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UW-Madison ECON 102 - Homework #4

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Economics 102Spring 2011Answers to Homework #4 Due Wednesday 3/30/2011Directions: The homework will be collected in a box before the lecture. Please place your name, TA name, and section number at the top of the homework (legibly). Make sure you write your name as it appears on your ID so that you can receive the correct grade. Please remember the section number for the section you are registered for, because you will need that number when you submit exams and homework. Late homework will not be accepted so make plans ahead of time. Please show your work. Good luck!1. The economy of country A is described by the production function:Y KL=In this equation the symbol Y stands for real GDP, K for capital and L for the employed individuals in this economy. The labor force at the beginning of time, period 0, is 100 individuals. Capital in each period is constant and equal to 100 units. Assume this economy is under full employment. (Suggestion: Use Microsoft Excel or some other computer program to do the calculations and the graphs of the questions posed below rather than a regular calculator.)a. Complete the following chart using the above information. Calculate real GDP with atleast 3 decimal places. Hint: this will be far easier if you use Excel to do thesecalculations and you may present your own Excel Chart as the answer to this question. Aggregate VariablesCapital(K)Labor(L)Change in LReal GDP(Y)change in YMPLPeriod 0 100 100 NA 100 NA NAPeriod 1 100 200 100141.421356241.42135620.292893Period 2 100 250 50158.11388316.69252680.105573Period 3 100 370 120192.353840634.23995760.178005Period 4 100 480 110219.08902326.73518240.122029Period 5 100 550 70234.52078815.4317650.065801Period 6 100 640 90252.982212818.46142480.072975Period 7 100 800 160282.842712529.86049970.105573Period 8 100 890 90 298.32867 15.48596 0.05190178 53 9Period 9 100 970 80 311.4482313.11955220.042124Period 10100 1050 80324.037034912.58880490.03885Or see the excel file.b. Graph Real GDP and the labor level for this economy. Please measure real GDP on the y-axis and labor on the x-axis, when the amount of capital is fixed at 100 units. For this question it is fine to present your answer as an Excel Graph (hint: this is much easier to do with a program like Excel).c. Graph the marginal product of labor each period for this economy. Measure the MPL on the y-axis and the amount of labor on the x axis. You will find this easiest to do using Excel and its graphing feature. Does the MPL you have calculated and graphed exhibit the property of diminishing return to labor? If not, why?2Answer:First, notice that this production function is the same as in the exercise 3, handout 6. The output Y depends on two variables: L and K. To see the relation between output (Y) and labor (L), we can divide the function Y KL= by K on both sides. So the function becomesY LK K=. And if we measure Y/K on the y-axis and L/K on the x-axis, the function is y x=, where y=Y/K, x=L/K. The graph of y x= is:And this production function exhibits diminishing return to labor. Another judgment is that the marginal product per labor, MPL, should decreases with the amount of labor. Thereason why that MPL is not a downward line here is that the change of labor each period is not the same. If the labor change in each period is the same, the MPL is a smooth downward line. To see this, let’s draw the graph using Excel for when the labor increases by 100 individuals each period.Aggregate VariablesCapital(K)Labor(L)Change in LReal GDP(Y)change in Y MPLPeriod 0 100 100 NA 100 NA NAPeriod 1 100 200 100 141.42135 41.4213562 0.29289362 4 3Period 2 100 300 100173.205080831.783724520.183503Period 3 100 400 100 20026.794919240.133975Period 4 100 500 100223.606797723.606797750.105573Period 5 100 600 100244.948974321.342176530.087129Period 6 100 700 100264.575131119.626156830.07418Period 7 100 800 100282.842712518.267581370.064586Period 8 100 900 100 30017.157287530.057191Period 9 100 1000 100316.22776616.227766020.051317Period 10 100 1100 100331.66247915.434713020.046537When the change in labor is constant in each period the MPL curve decreases smoothly as the level of hired labor increases. When you want to analyze the MPL you should make sure that the change in labor is the same for each period. d. Suppose now that the capital increases to a constant level of 200 units from period 0.Nothing else changes about this problem. Present real GDP under both situations in thesame graph: situation one, where capital is equal to 100 units and situation two, wherecapital is equal to 200 units. On your graph measure real GDP on the y-axis and the timeperiods on the x-axis. Remember to note clearly the label of each curve.4e. Suppose now that the technology changes, so that the productivity doubles. That is, theproduction function is now 2Y KL=, and all other conditions remain the same. Theamount of capital is fixed at 100 units. Present real GDP under both situations in the samegraph: situation one, where Y KL=; and situation two, where 2Y KL=.Which production function has higher marginal return to labor? Why? (you can answerthis question by analyzing the graph you have drawn) The new technology has higher marginal return to labor.5Question 2 Suppose that the demand for labor and supply of labor in Fantasyland are given by the following equations:Demand for labor: w = 170 – LDSupply of labor: w = 8 + LSwhere w is the wage per week and L is the number of labor units hired per week. Assume thatone labor unit is equal to one worker.a. Suppose we have full employment, what is the equilibrium level of employment per week and the equilibrium wage rate per week in Fantasyland? Answer: 170-L=8+L => L=81, W=$89b. Suppose the aggregate production function in Fantasyland is given by the following equation: 2Y KL= where Y is real GDP in unit of dollars, K is capital and L is labor. Furthermore, suppose that Fantasyland has 100 units of capital. Given this information, what is the full employment output for this economy?Answer: 2 100*81 180Y = =dollars.c. Suppose we define labor productivity for Fantasyland as output divided by the level of labor used in a week. What is the value of labor productivity for Fantasyland if it produces the full employment level of output?Answer: 180/81=2.22 dollars per unit of labor.Suppose now, that the government implements a minimum wage rate of $95 per week. Use this information to answer parts (d) through (f).6d.


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