Economics Answer Note #119

Question 1 Scottish Rower are a firm that provide a small amount of gas to the UK gas market. They aim to produce 30 terawatts per hour using the following production function: 𝑞 = 𝑞0.7𝑞 0.3 , where 𝑞 is their output of gas to final consumers, 𝑞 is the amount of wholesale gas and 𝑞 is the amount of capital. (a) The price of gas has recently increased from £150 terawatts per hour to £300 terawatts per hour. The rental rate has remained at £50. Use the Lagrangian method to find out the impact that the change in price has on the amounts of wholesale gas and capital that the firm uses. (12 marks) (b) Explain the intuition behind the result in (a). Explain the barriers to Scottish Rower being able to make this change. Answer (a) The change in price from £150 terawatts per hour to £300 terawatts per hour will have a significant impact on the amounts of wholesale gas and capital used by Scottish Rower, as determined by the Lagrangian method. The Lagrangian method is a mathematical optimization technique that helps to find the optimal values of multiple variables subject to constraints. In this case, the variables are the amounts of wholesale gas and capital used, while the constraints are the production function and the given price and rental rate. To find the optimal values, we need to construct the Lagrangian function as follows: 𝑞(𝑞,𝑞,𝑞,𝑞) = 𝑞−𝑞0.7𝑞0.3 + 𝑞(𝑞𝑞−𝑞−𝑞𝑞) Where 𝑞 is the Lagrange multiplier, 𝑞 is the price of gas (£300 terawatts per hour) and 𝑞 is the rental rate (£50). Next, we take the partial derivatives of the Lagrangian function with respect to each variable and set them equal to 0, in order to find the critical points: ∂𝑞/∂𝑞 = 1+𝑞𝑞 = 0 ∂𝑞/∂𝑞 = −0.7𝑞−𝑞 = 0 ∂𝑞/∂𝑞 = −0.3𝑞−𝑞𝑞 = 0 Solving the system of equations, we get: 𝑞 = 1/0.7 × 𝑞 = 1.43𝑞 𝑞 = −1/0.7𝑞𝑞 = −1.43𝑞𝑞 𝑞 = −1/0.3𝑞𝑞 = −3.33𝑞𝑞 Plugging in the given price and rental rate, we can solve for 𝑞 and get: 𝑞 = 𝑞/(1/0.7𝑞+1/0.3𝑞) = £300/(2.2833) = £131.42 Substituting 𝑞 back into the equations, we can find the optimal values of 𝑞, 𝑞 and 𝑞 as: 𝑞* = 1.43𝑞 = £188.05 𝑞* = −1.43𝑞𝑞 = −£197.18 𝑞* = −3.33𝑞𝑞 = −£6566.37 Therefore, the firm will use £188.05 worth of wholesale gas and £6566.37 worth of capital to produce £197.18 worth of final gas output, in order to maximize their profits, given the new price and rental rate. (b) The result in (a) can be explained intuitively by the concept of cost minimization. When the price of gas increases, the firm needs to adjust its inputs (wholesale gas and capital) in order to produce the same amount of output and minimize its costs. The Lagrangian method helps the firm to find the optimal combination of inputs that will result in the same output, while minimizing their costs. However, there are certain barriers that can prevent Scottish Rower from making this change. One major barrier is the availability of resources. As a small firm, Scottish Rower may not have enough funds or access to wholesale gas and capital in the required amounts to produce 30 terawatts per hour. This may limit their ability to make this change and increase their production. Another barrier could be technological limitations. The production function assumes a fixed technological state, where output can only be increased by increasing inputs in the given ratio. If Scottish Rower does not have access to more advanced technology, they may not be able to produce 30 terawatts per hour even with the suggested optimal inputs. Lastly, regulatory barriers could also limit the firm's ability to increase its production. The firm may be subject to certain regulations or restrictions on the amounts of gas they can produce, which would prevent them from taking advantage of the higher price and producing more gas. In conclusion, the Lagrangian method helps to identify the optimal combination of inputs for Scottish Rower to produce 30 terawatts per hour, taking into account the increase in gas price. However, there are certain barriers, such as availability of resources, technological limitations, and regulatory restrictions, that may prevent the firm from increasing their production and taking full advantage of the increased price. Question 2 Given the first order condition, Z xx =Z xy =0 and that Z xx .Z m >0 for minimum and z xx z xy (Z -y^ prime )^ 4 Describe comprehensively the convenient test for the second order condition under: |H| = det [[Y_{21}, Y_{12}, Y_{23}], [Y_{21}, Y_{22}, Y_{22}], [Y_{31}, Y_{22}, Y_{33}]] Answer Given the first order condition of a function, it is important to check for the second order condition to ensure that the point we have found is indeed a minimum. In this case, we are given the condition Z xx = Z xy = 0 and Z xx .Z m > 0. To test for the second order condition, we need to look at the determinant of the Hessian matrix, given by |H| = det [[Y_{21}, Y_{12}, Y_{23}], [Y_{21}, Y_{22}, Y_{22}], [Y_{31}, Y_{22}, Y_{33}]]. The Hessian matrix is a key tool in determining the nature of the critical point found, whether it is a maximum, minimum or saddle point. To better understand the role of the Hessian matrix in checking for the second order condition, we must first define what the Hessian matrix represents. The Hessian matrix is a square matrix of second order partial derivatives of a function, and it is used to determine the second order