Suppose that at a wage rate of $10, a firm which is using its resources efficiently has a marginal rate of technical substitution of 4, and at a $12.50 wage rate the marginal product of capital is 5. Then if there is an increase to a $12.50 wage rate and the firm makes the necessary adjustments to continue efficient resource utilization the marginal product of labor must be 25." Is this statement is true or false? First indicate the correct response and then explain your answer.
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- Consider two individuals with endowments of T 5 60 hours (per week) of leisure, nonlabour income of Y, and a wage of $7.50 per hour. At this wage, assume that workers are constrained by their employers to work 40 hours per week, or not at all. a. On a carefully labelled diagram, show the equilibrium for a worker for whom 40 hours is the optimum labour supply and a worker who would like to work 20 hours, but still prefers the 40-hour week to not working. Compare the mar- ginal rates of substitution for these individuals at 40 hours per week. b. The average part-time wage is $7 per hour, in contrast to $7.50 wage for full-time workers. Using the above model, provide an explanation for this difference in wage rates.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…Let’s say we extend the the standard 1 person static labor supply problem instead to a family with two people making money. In the family of two money makers, given as person 1 and person 2, the given utility function is u(c1 + c2, l1, l2). Utility is defined as a function of the sum of goods consumed by the two people in the family, and of each person’s leisure time. Each person has a time endowment of “Ti” and gets a wage “wi”, where i = 1, 2. The two person family has a non-labour income defined y (which means this is the income of the family whether or not they work) and face price “p” for consumption goods. How can I utilize the Slutsky equation to show how the labour supply of 2 changes with w1? Thank you!
- In a standard consumer optimization problem in microeconomics, a consumer purchases any pair of goods untilU1U2=p1p2(?), at an interior solution, where Ux is the marginal utility of good x for the consumer and px is the price of good x, where x={1,2}. What is the corresponding optimality condition for a worker-consumer in his labour-leisure optimization problem and why is it just a variation on the standard optimality condition (?)?Suppose 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 consumer with the following utility function for consumption and leisure: U (R, C ) = 160 ln N + Y where N is the hours of leisure (“recreation”) consumed per day (24 maximum) and Y is dollars spent on consumption (p = 1). The consumer has an hourly wage w. (a) Assume the consumer derives all income from work at a wage rate w. Derive the labor supply function, LS(w). (b) For what values of w does the consumer work zero hours? (Hint: does a corner solution arise?) (c) Suppose that w = 10. How many hours does this consumer work? If the wage rate increases to w′ = 16, how many hours do they work? What is the total effect on the supply of labor?
- 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?Assume the following production function (photo aswell) q = [0.3k^(-1)+ 0.7l ^(-1)] ^(-1) a) Determine the rate of technical substitution, RTS. b) Determine the elasticity of substitution. c) What would happen to the capital intensity if the relative factor price w/v were to increase by 10%? d) If the elasticity of substitution were higher than what you calculated in (b), would the effect of a 10% increase in the relative factor price w/v on the capital intensity be higher or lower than what you calculated in (c)? Explain your reasoning.Suppose Carl’s wage-schooling locus is given by Years of Schooling Earnings 10 $30,000 11 $35,000 12 $40,000 13 $45,000 14 $50,000 15 $55,000 16 $60,000 17 $65,000 Derive the marginal rate of return schedule. When will Carl quit school if his discount rate is 14 percent? What if the discount rate is 9 percent?
- In the specific factor model with two goods 1 and 2 and labour as the mobile factor, in the autarky equilibrium it must hold that A P1MPL1=P2MPL2 for marginal labour productivities MPL1 and MPL2 B MPL1=MPL2 for marginal labour productivities MPL1 and MPL2 C the wage w is given by w=MPL1/MPL2 D the wage w is given by w=P1/P2With 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.Consider a worker who is endowed with T hours of time and investment income V per week, faces a wage rate w per hour, and whose preferences over the consumption of goods, the price of a unit of which is p, and leisure can be represented by the utility function U(C,L) = C1/2L1/2. Find the worker’s labour supply. When is it positive? Will the worker ever work more than half their time endowment? (25%)