## Understanding Production Functions: Poland Spring Case Study### Given Production Function for Bottled Water:The production function provided to Poland Spring for bottled water can be expressed as:\[ Q = K + 10L^{1/2}\]where:- $ Q $ is the quantity of bottled water produced.- $ K $ represents units of capital.- $ L $ represents units of labor.### (i) Marginal Product of Capital and Labor**Question:**What is the marginal product of capital and the marginal product of labor given this production function?**Answer:**- The **Marginal Product of Capital (MPK)** is the additional output produced by using one more unit of capital while keeping labor constant. From the production function $ Q = K + 10L^{1/2} $:\[MPK = \frac{\partial Q}{\partial K} = 1\]- The **Marginal Product of Labor (MPL)** is the additional output produced by using one more unit of labor while keeping capital constant. From the production function:\[MPL = \frac{\partial Q}{\partial L} = \frac{10}{2}L^{-1/2} = 5L^{-1/2}\]### (ii) Marginal Rate of Technical Substitution**Question:**What is the marginal rate of technical substitution of labor for capital?**Answer:**The **Marginal Rate of Technical Substitution (MRTS)** is the rate at which labor can be substituted for capital while keeping output constant. The MRTS is given by the ratio of the marginal products of labor and capital:\[MRTS_{L,K} = \frac{MPL}{MPK} = \frac{5L^{-1/2}}{1} = 5L^{-1/2}\]### (iii) Short-Run Costs**Question:**Initially, Poland Spring has 0 units of capital and pays $2 per unit of capital and $3 per unit of labor. What are Poland Spring’s short-run total costs as a function of the quantity it produces? What are Poland Spring’s short-run average costs and marginal costs?**Answer:**Given:- $ K = 0 $ units of capital.- Cost of capital $ = $2 $ per unit.- Cost of labor \( = $3 \

We use the following formulas: 1) Marginal Product of Capital (MPK)= ∆Q∆K = dQdKThe marginal product…

3. Suppose that Poland Spring has the following production function for bottled water: Q=K+10L 1/2 (1) What is the marginal product of capital and marginal product of labor given this production function? (ii) What is the marginal rate of technical substitution of labor for capital? (iii) Initially, Poland Spring has 0 units of capital and pays $2 per unit of capital and $3 per unit of labor. What are Poland Spring's short-run total costs as a function of the quantity it produces? What are Poland Spring's short-run average costs and marginal costs?