A household of a wife and husband get utility from hours spent watching HBO (Z). To produce one unit of hours spent on watching HBO, this household need to subscribe to HBO that cost X, and spent time watching show together (t,, tu). The home production function is Z = X · tuth: Moreover, the utility function is linear (i.e. U(Z) =Z)). Assume that each member of the household either spent time watching HB0 or out working in the labor market and get wage (wh, Ww). If you were a household planner, how do you think they should decide the optimal allocation of working hours? In a simple case, lets assume T = 1, and there is no out-out-labor market income (i.e. V = 0).
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- Suppose the price of labor used by a cost-minimizing firm decreases. The firm responds to the price change by changing its demands for certain inputs, but keeps its output constant. What happens to the firm’s use of labor? What happens to the firm’s production costs? Graphically show the new optimal bundle and associated isocost curve on the graph below.Gıven two inputs X1 and X2 with Prices W1 and W2, and a production function Y=F(X1X2) show that the marginal rate of substitution between the two inputs is equal to the ratio of their prices.A college football coach says that given any two linemen A and B, he always prefers the one who is bigger and faster. Is this preference relation transitive? Is it complete? The production function for puffed rice is given by q , where q is the number of boxes produced per hour, K is the number of puffing guns used each hour, and L is the number of workers hired each hour. Calculate the q = 1,000 isoquant for this production function and show it on a graph. If K = 10, how many workers are required to produce q = 1,000? What is the average productivity of puffed-rice workers? Suppose technical progress shifts the production function to , answer parts a and b for this new situation.
- Suppose that the production function for a phone is ? = 20K^0.5L ^0.5 . The marginal product of labor is 10(k/l)^0.5 , and the marginal product of capital is 10(L/K) ^0.5 . Suppose that labor can be hired for $6 and capital can be hired for $9. a. When the firm is producing 49 units at lowest cost, what will the firm’s marginal rate of technical substitution be? b. Solve for the lowest-cost combination of labor and capital that will allow the firm to produce 49 phones. Fractional units of labor and capital are allowed. c. What is the minimum cost of producing 49 phones?Assume an economy consists of two agents, a and b. Individual a has available 1 unit of time which he divides between leisure ra and labor ℓa . Likewise, individual b has available 1 unit of time which he divides between leisure rb and labor ℓb. Commodity x is produced by one competitive firm according to the production function: x=√L , where L denotes labor hired. The firm is fully owned by agent a. The utility function of the two individuals are: u(xa,ra)=xara and u(xb,rb)=xbrb . Taking the price of labor as the numéraire (i.e. w=1), find the competitive equilibrium for this economy.A firm is attempting to maximize output given a budget. Draw a graph that illustrates the output-maximizing input combination. Suppose the cost of capital decreases. Show the new isocost line and the new output-maximizing combination of inputs.
- There are two firms in the economy. Each firm employs positive amounts of capital and labour. The technology satisfies diminishing marginal rate of technical substitution of labour for capital. Currently, A’s marginal rate of technical substitution of labour for capital is 4 while B’s marginal rate of technical substitution of labour for capital is 2. A. Is the current production of the two firms efficient? If not, describe an exchange of inputs that would improve efficiency. B. Can these production levels of the two firms be observed in a perfectly competitive equilibrium of a production and exchange economy? Explain.Joe owns a small coffee shop. His production function is q = 2K0.5 L where q is the number of cups of coffee produces, K is the number of coffee machines, and L is the number of employees. a) If K=1 and L=2, the marginal rate of technical substitution is _________. b) If, starting from 1 machine and 2 employees, Joe hires one more employee, then he can produce the same quantity of cups of coffee with _________ less machines.Suppose the utility function of a person consuming two commodities X and Y with income Birr 600 is given by U =2xy. If the per unit price of X is Birr 20 and per unit price of Y is Birr 40. a) Calculate the utility maximizing level of consumption of X1 and X2. b) Find the MRSX, Y at the optimum.If the production function of a firm is given by Q=,and the input prices are r = Birr 8 per unit and w = Birr 2 per unit,
- a) Consider a couple whose behaviour follows the unitary household model. Their preferences can be represented by the utility function: U(CM; CH) = min (CM, CH), where CM, denotes market goods and CH denotes home production. Each spouse can work up to 50 hours per week, and those 50 hours can be divided between market work and home production. Joe and Anna are each paid £20 per hour for market work. Joe produces £20 of home production per hour, while Anna produces £30 per hour of home production. How many hours are each of the spouses allocating to home production and market work? b) Suppose that Anna is offered a pay raise, so that her hourly market wage increases to £25, and nothing else changes. Will that change the identity of the spouse who works more hours on the market? Explain your answer. [Hint there is no need to calculate the full solution to this case]b) There are two firms in the economy. Each firm employs positive amounts of capital and labour. The technology satisfies diminishing marginal rate of technical substitution of labour for capital. Currently, A’s marginal rate of technical substitution of labour for capital is 4 while B’s marginal rate of technical substitution of labour for capital is 2. Is the current production of the two firms efficient? If not, describe an exchange of inputs that would improve efficiency. Can these production levels of the two firms be observed in a perfectly competitive equilibrium of a production and exchange economy? Explain c) State the first theorem of welfare economics of a production and exchange economy. Which conditions must be satisfied for the Pareto efficiency of an allocation of a production and exchange economy (assuming that all goods are used, produced and consumed in strictly positive quantities)? Discuss and explainDefine marginal rate of substitution.