w wages and r in rent, respectively. a) What is George's objective function b) What is George's constraint? c) Which variables are endogenous, and which variables are exogenous? d) What is George's constrained optimization problem?
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- Consider a Cobb-Douglas production function:f(l, k) = Alα k1−α,where A is the total factor of productivity (a constant greater than 1), 0 < α < 1, lrepresentslabor, and k represents capital. The following sub-questions will guide you through showing thatthe elasticity of substitution is constant.a) Find the marginal product of labor. Verify that this production function exhibits diminishingmarginal productivity of labor. b) Find the marginal product of capital. Verify that this production function exhibits diminishingmarginal productivity of capital. c) Find the marginal rate of technical substitution. Write your answer as MRT S = . . . d) In part (C), you should’ve found the MRTS as a function of the input ratio, kl. Take theabsolute value of both sides and solve for the input ratio, so that the expression gives theinput ratio as a function of MRTS (i.e. kl = . . .). Take the log of both sides, then take thederivative with respect to the log of MRTS. Is the elasticity of…With its current levels of input use, a firmʹs Marginal Rate of Technical Substitution is 5 (when capital is on the vertical axis and labour is on the horizontal axis). This impliesA. the firm could produce 5 more units of output if it increased its use of capital by one unit (holding labour constant).B. the marginal product of labour is 5 times the marginal product of capital.C. the marginal product of labour is 1/5 times the marginal product of capital.D. if it used one more unit of both capital and labour, the firm could produce 5 more units of output.Answer the Constrained Optimization: Cobb-Douglas Production Function:3. Solve for the formulas of the Marginal Product of Labor (MPL), and Marginal product of Capital (MPK)4. Using your knowledge of the tangency condition in Producer’s theory, find the combination of K and L that the firm should use to produce the maximum possible output. Do not solve the 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
- Suppose that a firm has production function F(L, K) = L^1/8 K^3/8 for producing widgets, thewage rate for labor is w = $16, and the rental rate of capital is r = $3.a) Suppose that the firm has received an order for Q = 100 units of output. Neatly specify this firm’s costminimization problem, using the particulars associated with this problem.b) Give two equations that an interior solution satisfies, tailoring your equations to the particulars of thisproblem.c) Solve the two equations for the firm’s optimal choice. Show your work.d) Determine this firm’s minimum cost of producing 100 units.e) Now suppose that the firm’s production goal is left as the variable Q. Come up with the firm’s costfunction C(Q). Show your work.Suppose there are two primary factors of production, labour (L) and capital (K), and there are nosecondary factors of production. The production function for this economy is simply Y = cKL,where c is a constant and α= 0.5. Which of the following is true? 1.This economy exhibits constant marginal product of labour 2.This economy exhibits decreasing returns to scale 3.This economy exhibits constant marginal product of capital 4.This economy exhibits diminishing marginal product of labourSuppose that the production function is Yt = AtKt^(1/2) Lt^(1/2) where Yt is output, Kt is the stock of capital and Lt is amount of labor firms hire. Assume that Kt = 100, At = 2. (a) Firms in this economy maximize their profits, given by revenue net of labor costs: Yt −WtLt . Derive the firm’s labor demand curve. (b) Workers in this economy maximize their utility, given by U(Ct , Nt) = log(Ct) − Nt , where Ct is consumption and Nt is the amount of labor workers supply. Their budget constraint is Ct = WtNt . Derive the workers’ labor supply curve. (c) Calculate the equilibrium wage rate and employment in this economy using the expressions for labor demand and supply you derived above. Also calculate the total amount of output the economy produces. (d) Suppose that capital doubles, to Kt = 200. Calculate what happens to the wage rate, employment and output. Illustrate the effects of this increase in capital graphically, using a labor demand/supply diagram.
- Consider a firm for which production depends on two normal inputs, labor and capital, with prices w and r, respectively. Initially the firm faces market prices of w = 6 and r = 4. These prices then shift to w = 4 and r = 2.a. In which direction will the substitution effect change the firm’s employment and capital s tock?b. In which direction will the scale effect change the firm’s employment and capital stock?c. Can we say conclusively whether the firm will use more or less labor? More or less c apital?Suppose that a firm has production function F(L, K) = L2/3 K1/3 for producing widgets, thewage rate for labor is w = $400, and the rental rate of capital is r = $25. a) Suppose that the firm has received an order for Q = 120 units of output. Neatly specify this firm’s costminimization problem, using the particulars associated with this problem.b) Give two equations that an interior solution satisfies, tailoring your equations to the particulars of thisproblem.c) Solve the two equations for the firm’s optimal choice. Show your work.A firm produces at an output level where the marginal product of labor (MPL) is 50 units and the marginal product of capital (MPK) is 10 units. Suppose that the wage rate (PL) is $25 and the rental price of capital (PK) is $40. a.Is this firm maximzing profit? b.What should the firm do if it is not alocating its budget efficiently?
- Consider a firm for which production depends on two normal inputs, labor and capital, with prices w and r,respectively. Initially, the firm faces market prices of w=$5 and r=$15. Assume the firm has a cost budget of$1,500.a. Using the isoquant-isocost model, graphically show the optimal level of employment for this firm in thelong run.b. Suppose the government now imposes a minimum wage of $10 for workers. Using the same graph aspart a, graphically show the impact of the minimum wage on the optimal level of employment in thelong run.c. Refer to the initial situation described in part a. Now suppose a new innovation causes the price ofcapital to fall to $10. Using a new isoquant-isocost model, graphically show how this change impacts theoptimal levels of employment and capital in the long run. Clearly identify the resulting scale andsubstitution effects caused by the lower cost of capital. Could you please answer this question using the graphs please.Suppose that a salesperson earns a basic monthly salary of $800 plus a commission rate is 15% and the possible bonuses are lump-sum amount of $1000 if her monthly sales exceed $10,000 and a further lump-sum of $2,500 if her monthly sales exceed $15,000. Find the function that relates sales to earnings for this salesperson and graph it. At which points is the function discontinuous? Interpret the incentives created by this pay scheme?onsider a producer with budget C = 200 who can buy labor L at a wage w = 10 and capital K at a price r = 5. The producer has the following production function F(K,L) = 3K1/3 L2/3 . a. Does F(K,L) exhibit increasing, decreasing, or constant returns to scale? Show your work. Consider a new production function F(K,L) = K1/3 L2/3 , where is a? ? positive constant not equal to 3. Does the new production function exhibit increasing, decreasing, or constant returns to scale? b. Using F(K,L) = 3K1/3 L2/3, find the optimal bundle. Show your optimal bundle on a plot (this should include isoquant and isocost curves). c. Suppose the budget is reduced to C = 100 and then C = 50. Find the new optimal bundles. Plot these bundles, along with the initial optimal bundle, on one plot. Draw the expansion path PLEASE SHOW GRAPHS do not use chat gpt