Microeconomic Theory
12th Edition
ISBN: 9781337517942
Author: NICHOLSON
Publisher: Cengage
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Chapter 16, Problem 16.10P
To determine
To discuss:The signs of cross substitution effects.
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Construction workers in the town of Cortland, NY have Cobb-Douglas utility for labor and consumption,
UL=(20-L)C,
MRS=C20-L
where L is the number of hours of labor supplied in a day and C is the dollar amount of consumption goods purchased with wage income (pc=$1 ). R=20-L is the number of hours of leisure that the worker has during the day.
Derive the labor supply function for workers in Cortland.
Draw the labor supply curve on a graph with L on the horizontal axis and w on the vertical axis.
Tatum is a mother of Anny (1 year old). Tatum derives utility from income Y (i.e. a disposable income that she can spend on consumption goods other than childcare) and leisure L according to the utility function U(Y,L)=Y*L . She has non-labour income of $300 per day. Her time endowment is 16 hours per day that can be spent either on Leisure (which mostly consists of caring for Anny) or labour market work. If Tatum works she has to leave Anny in the child care. Tatum’s wage is $30 per hour, while the childcare cost $5 per hour. Tatum only uses childcare when she works.
Compute how many hours Tatum will work under these circumstances. Round your answer to the second decimal point.
What is the conditional demand for input 1? Use cost function c(w1;w2; y) = w11/2w21/2y2
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- A worker has 110 hours available in a week that can be used for leisure (L) or work (h). The utility function is U = (1 - α)ln(C) + α ln(L), where C is consumption. a) The price per unit of consumption is 1, the hourly wage is w, and the worker has a non-labor income of V. Show that the labor supply is: h* = (110(1-a)- (av)/w). Also, find the demand for consumption and leisure. b) What is the effect on labor supply of i) an increase in the hourly wage and ii) an increase in non-labor income? c) Set α = ½. What are C, L, and h when w = 200 and V = 10000? What is the reservation wage? d) What is the effect on labor supply of i) a 30% income tax and ii) a 10% wealth tax (on V)? e) What is the labor supply if V increases to 11600? f) An increase in V to 11600 gives the worker the same utility as w = 250 and V = 10000 (you do not need to show it). What are the income, substitution, and total effects on labor supply of an increase in wage from 200 to 250 while V remains at 10000?…arrow_forwardConsider a single mother with the utility function U = 2/3 log(x) + 1/3 log(), where x is consumption and is leisure. The mother can work up to 100 hours per month. Any of the 100 hours that are not worked are leisure hours. She earns a wage of $10 per hour and pays no taxes. The consumption price is normalized to $1. To be able to work, she has to incur a child care cost of $5 for every hour worked. a. Suppose that there is no tax and welfare benefits. How many hours will she work and what will be her consumption level? Draw the graph depicting her budget set with consumption on the vertical axis and leisure on the horizontal axis. b. Suppose that the government introduces a negative income tax (NIT) that guarantees an income of $200 per month. The benefit is taken away one for one as earnings increase. Draw the new budget set. Compute the new number of hours worked and consumption level. Has consumption increased and is the mother better off? Why or why not? c. Now…arrow_forward7arrow_forward
- Assume an individual has a utility function of this form U(C, L) = 20 + 4(C*L)1/2 This utility function implies that the individual’s marginal utility of leisure is 2(C/L)1/2 and her marginal utility of consumption is 2(L/C)1/2. The individual has an endowment of V=$80 in non-labour income and T = 16 hours to either work (h) or use for leisure (L). Assume that the price of each unit of consumption good p=$1 and the wage rate for each hour of work w=$10. a. What is this individual’s optimal amount of consumption and leisure? b. Assume a cash grant welfare program is instituted which pays M = 20 dollars for individuals who do not work. Compute the new optimal labour supply for this individual under the welfare program. Assume that prior to the welfare program, p =$1, w =$10, and V =$80 (as in part c). Does the individual accept the welfare program and not work? Show why or why not.arrow_forwardAssume an individual has a utility function of