Final output in the economy is pro-duced using capital K and labor L. The production function is:Y = ĀK/ L^/5Assume that the supply of all inputs are exogenous and equal to L and K. Perfectly compet-itive firms are price-takers and choose how much capital and labor to demand by maximizingprofits. Let w and r denote the wage and rental rate of one unit of labor and capital respec-tively. (c) Show that the solution for output per capita can be written as y = Ak/5. ObservedGDP per capita is 3 in this economy. Capital per person is 32. If Ā = 1, what doesthis model predict for GDP per person? Does this model over-predict or under-predictGDP? Show your working.(d) If the model is to explain the observed value of y, what must TFP (Ā) be equal to?Briefly explain why TFP might vary across countries.-

(c) Y=AK1/5L4/5 Output per capita measures the output produced by per person in a nation. It is also…

(c) Show that the solution for output per capita can be written as y = Ak/5. Observed GDP per capita is 3 in this economy. Capital per person is 32. If A = 1, what does this model predict for GDP per person? Does this model over-predict or under-predict GDP? Show your working. (d) If the model is to explain the observed value of y, what must TFP (A) be equal to? Briefly explain why TFP might vary across countries.-

(c) Show that the solution for output per capita can be written as y = Ak/5. Observed GDP per capita is 3 in this economy. Capital per person is 32. If A = 1, what does this model predict for GDP per person? Does this model over-predict or under-predict GDP? Show your working. (d) If the model is to explain the observed value of y, what must TFP (A) be equal to? Briefly explain why TFP might vary across countries.-

