(1) Suppose the production function is instead y3ga+ɛ, where g > 0 indicates the productivity of a person with ability g. Solve the basic principal-agent model where the cost function is c(a)=D0.5a?.
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- The productivity of a certain country with the utilization of x units of labor and y units of capital is given approximately by the function f(x,y)=100x0.65y0.35. (A) Find fx(x,y) and fy(x,y). (B) If the country is now using 300 units of labor and 200 units of capital, find the marginal productivity of labor and the marginal productivity of capital. (C) For the greatest increase in the country's productivity, should the government encourage increased use of labor or increased use of capital?Which of these production functions have diminishing marginal returns to labor? a)F(K,L) = 2K+15L b)F(K,L)=√KL c) F(K,L) = 2√K+15√LThe Cobb - Douglas production function is a classic model from economics used to model output as a function of capital c. and labor. It has the form f(L,C) = c_L" C² where co, c₁, and c₂ are constants. The variable L represents the units of input of labor and the variable C represents the units of input of capital. (a) In this example, assume c₁ = 5, c₁ = 0.25, and c = 0.75. Assume each unit of labor costs $25 and each unit of capital costs $75. With $80,000 available in the budget, develop an optimization model for determining how the budgeted amount should be allocated between capital and labor in order to maximize output. Max s.t. 25L +75C 80000 L,C>0 (b) Find the optimal solution to the model you formulated in part (a). What is the optimal solution value (in units)? (Hint: When using Excel Solver, use the bounds 0 L3,000 and 0 C1,000. Round your answers to the nearest integer when necessary.) units at (L,C) = (x)
- 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,…Production Function. Consider the Cobb-Douglas production function discussed in class:F(K, L) = AK1/3 L2/3. Suppose that parameters are initially A = 1, K = 150, and L = 10. b) State the equation relating outper per worker and capital per worker. Calculate outputper worker and capital per worker in this case and verify that they satisfy this equation.A manufacturer named Kjell is facing the following product function.1 x = f(N, K) = 100Nº.8K0,2 The price for labor is NOK 500 per unit. The price for capital is NOK 125 per unit. a) Does the product function provide constant, increasing, or decreasing scale benefits? b) Kjell wants to produce 400 cakes at the lowest possible cost. How much labor and does he need capital? What will be the cost? c) The bank is not impressed by Kjell's production technology. He therefore receives a loan of only NOK 1,875. How many cakes can Kjell produce a maximum of NOK 1,875. d) Make a factor diagram (N, K). Note Kjell's adaptation from problem b) and c). Draw the isocost lines and the substituent template. e) A short-term strike leads to the price of labor rising to NOK 750. Kjell still has only NOK 1,875. How will Kjell change his adaptation from c)?How many cakes does he produce now? f) Kjell eventually starts making money, and wants to expand production to 650 cakes. What will be Kjell's new…
- The Cobb - Douglas production function is a classic model from economics used to model output as a function of capital and labor. It has the form f(L, C) = COLc1Cc2 where c0, c1, and c2 are constants. The variable L represents the units of input of labor and the variable C represents the units of input of capital. (a)In this example, assume c0 = 5, c1 = 0.25, and c2= 0.75. Assume each unit of labor costs $25 and each unit of capital costs $75. With $70,000 available in the budget, develop an optimization model for determining how the budgeted amount should be allocated between capital and labor in order to maximize output. Max s.t. = 0 (b) Find the optimal solution to the model you formulated in part (a). What is the optimal solution value (in units)? (Hint: When using Excel Solver, use the bounds 0 < = L < = 3,000 and 0 <= C <= 1,000. Round your answers to the nearest integer when necessary.)________ units at (L, C) = Please use Excel to get answers. Show steps on Excel Please!Q(K, L) = 4K²L2 Type your answers in all of the blanks and submit The production function of good x is as follow: a. What is the marginal production of capital with 16 labor and 4 capital? Type your answer her b. Calculate the slope of the isoquant when the entrepreneur is producing efficiently with 10 labors and 400 units of capital. Slope = Type your answer her (absolute value.) c. If we increase the amount of labor we use in our production process from 10 to 1000 units, how much change in capital is needed for us to stay on the same isoquant? Change in K= Type your answer here (Actual value, negative if decrease and positive if increase) d. Is diminishing return a characteristic of this production function? For labor? (yes/no) Type your answer her ;For Capital (yes/no)? Type your answer hereThe Cobb-Douglas production function is a classic model from economics used to model output as a function of capital and labor. It has the form f(L, C)=²1C²2 where co. ₁, and care constants. The variable L represents the units of input of labor and the variable C represents the units of input of capital. (a) In this example, assume co5, c, 0.25, and c₂-0.75. Assume sach unit of labor costs $25 and each unit of capital costs $75. With $70,000 available in the budget, devalop an optimization model for determining how the budgeted amount should be allocated between capital and labor in order to maximize output. Max s.t. L, CZO € 70,000 (b) Find the optimal solution to the model you formulated in part (a). What is the optimal solution value (in units)? (Hint: When using Excel Solver, use the bounds 0S LS 3,000 and 0 s Cs 1,000. Round your answers to the nearest integer when necessary.) units at (L. C)=(
- Suppose the Cobb-Douglas production function is given as: , where b0 = level of technology, b1 = proportion with which labor is used in production, b2 = proportion with which capital is used in production. Then, a. Find marginal product of factors (L and K). b. Compute marginal rate of substitutions (MRSLK and MRSKL). c. Find the elasticity of substitution for this production function. d. Compute elasticity of output with respect to labor and capital.Introduction to Calculus in Economics: Calculus is a powerful tool used in economics. One of the initial applications areas is the study of a firm, a topic in microeconomics. An important function is the cost function function C(x), the cost of producing items (of whatever they are selling). This question deals with just the cost function C'(x). Problem Set question: The cost, in dollars, of producing a units of a certain item is given by C(x) = 10x10√x - 8. Find the production level that minimizes the average cost per unit. The number of units that minimizes the average cost is NumberFactor-Factor Decision Making. Using the following information, answer the questions. y = 100 Px1 = $5 Py $3 Px2 = $2 Given: TFC = $50 Labor Water x2 MRS. 3 4 8. 12 MRS, and insert above. 1. Calculate the What is the price ratio that should be used to determine cost minimization combination? 2. 3. What is the least cost combination of inputs? 15 4. What is the total cost of producing 100 units of output? 5. How much profit was made? 964m