condition in optimization problems. It is represented in the following form: H = [Z xx , Z xy , Z xz ; Z yx , Z yy , Z yz ; Z zx , Z zy , Z zz] Where Z xx , Z xy , Z xz represent the second-order partial derivatives of the function with respect to x, y, and z variables, respectively. Now, to test for the second order condition under the given conditions, we need to first calculate the determinant of the Hessian matrix. If the determinant, |H|, is positive, then the critical point is a minimum. If it is negative, then the critical point is a maximum. And if the determinant is equal to 0, the test is inconclusive and further analysis is needed. In this case, we are given that Z xx = Z xy = 0 and Z xx .Z m > 0. Using this information, we can rewrite the Hessian matrix in the following form: H = [0, 0, Z xz ; 0, 0, Z yz ; Z zx , Z zy , Z zz] Thus, the determinant of the Hessian matrix is: |H| = Z xz Z yz Z zz – (Z zx )^2 Since we know that Z xx .Z m > 0, we can conclude that Z zz must be positive, as both Z xx and Z m are assumed to be positive for a minimum. Therefore, the sign of the determinant of the Hessian matrix depends on the signs of Z xz and Z yz . If Z xz and Z yz are both positive, then the determinant will be positive, and the critical point will be a minimum. If they are both negative, then the determinant will be negative, and the critical point will be a maximum. And if they have opposite signs, then the determinant will be negative, and the test will be inconclusive. In summary, the convenient test for the second order condition under the given conditions is to calculate the determinant of the Hessian matrix and analyze the signs of the second order partial derivatives Z xz and Z yz . This test will allow us to determine the nature of the critical point found, whether it is a minimum, maximum or saddle point. Question 3 ба. A firm's production function is given by Q=2K^ aleph L ^ n , Unit capital and labour costs are 44 and N3, respectively. Using the substitution method of optimisation, find the values of K and I which minimise total input costs if the firm is contracted to provide 160 units of cutput. Show that the second order condition is satisfied. Answer In order to find the values of K and L that will minimize total input costs, we will use the substitution method of optimization. This method involves isolating one variable in the production function and then substituting it into the cost function to find the minimum cost. In this case, we will isolate L first by dividing both sides of the production function by K^ aleph: Q / K^ aleph = 2L ^ n Next, we will substitute this expression for L into the cost function, which is given by the formula C = 44K + N3L. This gives us: C = 44K + N3( Q / K^ aleph )^(1/n) Now, we can minimize the cost function by taking the derivative with respect to K and setting it equal to 0: dC/dK = 44 - ( Q / K^ aleph )^(1/n) * (nQ / K^(n+1) * aleph ) = 0 Simplifying the expression, we get: 44 = nQ / K^(n+1) * aleph Solving for K, we get: K = ( nQ / 44 * aleph )^(1 / (n + 1)) Plugging in the given values of Q = 160 and n = 3, we get: K = ( 3 * 160 / 44 * aleph )^(1/4) Now, we can find the value of L by substituting this value of K into the production function and solving for L: 160 = 2( ( 3 * 160 / 44 * aleph )^(1/4) )^aleph * L ^3 L = ( 80 / aleph )^(1/3) Thus, the values for K and L that will minimize total input costs are: K = ( 3 * 160 / 44 * aleph )^(1/4) L = ( 80 / aleph )^(1/3) In order to show that the second order condition is satisfied, we need to take the second derivative of the cost function with respect to K and show that it is positive. The second derivative is given by: d^2C / dK^2 = -n(n + 1)Q / K^(n + 2) * aleph Let's plug in the values of Q = 160, n = 1 and K = ( 3 * 160 / 44 * aleph )^(1/4) to get: d^2C / dK^2 = -n(n + 1) * 160 / ( 3 * 160 / 44 * aleph )^(n + 2) * aleph d^2C / dK^2 = -4 * 160 / ( 3 * 160 / 44 * aleph )^3 * aleph d^2C / dK^2 = -2560 / 17896 * aleph^3 Since aleph is a positive constant, we can see that the second derivative is indeed positive. Therefore, the second order condition is satisfied and the values of K and L that we have found will indeed minimize the total input costs for the firm. Question 4 Part IV: Workout Questions: Attempt all question with necessary steps (has 10 points) 1. Given the dat for 10 workers on Y (output) and X (labor hours worked) as follows. Then, answer the following question: XY = 789 X2 = 668 Σ γ2 = 952 X = 8 Y = 9.6 A. Find the intercept and coefficients of the regression equation and interpret the coefficient of result? (2pt) B. Determine the estimated regression equation of Y on X? (2pt) C. Suppose the Labure's worked is 20 hrs., what is the predicted output? (1pt) D. Find the coefficient of determination (R2) and interpret the result? (2pt) E. Calculate the variance of the error term? (2pt) Answer To answer these workout questions, we will rely on the principles of linear regression, a commonly used statistical method in economics. Linear regression allows us to examine the relationship between two variables and make predictions based on that relationship. A. The intercept and coefficients of the regression equation can be found using the formula: Y = α + βX, where α is the intercept and β is the coefficient. In this case, we have XY = 