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter9: Production Functions

Section: Chapter Questions

Problem 9.9P

See similar textbooks

Similar questions

Consider an economy with production function given by Y = AK0:5L0:5 where A is the total factorproductivity (TFP), K is the capital stock and L is the labor input. For simplicity assume capital is xed and equal to 1. Assume A=100.a. Write the rm's problem of choosing labor demand. Derive the demand for labor as a functionof the real wage.b. Assume labor supply is inelastic and xed at L= 100. Find the equilibrium values of the wageand the employment level for this economy. Display graphically the labor supply and the labordemand curves. Carefully label your graph.c. Suppose the economy faces a positive productivity shock and TFP is now A=200. Displaygraphically the new labor demand function. What are the equilibrium values of employment andthe real wage?d. Compute the total output when A=100 and when A=200. What is the output's growth rate?Compare that growth rate with the growth rate in A. How does the growth rate of output percapita compares to the growth rate in A? Explain…
It deals with the Production Function, Marginal Rate of Technical Subsition . Problem: The Production Function for a Product is given by q=10 K ^1/2 L^1/2 where K is capital and L is Labor and q is output . 1a. Find the Marginal Product of labor and the Marginal Product of Capital 1b. Find the Marginal Rate Of Technical Substitution between Labor and Capital 1c. Denote the wage of Labor by w and the renal of cpaital by r . What is thre cost of the minization condition for a firm . Show it digrammatically.
Cowles Foundationfor Research in Economicsat Yale UniversityCowles Foundation Discussion Paper No. 2021ARE WE APPROACHING AN ECONOMIC SINGULARITY?INFORMATION TECHNOLOGY AND THE FUTUREOF ECONOMIC GROWTHWilliam D. NordhausSeptember 2015An author index to the working papers in theCowles Foundation Discussion Paper Series is located at:http://cowles.yale.edu/This paper can be downloaded without charge from theSocial Science Research Network Electronic Paper Collection:http://ssrn.com/abstract=2658259Electronic copy available at: http://ssrn.com/abstract=2658259Are We Approaching an Economic Singularity?Information Technology and the Future of Economic GrowthWilliam D. Nordhaus1September 1, 2015AbstractWhat are the prospects for long-run economic growth? The presentstudy looks at a recently launched hypothesis, which I label Singularity.The idea here is that rapid growth in computation and artificialintelligence will cross some boundary or Singularity after whicheconomic growth will…
A producer has the following technology.y= 6K^(1/2)L^(1/2) a) Prove formally that the production function exhibits constant returns to scale (use “λ” argument). b) Find analytically MPL and MPK. Is MPL increasing, decreasing, or constant inL? Is MPK increasing, decreasing, or constant in K? c) Short-run: Given stock of capital ̄K= 1 find labor demand (formula) of a competitive firm. Find equilibrium real wage rate if labor supply is given by Ls= 9 (one number). d) Assume again ̄K= 1 and that Ls= 9. The government adopts a real minimum wage of wmin/p=(3/2). Find labor demand (one number) and the unemployment rate (one number). Please depict the equilibrium on a graph with the real wage on the vertical axis and labor on the horizontal axis, indicating the equilibrium quantity of labor, wage, and unemployment, as well as the relevant curves. e) Find the cost function given prices of inputs wK= 4 and wL= 1 (formula). Plot the cost function on a graph, indicating the slope of the cost…
Consider the following one-period model. Assume that the consumption good is produced by a linear technology: Y = zND where Y is the output of the con- sumption good, z is the exogenous total factor productivity, ND is the labour hours. Government has to finance its expenditures, G, using a lump-sum tax, T, on the rep- resentative consumer. There is no other tax in the economy. The firm is owned by the representative consumer who is endowed with h hours of time she can allocate between work, NS and leisure, l. Preferences of the representative consumer are: U (c, l) = α ln c + (1 − α) ln l (1) where 0 < α < 1 is a parameter. (a) Write down the definition of a competitive equilibrium for the above economy 1 (b) Solve for the leisure, l, the consumption, c, employment, N, wage rate, w, lump-sum tax, T , and output, Y in equilibrium. (c) Solve for the optimal allocation of leisure, l, the consumption, c, employment, N, output, Y . Contrast these quantities with those…
Consider the following one-period model. Assume that the consumption good is produced by a linear technology: Y = zND where Y is the output of the con- sumption good, z is the exogenous total factor productivity, ND is the labour hours. Government has to finance its expenditures, G, using a lump-sum tax, T, on the rep- resentative consumer. There is no other tax in the economy. The firm is owned by the representative consumer who is endowed with h hours of time she can allocate between work, NS and leisure, l. Preferences of the representative consumer are: U (c, l) = α ln c + (1 − α) ln l (1) where 0 < α < 1 is a parameter. Answers for part b below: A consumer's choice of optimizing its consumption and labor hours (h - l ) is given by the point where , MRS(c , l) = wage rate Now , MRS (c, l) = MU(c) /MU(l) MU(c) = dU/d c = a/c MU(l) = dU /dl = (1-a )/l MRS = a (l)1−a (c)a (l)1-a (c) Putting this in optimal condition we have : a (l)(1−a) (c)a (l)(1-a) (c) = w --- (i) l = w…
Consider a Cobb-Douglas production function, Y=zKα(Nd)1−α, with 0< α <1. (a) Show that this production function exhibits constant returns to scale. (That is, show that when K becomes xK and Nd becomes xNd, Y becomes xY for x > 0. (b) What is MPN? Is it increasing or decreasing in Nd?
Suppose that government wishes to promote more capital intensive agriculture and subsidises agricultural capital equipment in order to achieve this. a) Using an isoquant map of the tradeoff between capital and labour, show how the reduced cost of capital will affect the individual farmer’s decision problem. b) Using a production function approach with labour as the variable input, show how the reduced cost of capital will affect the individual farmer’s decision problem. c) What overall effect on agricultural employment would you expect, given your answers in (a) and (b)?
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)
Please no written by hand and no image Suppose that the production function is given by Y=AK0.4N0.6. What is the percentage change in output if both capital and labor rise by 42%? Write the answer in percent terms with up to two decimals (e.g., 10.22 for 10.22%, or 2.33 for 2.33%).
hy does a measure of labor productivity—the output produced per worker– rise for many firms during recessions? During the boom years period of 2005through November 2007, the annual average output per worker was lower in U.S.manufacturing than during the Great Recession of 2007–2009 as well during therelatively low-demand years since then through 2013.Firms produce less output during recessions as demand for their products falls.Consequently, firms typically lay off workers during recessions. Thus, whetheroutput per worker rises or falls depends on whether output or employment fallsby more. The labor productivity pattern over the business cycle differs across in-dustries. If we know about a firm’s production process, can we predict whetheroutput produced per worker will rise or fall with each additional layoff?
Production Function. Consider the Cobb-Douglas production function discussed in class:F(K, L) = AK^1/3 L^2/3. Suppose that parameters are initially A = 1, K = 150, and L = 10. d) Suppose that the quantity of labor L doubles. Calculate Y, w, r, Y/L, and K/L. Com-ment on how and why these numbers changed relative to (c) and why they did so. E) Suppose that the quantity of capital K doubles as well. (So now both K and L aretwice their previous value). Calculate Y, w, r, Y/L, and K/L. Comment on how thesenumbers changed relative to both their initial values, and their values in (d).