3. Prove the following proposition, that an increase in w₁, the price of input i, all else constant, weakly decreases x,, the unconditional input demand for input i. Proposition. Suppose w' and w differ only in the i-th component, where w > W₁. The market price of output is fixed at p. Suppose x solves the profit maximization problem at (p, w), and x' solves the profit maximization problem at (p, w'). Then x₁ = x₁.
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- 1. A firm’s production function is , where Ldenotes the size of the workforce. Find the value of MPLin the case when: (a) L=1, (b) L=10, (c) L=100, (d) L=1000 Does the law of diminishing marginal productivity apply to this particular function? 2. Show that the price elasticity of demand is constant for demand functions of the form where A and n are positive constants. 3. The demand and total cost functions of a good arerespectively and a) Find expressions for TR, (profit) , MR, and MC in terms of Q. b) Solve the equation and hence determine the value of Q which maximizes profit. c) Verify that, at the point of maximum profit, MR=MC. 4. The cost of building an office complex, x floors high, in a prime location in Accra is made up of three components: (a) GH¢10 million for the land (b) GH¢1/4 million per floor (c) Specialized costs of GH¢10000x per floor. How many floors should the office complex contain if the average cost per floor is to be minimized? 5. The supply…Q5 A firm's production function is Q is equal to 5L2Ka) Find out the MPL and MPb) Does this production function exhibit constant, increasing, or decreasingreturns to scale?c) What is the marginal rate of technical substitution of L for K for thisproduction function Q6.Suppose the firm sells its output according to the following demand scheduleMarginalTotal Product Total Revenue eLabor Product Price Revenue Product$3.50192c2.80182.30291.80391.65471.5021.4053Fill in the remaining two columns of the table. How many workers wilhired at a wage of $72No chatgpt. Considering demand for inputs, does the relative slope of the ordinary demand for an input and the conditional demand for the same input depend on whether the input in question is a normal or an inferior input? Discuss. Should I include any mathematics expressions and diagrams for a good answer?
- Answer the Constrained Optimization: Cobb-Douglas Production Function:1. Based from the factor shares of the two inputs, what will happen to the number of output ifit the firm decides to triple both the amount of labor and capital?2. State the optimization problem of the firm.3. Solve for the formulas of the Marginal Product of Labor (MPL), and Marginal product ofCapital (MPK)4. Using your knowledge of the tangency condition in Producer’s theory, find the combinationof K and L that the firm should use to produce the maximum possible output. Do not solvethe problem using the Lagrangian method.Note: The tangency conditions just states that the slope of the production function must beequal to the slope of the isocost function.5. What is the maximum possible output that the firm could earn given the constraint it faces?2. Assume the technology is described by y =(x,x2)i and that prices of inputs are wi w: respectively. (a) Find the conditional factor demands for inputs x, and x2.2/3- 1/3 Q1. Consider the production function Q = 9LK" a. What is the level of output when L 1000 and K = 216 b. Use marginal analysis to estimate Q(1000,216) and Q(1000,217.5) c. Compute.these two values of Q to three decimal places and compare these values with your estimate in (b) above. d. The demand funetjon Q = K,P,""P;"1" is ealled a constant clasficity demand function. Compute the three clasticities (Own price, Cross Price and Income) and show that they are all constant. %3D a12,b1
- 13) You begin to work at a factory that produces Aguardiente (Firewater, a type of alcoholic beverage), the function of production its represented by the technology: Q(L, K, Z) = L³K+zł Where with offers of input perfectly elastic A) Salary w is the price of labor L. B) Interest r is the price of capital K. C) Rent q is the price of drinkable alcohol Z. Calculate The function of demand for each input The function of offer for the Firewater The function of total costs for the Firewater The maximum benefit.13) You begin to work at a factory that produces Aguardiente (Firewater, a type of alcoholic beverage), the function of production its represented by the technology: Q(L, K, Z) = L³ K+z! Where with offers of input perfectly elastic A) Salary w is the price of labor L. B) Interest r is the price of capital K. C) Rent q is the price of drinkable alcohol Z. Calculate The function of demand for each input The function of offer for the Firewater The function of total costs for the Firewater The maximum benefit.Consider a firm that produces output using industrial robots and skilled labor. Suppose it is the case that a reduction in the price of industrial robots causes the firm's labor demand curve for skilled workers to increase (i.e., to shift to the right). This implies that: There is diminishing marginal product of robots. O There is diminishing marginal product of skilled labor. O Robots and skilled labor are complements. O Robots and skilled labor are substitutes. The unit cost of skilled labor is below the unit cost of robots.
- 1. Juan Valdez owns a coffee farm in Colombia. His production function is: f (x1, x2) = (x1 – 1)0.25 x9-5. Assume the price of input 1 is r and the price of input 2 is w.. (a) Write down an expression for the technical rate of substitution. (b) Find Juan's demand for inputs conditional on the quantity y of coffee Juan wants to produce. (c) Find Juan's cost function. (d) What is the supply function of Juan's firm? 2. Show that the profit function is convex in (p, w). 3. Find the profit function for the Cobb-Douglas production function f(¤1, 12) = Ax†' x" with A > 0, a1, ¤2 > 0 and a1 + a2 0, B > 0, 0 < a < 1, and 0 + p< 1. 6. Find the profit function for the CES production function. 7. Verify Hotelling's Lemma for the CES production function with B < 1.4 The prices of inputs x1; x2; x3; x4 are p1 = 2; p2 = 2; p3 = 5; and p4 = 2. If the production function is given by f(x1, x2, x3, x4) = min{x1, x2} + min{x3, x4}, what is the minimum cost of producing one unit of output?.4. What must the relationship between marginal productivities and input prices if a firm is not choosing the optimal bundle of inputs but instead is hiring too much labor relative to the optimum?