789 and X2 = 668, so we can substitute these values into the formula as follows: Y = α + β(668/X). To find the intercept, α, we can use the formula: α = Y - βX. Substituting the values of X (=8) and Y (=9.6), we get: α = 9.6 – (668/8) = 9.6 – 83.5 = 73.9. Therefore, the intercept is -73.9. To find the coefficient, β, we can use the formula: β = ΣXY/ΣX2, where ΣXY is the sum of the product of X and Y values, and ΣX2 is the sum of the squared X values. In this case, we have ΣXY = 789, ΣX2 = 668 and X = 8. Substituting these values into the formula, we get: β = 789/(668*8) = 0.15. Therefore, the coefficient is 0.15. Interpretation of the coefficient: The coefficient measures the effect of a unit increase in the independent variable, X, on the dependent variable, Y. In this case, the coefficient of 0.15 means that for every unit increase in labor hours worked (X), the output (Y) is expected to increase by 0.15 units. B. The estimated regression equation of Y on X is Y = -73.9 + 0.15X. This equation can be used to make predictions about the output (Y) for a given value of labor hours worked (X). C. To find the predicted output for 20 labor hours worked, we can simply substitute X = 20 into the estimated regression equation: Y = -73.9 + 0.15(20) = 0. The predicted output for 20 labor hours worked is 0. D. The coefficient of determination (R2) measures the strength of the relationship between the independent and dependent variables. It ranges from 0 to 1, where 0 means no relationship and 1 means a perfect relationship. To find R2, we can use the formula: R2 = (ΣXY)2/(ΣX2ΣY2). In this case, we have ΣXY = 789, ΣX2 = 668 and ΣY2 = 952. Substituting these values into the formula, we get: R2 = (789)2/(668*952) = 0.928. Therefore, the coefficient of determination is 0.928. Interpretation of the coefficient of determination: This means that 92.8% of the variation in the output (Y) can be explained by the variation in the labor hours worked (X). The higher the coefficient of determination, the better the estimated regression equation is at predicting the dependent variable. E. The variance of the error term measures the amount of variation in the dependent variable that is not explained by the independent variable. To calculate the variance of the error term, we can use the formula: σ2 = Σ(Y – Ŷ)2/N, where Y is the observed value, Ŷ is the predicted value, and N is the sample size. In this case, we have Y = 9.6, Ŷ = -73.9 + 0.15(8) = -72.7, and N = 10. Substituting these values into the formula, we get σ2 = Σ(9.6 – (-72.7))2/10 = 1511. Question 5 A project requires 600 linear feet of fencing for a storage. the purchase price of fencing is $4.40/lf. the initial salvage value of a new fence starts at $1.00/lf and is reduced at the rate of 25% per year. A fence can be rented for $1.50/lf per year but has no salvage value. The interest rate is 4%. The project duration (years) at which is first becomes cheaper to purchase rather than to rent is? Answer To determine the project duration at which it becomes cheaper to purchase the fencing rather than renting, we must compare the costs of both options over time. In order to do so, we will use the Net Present Value (NPV) method, which takes into account the time value of money and allows us to compare the present value of future cash flows. First, we must calculate the present value of purchasing the fencing for a duration of n years. The cost of purchasing 600 linear feet of fencing is $4.40/lf, which amounts to $2,640. This cost will only be incurred once at the beginning of the project. Additionally, the salvage value of the fence decreases by 25% each year, meaning that it will have a salvage value of $0 after 4 years. Therefore, the present value of the salvage value can be calculated using the following formula: PV = SV/(1+r)^n where r is the discount rate (which in this case is 4%) and n is the number of years. Using this formula, the present value of the salvage value after 4 years would be $0.81. Therefore, the total present value of purchasing the fencing for a duration of 4 years is $2,639.19 ($2,640 purchase cost + $0.81 salvage value). Next, we must calculate the present value of renting the fencing for a duration of n years. Each year, the cost of renting the fence would be $1.50/lf, which amounts to $900 per year. This cost will be incurred every year for the duration of the project. Therefore, the present value of renting the fencing can be calculated using the following formula: PV = (C/r)(1-1/(1+r)^n) where C is the annual cost, r is the discount rate, and n is the number of years. Using this formula, the present value of renting the fencing for a duration of 4 years would be $3,324.49. In order to determine the project duration at which it becomes cheaper to purchase rather than rent, we must find the point at which the present value of purchasing is lower than the present value of renting. Therefore, we can set up the following equation: $2,639.19 = $3,324.49/(1+r)^n (since 4 years is the highest duration given the salvage value of $0) Solving for n, we get a duration of approximately 2.33 years. This means that if the project duration is less than 2.33 years, it would be cheaper to rent the fencing. However, if the project duration is longer than 2.33 years, it would be cheaper to purchase the fencing.