Consider a production function for an economy with 2 factors of production L, K: Y=10(KL)1/2 where Y is real output, L is labour, K is capital. In this economy the factors of production are in fixed supply with L = 10, K = 10. What type of returns to scale does this production function exhibit. Demonstrate by example. If the economy is competitive what is the total income that will go to the owners of capital?
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Consider a production function for an economy with 2 factors of production L, K: Y=10(KL)1/2 where Y is real output, L is labour, K is capital. In this economy the factors of production are in fixed supply with L = 10, K = 10. What type of returns to scale does this production function exhibit. Demonstrate by example. If the economy is competitive what is the total income that will go to the owners of capital?
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- Suppose the long-run production function for a competitive firm is f(L,K)= L 1/3 K 1/4 , where L is the amount of labor and K is the amount of capital. The cost per unit of labor is w and the cost of capital is r, which is the interest rate. Fixed costs are zero. .a. Find the cheapest input bundle, i.e. amount of labor and capital, that yields the given output level of y. .b. Draw the conditional input demand functions for labor and capital in the L-y and K-y spaces. .c. Write down the formula and draw the graph of the firm’s total cost function as a function of y, using the conditional input demand functions. What is the relationship between the returns to production scale and the behavior of the total costs? .d. Write down the formula and draw the graph of the average cost and marginal cost functions, as functions of y.Consider a competitive, closed economy with a Cobb-Douglas production function with parameter α = 0.25. The parameter A is equal to 60. Assume also that capital is 100, labor is 100. Calculate GDP (Y) for this economy. Does the production function exhibit constant returns to scale? Demonstrate with examples. Determine if the production function exhibits diminishing marginal returns to capital. Demonstrate with calculus What is the real wage in this economy? What share of GDP will go to labor in this economy?Consider a competitive, closed economy with a Cobb-Douglas production function with parameter α = 0.25. The parameter A is equal to 60. Assume also that capital is 100, labor is 100. Does the production function exhibit constant returns to scale? Demonstrate with examples. Determine if the production function exhibits diminishing marginal returns to capital. Demonstrate with calculus
- Consider the following production function: f (A, B) = gamma multiply A^alpha multiply B^Beta. where A and B are the inputs and alpha, Beta, gamma are in the set (0,1). Let wA and wB the price of the two inputs. Assume wA, wB > 0. Is the production function separable?Does the production function exhibit constant returns of scale?Compute the cost function and the conditional input demand function.How do these three functions react to a change in wA? Suppose the price of both inputs double, what happens to the conditional input demand function? And to the cost function? Suppose the desired level of output double, what happens to the conditional input demand function? And to the cost function?…Suppose that a firm that produces face masks is in a long-run equilibrium setting where it has 3 units of capital and 3 units of labor, where MRTS = w/r, and where the firm is maximizing profits. Then suddenly in March 2020, the price of face masks increases due to increasing demand: In a graph with capital on the y-axis and labor on the x-axis, graph how the increase in price will change the firm’s input choices (labor and capital) in the short-run (when only labor can adjust). Will the new choice of inputs be on the expansion path? Suppose that new entry into the face mask production industry is impossible. In the same graph, graph how the increase in price will change the firm’s input choices in the long-run (when both labor and capital can adjust). Will the new choice of inputs be on the expansion path? Now suppose that entry into the face mask industry is free and that this is a constant cost industry. What will happen to the price in the long run? How will the firm’s input…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?
- Suppose that a firm that produces face masks is in a long-run equilibrium setting where it has 3 units of capital and 3 units of labor, and where MRTS = w/r. Then suddenly in March 2020, the price of face masks increases due to increasing demand: In a graph with capital on the y-axis and labor on the x-axis, graph how the increase in price will change the firm’s input choices (labor and capital) in the short-run (when only labor can adjust). Will the new choice of inputs be on the expansion path? Suppose that entry into the face mask production industry is very costly. In the same graph, graph how the increase in price will change the firm’s input choices in the long-run (when both labor and capital can adjust). Will the new choice of inputs be on the expansion path? Now suppose that entry into the face mask industry is free and that this is a constant cost industry. What will happen to price in the long run? How will the firm’s input choices (labor and capital) change in the…Assume that a competitive economy can be described by a constant returns to scale, Cobb–Douglas, pro- duction function. Suppose that in this economy all factors of production are fully employed. Holding other factors constant, including the quantity of capital and technology, carefully explain how a one-time, 10-percent decrease in the quantity of labor will change each of the following: the level of output produced (That is, will output increase or decrease (by more or less than the change in labor) or remain the same? Explain); the real wage of labor; the real rental price of capital; labor’s share of total income.Heterogeneity and growth, donated by Rodolfo Manuelli Consider an economy populated by a large number of households indexed by i. The utility function of household i is Where 0 i0 > 0. Let total capital in the economy at time t be denoted kt and assume that total labor is normalized to 1. Assume that there are a large number of firms that produce output using capital and labor. Each firm has a production function given by F(k, n) which is increasing, differentiable, concave and homogeneous of degree one. Firms maximize the present discounted value of profits. Assume that initial ownership of firms is uniformly distributed across households. a. Define a competitive equilibrium. b. i) Economist A argues that the steady state of this economy is unique and independent of the ui functions, while B says that without knowledge of the ui functions it is impossible to calculate the steady-state interest rate. ii) Economist A says that if k0 is the steady-state aggregate stock of capital,…
- True or False, Explain Why 1. A production function is characterized by ? = 10 + 5L, where q is output per hour and L is labor input per hour. If workers earn $10 per hour, the marginal cost of the 5th unit of output is $10. 2. The producers’ surplus in the short-run reflects what the firms gain, while the producers ‘surplus in the log-run reflects what the input owners gain. 3. A monopoly is a price maker, thus its price can never be equal to its marginal revenue. 4. For a monopolistic competitive firm, if a government imposes a lump-sum tax on a firm, the policy will never affect its profit maximizing output and price.How does any one economic variable respond to changes in another? Why would a profit-maximizing firm expand the use of each input until its marginal revenue product equals the price of the input?Suppose a firm has a production function with two inputs, capital (K) and labor (L). The production function takes the form: Q = L2K2. Further, let the wage be given by w, the rental rate of capital be given by r, and suppose that the firm wishes to produce Q0 units of output. Determine the elasticity of substitution for this production function. Explain your answer. Determine the returns to scale for the production function. Solve for the long-run optimal input demand functions for capital and labor as a function of exogenous variables only. Derive an expression for long-run total cost as a function of the exogenous variables. Let w=16, r=25, and Q0 = 10,000. Solve for the long-run cost-minimizing input combination.