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![4) Find the elasticity of substitution (o) for the following production functions.
a.) q = 10 K0.²L0.8
b.) q = 2 [0.2K-2 + 0.8L¬2 ]-0.5](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fea0084d8-63e7-4f56-975c-add2c9289a98%2Fbdca8cd4-dcf6-4d19-afa4-627660f31eec%2Fdenp8od_processed.jpeg&w=3840&q=75)
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- (a) For the cost function C(w1, w2, y) = 2y²w} w, calculate the Allen elasticity of substitution between the two inputs at the cost-minimizing input point (xf(w1, w2, y), a(w1, w2, y)). (b) Consider the production function f(r, y, z) = Vry + rz+ yz. Find the scale elasticity SE at (x, y, z) = (1,2, 3), (5, 1,6), (6, 6, 6) and determine if the pro- duction function is IRTS, CRTS, or DRTS locally at each point. (c) A profit maximizing firm in the market operates where the production exhibits decreasing return to scale (DRTS). Is this market in its long-run equilibrium? Justify your answer. (d) Suppose that there are the infinite number of potential firms that produce the identical output good y under the cost function C(y) = + 3. Assume free entry and exit. Find the long-run equilibrium output price p, the amount of the output that each firm in the market produces in the long-run equilibrium, and the value of profit that each firm earns in the equilibrium.Q3. Refer to the diagram. Using the midpoint formula, calculate the price elasticity of demand between the prices of $15 and $12. Accordingly, state whether demand is elastic or inelastic between these two points. P$/unit 15 12 D 18 22 Q (units/week) Ep = Damand ie ... Q4. For each case below, answer the bolded questio Classification of the Case Calculations product(s) if requested to do so 1. Suppose that a 2% increase in income in the economy decreases the quantity of gadgets demanded by 1% ar every E,= Gadgets are possible price. Find the income elasticity of demand and dassify the product accordingly (state whether gadgets is a normal, necessity, luxury or an inferior product). 2. A firm finds that its price elasticity of demand is 4.0. Currently, the firm is selling 2000 units per month at $5 per unit. Price must be lowered by= If it wishes to increases its quantity sold by 10%, by how much it must lower its price? 1 Suppose legalization-and subseque nt regulation-of products Xand…The market for gravel has the following demand and supply relationships: Supply function: Q = 100P - 1,000 Inverse demand function: P = 50 - 0.01*Q + PX, where P represents price of gravel per ton in dollars, Q represents sales of gravel per week in tons, and PX is the price of some other product X in dollars per unit. Let PX = $50/ton In a diagram, qualitatively describe the change that would occur in the market for gravel (i.e. equilibrium price and quantity) if a new discovery has just made the production of product X cheaper. Briefly explain whether it is a movement along or shift of demand curve and supply curve for gravel. In addition to the new discovery regarding product X in previous question), suppose now workers producing gravel ask for sick leave due to COVID. Use supply and demand analysis to predict how these two shocks will affect equilibrium price and sales. Illustrate your results in a diagram. Is there enough information to determine if market prices will rise or…
- Consider the following production function for shirts: q = v6 L3/4K/4, where L is worker-hours, and K is sewing machine-hours. a. Compute the marginal products of labor and capital, the average product of labor, and the marginal rate of technical substitution of labor for capital (i.e. how many units of capital are needed to make up for the loss of one unit of labor)? b. Are there diminishing returns to labor (that is, does the marginal product of labor decrease when labor L increases)? What about to capital? Is there diminishing marginal rate of technical substitution (MRTS)? с.Quèstion 13 Suppose that the price changed from P1 to P2 in the graph below. Sdomestic A P1 G H P2 B K Ddomestic Q1 QO Q2 Using capital letters and reading from left to right and top to bottom (e.g. AGH or HELJK identify the following: Total variable production cost of producing Qo units (at price P1): New variable production cost of producing Q1 units (at price P2): Change in total variable production from reducing production for Qo to Q1Juan Valdez owns a coffee farm in Colombia. His production function is: f(x1,x2)=(x1−1)^0.25 x2^0.5 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?
- The quantity demanded x each month of Russo Espresso Makers is 250 when the unit price p is $150; the quantity demanded each month is 1000 when the unit price is $120. The suppliers will market 750 espresso makers if the unit price is $50. At a unit price of $80, they are willing to market 3000 units. Both the demand and supply equations are known to be linear. (a) Find the demand equation.p = (b) Find the supply equation.p = (c) Find the equilibrium quantity and the equilibrium price. equilibrium quantity units equilibrium price $2) Imagine that you have started a firm and own the patent for a new technology that allows for the 3D printing of food. This is a huge nutritional breakthrough since the printer can provide nutritional food for years. Since you are the first with this technology and have patented it, your firm is the only firm that can produce and sell this type of printer. There are no close substitutes for your printer and your data indicates weekly demand is given by the equation Qa = 9000 - 2P. a) Your production analysis indicates that your firm's cost function is C(Q) = F +Q². Illustrate the market, 2 showing the inverse demand curve, the MR curve and the MC curve. Then, compute and illustrate your firm's profit maximizing price and quantity of the printer. How low must the fixed cost be to ensure that the firm is making positive profits? Make sure to clearly label all relevant curves. Finally, what is the markup on the printer (measure markup in the following way: Price/MC)? b) Redraw the…1. Demonstrate that ɛ = Ex,l + Ex,k Note that: aA /\ = Ək / k = Əl / 1 And Ex.l is the elasticity product of the factor l. Similar for the factor k. 2. Find the o of: (with the process) %3D x (L, K) = aLP + bKP)ɛ/p If p<1 and a = 1; b = 1 %3D
- When sold for $790.00, a certain desktop has an annual supply of 129.5 million computers and an annual demand of 155.5 million computers. When the price increases to $865.00, the annual supply increases to 147.5 million computers, and the demand drops to 134.5 million computers. NOTE: Round slope and vertical intercept to 4 decimal places and use those rounded values to the end. (a) Assuming that the supply and demand equations are linear, find the supply and demand equations. Supply Equation p = Demand Equation p = esc (Note: The equations should be in the form p = mq + b where p denotes the price (in dollars) and q denotes the quantity (in billions). The slope and y-intercept should be accurate to 4 decimal places). (b) Find the Equilibrium price and quantity. Equilibrium price p = Equilibrium quantity q = 9- F2 A (Note: The equilibrium price should be accurate to 2 decimal places and quantity should be rounded to the nearest whole number, and the equilibrium price should include a…Constrained Optimization: Cobb-Douglas Production Function:1. Based from the factor shares of the two inputs, what will happen to the number of output if it 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?Production. Assume a production with two variable inputs (L, K). Show graphically the following and explain: (a) fixed-proportions production function (b) inputs are perfect substitutes
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