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- 5. Suppose the production function is given by f(E, K) = E'/³K²/3. The hourly wage rate is $2 and the price for a unit of capital is $32. The output level is ĝ = 144 and the price of the output is $30. (a) Write a firm's CMIN problem. (b) Impose FONC to find MP(E) = w and MP(K) = r. %3D4) Consider the following production function F(K, L) = 6KO 25 L05 a) Does it display constant, increasing, or decreasing returns to scale? b) Derive an expression for the marginal productivity of labour. c) Given that K=15,625; and the supply of labour is given by L = 9000, find the equilibrium real wage and the equilibrium level of employment. d) Calculate and sketch a graph of Aggregate Supply.1. Consider a static labour supply model for an individual. Assume that the person works a positive number of hours. Assume that the utility function is of the following form: U=x0.5 +8(T-h)(0.5) where x is consumption and his hours of work, and 8 is a parameter of the utility function. The person is paid a wage of w for each hour worked. The person has no other source of income other than employment earnings. Assume that the price of x equals 1. a) Specify the utility maximization problem, write down the Lagrange function, and solve for the first order conditions. b) Solve for the MRS condition between consumption and hours of work. c) Assume that 8=2 if the person has a child living at home, and 8=1 otherwise. Explain (and either show mathematically or graphically) how this will affect the optimal number of hours of work of the person.
- Q1. Suppose we are given the constant returns-to-scale CES production function q = [k + l]1/ where k represents capital and l represents labora. a. Show that MPk = (q/k)1 and MPl = (q/l)1 . b. Show that RTS = (k/l)1 ; use this to show that elasticity of substitution between labor and capital= 1/(1 – ). c. Determine the output elasticities for k and l; and show that their sum equals 1.Note: Output elasticity measures the response of change in q to a change in any input. Elasticity of output wrt k is eq,k = %q/%k = (q/k)*(k/q) or (q/k)*(k/q) or lnq/lnkSimilarly for elasticity of output wrt l, eq,ld. Prove that q/l = (q/l) and hence that ln(q/l) = ln(q/l)11. Suppose that the production function is Y=K^(0.5) N^(0.5). With this production function the marginal product of labor is MPN=0.5K^(0.5) N^(-0.5). The capital stock is K=100. The labor supply curve is NS=100(w^2), where w is the real wage and NS denotes the quantity of labor supplied. Compute the equilibrium level of employment (N). 12. Consider Question 11 again. Compute the level of full-employment output (Y). Approximate your answer to the next integer. 13. Consider Question # 11 again. Due to a temporary reduction in immigration, the economy's labor supply curve changes to NS=90(w^2). A policy maker suggests that in order to increase full-employment output in response to the reduction in immigration the government might temporarily subsidize the firm's labor cost by paying a fractions of the wage it pays to a worker. That is, if the real wage received by a worker is w, the firm only pays a portion (1-s)w with the remaining sw coming from the government. Compute the subsidy s…= $30 and wage rate Let the production function be given by: q = 10(L^2)K, with rental rate v W = $20. 1. How much K and L are employed to produce 11,250 units of output? 2. Suppose that the firm wants to increase output to 19, 440 units of output, now how much K and Lare employed to produce with minimize cost? 3. Suppose that in the short run the amount of capital is fixed at K = 10, what is the firm short run fixed cost? 4. Suppose that in the short run the amount of capital is fixed at K = 10, what is the short run total cost of producing 900 units of output?
- 4) If the marginal product of capital MPK > r, where r is the market rental rate for capital, the firm A. Should hire more capital until MPK = r. B. Should hire more labor until MPK = r. C. Should hire more labor until MPK = 0. D. Should fire more capital until MPK = r. %3D 5) Figure 1 below shows the dynamics of per-capita GDP for six countries in ratio scale from 1870 to 2010. From the Figure we learn that the level of per capita GDP A. Has almost doubled for the US from 1880 to 2020 B. Was higher in the U.S. compared to the UK in 1880 C. Was higher in China compared to Argentina in 2000. D. Increased by approximately 5 times in Japan from 1950 to 1970. Answer 64,000 U.S. 32,000 16,000 Argentina 8,000 U.K. 4,000 Japan 2,000 S. Africa China 1,000 1880 1900 1920 1940 1960 1980 2000 2020Q1. Suppose we are given the constant returns-to-scale CES production function q = [k + l]1/ where krepresents capital and l represents labora. Show that MPk = (q/k)1 and MPl = (q/l)1 .b. Show that RTS = (k/l)1 ; use this to show that elasticity of substitution between labor and capital= 1/(1 – ).c. Determine the output elasticities for k and l; and show that their sum equals 1.Note: Output elasticity measures the response of change in q to a change in any input.Elasticity of output wrt k is eq,k = %q/%k = (q/k)*(k/q) or (q/k)*(k/q) or lnq/lnkSimilarly for elasticity of output wrt l, eq,ld. Prove that q/l = (q/l) and hence that ln(q/l) = ln(q/l)Q2. Suppose the production of airframes is characterized by a CobbDouglas production function: Q =LK. The marginal products for this production function are MPL = K and MPK = L. Suppose the price oflabor is $10 per unit and the price of capital is $1 per unit. Find the cost-minimizing combination of labor and capital if the manufacturer wants to…A firm employs M workers per week and the length of the workweek at the firm is H Let MPM be the added output from an additional employee and let MPH be the added output from a one hour increase in the workweek at the firm. MEM and MEH are, respectively, the marginal expense of an added worker and of an added hour of work. At present, the cost of an added unit of output produced by hiring more workers is $5 while the cost of an added unit of output produced by employing existing workers for more hours is $3. Is the firm maximizing profits given its current employment level and workweek? If your answer is yes, explain why. If your answer is no, explain why not and discuss what adjustments the firm should make. It's a general question. There's no given output being produced by added workers and additional hours.
- 29) How many units of labor will the AquaPlant hire based on the figures from the table? 30) What is the economic concept associated with the decline in MPL as the number of labor units increase?) Suppose there are two inputs in the production function-labor and capitaland that these two inputs are perfect substitutes. Suppose that the price of capital is 750 GHC per machine per week and that the weekly salary of each . worker is 300 GHC . i) Draw the isocost line ii) With the aid of a graph, explain the effect of a wage decrease on flrm-level employment and output holding initial cost ouday constant.Question: Suppose the firm's only variable input is labor. Using the table, what level of output corresponds to where the firm's production fuction begins to rise at a diminishing rate? Also, at what level of output would the firm's average product of labor (APL) be maximized? Explain.