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- A firm’s production function is - y = f(X1, X2)= X11/2 + X1X2 , Where X1≥0, X2≥0 1. Write down the firm’s production possibility set, and its input requirement set. 2. Is this production function concave, quasi-concave? 3. Is this production function homogenous? 4. Find its returns to scale when X1=1, and X2=1A firm can manufacture a product according to the production function Q=3K1/2L1/2, and capital is fixed at 4. a) When the firm hires 16 units of labor. The average product of labor is......... b) when the firm hires 16 units of labor, the marginal product of labor is....... C) if the firm can sell its output at a price of $ 10 per unit and can hire labor at $ 10 per unit , it should hire?............units of labor maximize the profits.A firm can manufacture a product according to the production function Q = F(K, L) = K0.5L0.5. (a) What is the average product of labor, APL, when the level of capital is fixed at 36 units and the firm uses 16 units of labor? (b) What is the marginal product of labor, MPL, when capital is fixed at 36 units? (c) Suppose capital is fixed at 36 units. If the firm can sell its output at a price of $100 per unit of output and can hire labor at $40 per unit of labor, how many units of labor should the firm hire in order to maximize profits?
- Show that the Cobb-Douglas Production Function ?(?, ?) = ??^??^? with α+ β=1 satisfies Euler’s Theorem.Let y = f(x1, x2)=x11/2 + x1x2 be a firm’s production function, where x1≥0, x2≥0. Write down the firm’s production possibility set, and its input requirement set. Is this production function concave, quasi-concave? Is this production function homogenous? Find its returns to scale when x1=1, and x2=1.Assume that Donnell Corp. is currently producing 500 units of output per period, using 25 units of labor and 20 units of capital. Values for the marginal product of each input and the prices of the inputs are as follows: MPK = 100, MPL = 200, w = 2, and r = 3. Given the information above, which of the following is true? a. The firm is currently using the optimal levels of capital and labor. b. The firm should increase labor and reduce capital usage. c. The firm is not using the optimal levels of capital and labor, and it is impossible to determine the optimal levels from the given information.
- 1. Inputs K, L, R and M cost £10, £6, £15 and £3 respectively per unit. What is the cheapest way of producing an output of 900 units if a firm operates with the production function Q = 20K0.4L0.3R0.2M0.25? 2. Make up your own constrained optimization problem for an objective function with three variables and solve it. 3. A firm faces the production function Q = 50K0.5L0.2R0.25 and is required to produce an output level of 1,913 units. What is the cheapest way of doing this if the per-unit costs of inputs K, L and R are £80, £24 and £45 respectively?Economics Suppose that production for good X is characterized by the following production function, Q = 4K0.5L0.5, where K is the fixed input in the short run. If the per-unit rental rate of capital, r, is $12 and the per-unit wage, w, is $20, then the average total cost of using 25 units of capital and 49 units of labor is Multiple Choice $6.25. $9.14.Correct $10.07. incalculable since there is insufficient information to determine the average total costs.For function Y= 10+0.3x-0.006x2 suppose that price of the output is $5 and the price of input is $3 what level of input use will maximize the total value of the product
- A bitcoin miner, Alex, needs only electricity (E) and computer (K) to mine bitcoin. Assume that the production function for his bitcoin business is of Cobb-Douglas type,?(?,?)=???? with?+?< 1, resulting in strictly convex isoquants and is the same in South Korea and USA. Suppose that, similar to the podcast, the price per unit of electricity is higher in South Korea than in USA. Suppose that the price of a computer is the same in both countries. i. Determine whether it is more expensive to mine one bitcoin in South Korea than in USA based on the above assumptions by using appropriate diagram and explain your answer. Please keep electricity on the horizontal axis and computer on the vertical axis while drawing your diagram.Constrained Optimization: Cobb-Douglas Production Function A firm operates with a Cobb-Douglas Production function: Q = 12K 0.4L 0.4 where K is units of capital, and L is number of laborers. To produce an output, the firm must pay $40 per unit of capital, and $5 per laborer. However, the firm has a budget of $800 only to spend for labor cost and capital cost. 1. Using your knowledge of the tangency condition in Producer’s theory, find the combination of K and L that the firm should use to produce the maximum possible output. Do not solve the problem using the Lagrangian method. Note: The tangency conditions just states that the slope of the production function must be equal to the slope of the isocost function.Q2. Suppose the production of airframes is characterized by a CobbDouglas production function: Q =LK. The marginal products for this production function are MPL = K and MPK = L. Suppose the price of labor is $10 per unit and the price of capital is $1 per unit. Find the cost-minimizing combination of labor and capital if the manufacturer wants to produce 121,000 airframes.