) - **Labor:** \( MP_L = \beta K^\alpha L^{\beta-1} \) Diminishing returns occur when \( \alpha < 1 \) and \( \beta < 1 \). --- ## **2. Production Function: \( F(K, L) = K^{1/2} + L^{1/2} \)** ### **Definition** This production function expresses output as the sum of two inputs raised to the power of \( 1/2 \), indicating diminishing returns to each factor at an increasing rate. It reflects a production process that benefits from both inputs but at a decreasing rate. ### **Cost Minimization** The firm aims to minimize costs subject to producing a certain output \( Q \). The objective function is \( C = wL + rK \), where \( w \) and \( r \) are the prices of labor and capital, respectively. ### **Contingent Demands** The firm derives contingent demands for labor \( L \) and capital \( K \) based on the output level \( Q \) and input prices \( w \) and \( r \). Using the Lagrangian method, the optimal combination of inputs is determined. ### **Cost Functions** - **Total Cost:** Represents the total expenditure for producing output. - **Average Cost:** Total cost per unit of output. - **Marginal Cost:** The cost of producing one additional unit of output. This function and its derivatives provide insights into cost efficiency and production behavior.