MICROECONOMICS Solutions to Problem Set 3 1. (Labour Supply) A worker derives utility from consumption (that is, goods per day), denoted by Y, and leisure, denoted by L: U = U (Y, L) and faces the "bud- get" constraint Y = w (24 - L) (NB: In contrast to the lecture handout, initially the worker has no nonlabour income.) (a) Is w "real" or "nominal" wage?
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- Joko is a university student, working part-time at copying service center for Rp. 8/hour with zero non-labor income. (a) Graph Joko’s budget constraint and label the utility-maximing outcome if Joko opts to work 40 hours per week. (b) Suppose Joko’s parents decide to send him Rp.100/week. Graph Joko’s new budget constraint. (c) How many hours will Joko now have to work to maintain the same weekly income as in (a)?Consider a consumer with the following utility function for consumption and leisure: U(R,C) = 160 In 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 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?
- 4. Steve's utility funetion over leisure and consumption is given by u(L,Y) = min (3L, Y). Wage rate is w and the price of the composite consumption good is p = 1. (a) Suppose w = 5. Find the optimal leisure - consumption combination. What is the amount of hours worked? (b) Suppose the overtime law is passed so that every worker needs to be paid 1.5 times their current wage for hours worked beyond the first 8 hours. Will this law induce Steve to work more hours? If so, how many? If not, explain.Question 1 A firm wishes to produce to Q units of output using two inputs - ₁ and ₂. The prices of the inputs are w₁ and w₂ respectively. Assuming the firm wants to keep its current production target, show the effect on the firm's optimal input choice in a diagram if the price of factor x₁ increases. In a separate diagram, derive the conditional factor demand for input x₁, i.e., the relationship between price of x₁ and quantity demanded for ₁.21. Let U=x 2 +y 2 is the utility function of a worker who has 10 hours that to be allocatedbetween labour supply (L) and leisure (x). Let y is a consumption good whose price is 1.Wage rate (w) is Rs 1 and non-wage income is 20. Find out L.a) 10 b) 0 c) 5 d) 8 e) none 22. On the basis of the above question, hen w=0 and non-wage income is 40, find out L.a) 10 b) 0 c) 5 d) 8 e) none
- Consider worker 1 with non-labour income Y facing a wage offer w and a utility function defined over consumption and leisure. U(c,l) = lnC + 4lnl a) Derive worker’s income elasticity. Is leisure a normal or inferior good for this worker? b) Provide the functional form of the income effect from a marginal decrease in income. c) Provide the functional form of the substitution and total income effects of a marginal increase in wage.Emma is employed at a contract where he can vary his labour supply between 0 and 40 hours per week. She derives utility from consumption, c, and leisure, L. Her preferences are represented by the following utility function: U(c,L) = min(c, L). The unit price of consumption is p, and the hourly wage rate is w. (a) Derive an expression for Emma's labour supply. (b) The government contemplates introducing a consumption boosting policy including a monthly transfer of money for which Emma is eligible. At what size of the transfer will Emma decide to not work? Assume that the unit price of consumption is 30 and the hourly wage rate is 50(5) Steven has non-labor income each week of $500. He can work up to 100 hours per week foran hourly wage of $10 per hour. His utility for recreation (R) and consumption (C) is given byU(R, C) = 2R2C. What is Steven’s reservation wage if the price of consumption is unity?(a) $30(b) $40(c) $50(d) $60(e) None of the above
- 1 The consumption-leisure framework Suppose that the representative consumer has the following utility function over consumption (c) and labour (n): u(c, l) = ln c − A 1 + � n 1+� (1) where, as usual, c denotes consumption and n denotes the number of hours of labour the consumer chooses to work, The constants A and � are outside the control of the individual, but each is strictly positive. Suppose the budget constraint (in real terms) faced by the individual is given by: c = (1 − t) · w · n (2) where t is the labour tax rate, w is the real hourly wage rate, and n is the number of hours the individual works. Remember, as seen in class, n + l = 1 is always true. Using the static consumption-leisure framework, answer the questions that follow. 1. State the utility maximization problem, and state carefully the choice variables in the problem. [2] 2. Write the Lagrangian function for this problem. [2] 3. Using the Lagrangian function from above, derive the first order condition with respect…SUBSTITUTION AND INCOME EFFECTS OF A WAGE INCREASE When the wage rate increases from $10 to $30 per hour, the worker's budget line shifts from PQ to RQ. In response, the worker moves from A to B while decreasing work hours from 8 to 5. The reduction in hours worked arises because the income effect outweighs the substitution effect. In this case, the supply of labor curve is backward bending. Income (dollars per day) 720 240 w= $30 w= $10 12 16 www. 19 Substitution Effect Income Effect Derive the graph into a backward bending labor supply curve Q 24 Hours of leisureLabor Supply 1 LABOR (Hours) he income effect of a higher wage outweighs the substitution effect when wages are he income effect is the phenomenon that workers choose to work hours when they are given a raise, because WAGE RATE (Dollars per (unoy