this form U(C, L) = 20 + 4(C*L)1/2 This utility function implies that the individual’s marginal utility of leisure is 2(C/L)1/2 and her marginal utility of consumption is 2(L/C)1/2. The individual has an endowment of V=$80 in non-labour income and T = 16 hours to either work (h) or use for leisure (L). Assume that the price of each unit of consumption good p=$1 and the wage rate for each hour of work w=$10. a. How much utility does the individual receive if she consumes C = 100 and works h = 7 hours? b. Calculate the rate at which the individual is willing to sacrifice an additional leisure hour when she is already working 4 hours. c. What is this individual’s optimal amount of consumption and leisure?arrow_forwardLabor Market and Real Output: In the labor market, there is one representative worker and a representative firm. Worker supply hours of working (NS), and firm demand hours of work in order to produce output (Y). The production function that capture firm's production process is Y = f(N) = z√KN, where in the short run z=2 and K=100. Worker has a natural log utility function U(c, l) = In(c) + In(l). h=10, T T = 10. Please draw labor supply and labor demand in a graph and determine the equilibrium w*, N*, and Y*. If you cannot solve w*, N*, and Y*, you can just show where they are in the graph. (Note: MRS,c=and MPN: 2√N = Now, TFP (z) increases. What happen to the equilbirium outcomes (w*, N*, and Y*)? Please show it in an another graph. Are the changes the same as the example that we discussed in the class?arrow_forward
- 1arrow_forwardThe weekly preferences over consumption (C) and leisure(L) are defined by u(C, L) = √C + 3√L. The person receives a weekly allowance of m from The hourly wage is $18 per hour, and the person can work up to50 hours each week (T = z + L = 50), where z is the number of hours spent working). a)How many hours will the person work if her allowance is m= $450 per week b) What is the smallest allowance m for which the person will stopworking altogether (z∗ = 0) for a wage of w = 18?arrow_forwardConsider a consumer who could earn $400 per week and has 50 weeks available each year to allocate between work (H) and nonmarket time (L). They have no non-labour income. Their utility function is U = C2L , where C is the value of consumption goods. What is their optimal choice for the number of weeks in nonmarket time and consumption? Show this in a diagram. Suppose the government introduces a policy that (i) offers no benefits to people who do not work, (ii) offers a wage subsidy on earnings at a rate of 25%, with a maximum benefit of $5000, and (iii) the benefit is subject to reduction at a rate of 25% for every dollar earned above $20,000 in the year. Show the person’s new budget constraint in a new diagram, and discuss how the person’s optimal choice might change (you do not have to calculate this, but point to where it is likely on the new budget constraint). Discuss how income and substitution effects play a role.arrow_forward
- What is the optimal number of work hours for the student whose utility function for other goods (X) and leisure (L) is U (C,L) = CL, and who has $50 of nonlabor income per week and the possibility to work at $5 per hour. Assume that after studying for class & other activities, the student has only 50 hours per week remaining to choose between work and leisure.arrow_forwardSuppose you have the following indirect utility function: V(Pa, Py, I) = In PxPy What are marshallian demands for x and y? I (a) (9x9y) = (22) (b) (9,9y) = (In, In 2) (c) (9, 9y) = (exp(2p/py), exp(2ppy)) I (d) (9x, gy) = (2pr+py' px+2py) What is the expenditure function for the associated expenditure minimization problem? (a) E(pa, Py, U) = (P + Py) ln(U) (b) E(pa, Py, U) = √exp(U)Papy (c) E(pa, Py, U)= (p²+p²) In(U) (d) E(pa, Py, U) = exp(U)²papy What are the individual's Hicksian demands for goods x and y? (a) (h₂, hy) = ((BU)¹/², (PU) ¹/²) (b) (ha, hy) = (RU, DU) (c) (ha, hy) = ((2 exp(U))¹/², (exp(U))¹/²) -1/2 (d) (hx, hy) = ((P₂PzU)−¹/², (P₂PzU)-¹/2) Are x and y complements or substitutes?arrow_forwardSuppose the utility function of a person consuming two commodities X and Y with income Birr 600 is given by U =2xy. If the per unit price of X is Birr 20 and per unit price of Y is Birr 40. a) Calculate the utility maximizing level of consumption of X1 and X2. b) Find the MRSX, Y at the optimum.If the production function of a firm is given by Q=,and the input prices are r = Birr 8 per unit and w = Birr 2 per unit,arrow_forward
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