Question 1: Consider a representative agent with a utility function: c¹-01 U (c, l) = 1-o that he or she maximises subject to a constraint: c=w (hl) -T+π where W, h, l, c, T, and í are wages and hours of time available, leisure, consumption, taxes, and dividend income. Labour supply is NS = h – l. a) Write the Lagrangian of the representative agent's optimisation problem. 11-01 1-0
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- 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) noneLet the demand and supply function for a commodity be Qa= D(p,Y) Dp 0 Qs = S(p, w) Sp>0, Sw<0 where p is the price, Y in exogenous income, and w is the exogenous wage rate. a. Find dp and dp using the implicit-function theorem. dw dp dy b. Find and dp by totally differentiating the equilibrium condition. dwIn 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 (?)?
- Consider a Sraffa system that describes production with a surplus. Assume that the whole of the wage is variable. The number of commodities in the system is ?. Some are non-basic commodities. . State the economic meaning of the viability condition and express it mathematically. . Assume that the viability condition is satisfied and the rate of profit is given exogenously at a level that is lower than the maximum rate of profit. Explain, with the help of the relevant theorem, why the solution for prices must be nonnegative. NOTE: You need to use only three equations, one for specifying the numeraire, the second for expressing the final solution of the price equations, and third for expressing the condition for nonnegative prices.Consider the following model of labour supply. There is a representative worker with the following utility function: U(C,L) = C + 3L The budget constraint and time constraint are: C = wh + V h = T – L where w = 1,V = 500,T = 100. The notation is the same as question 1. a) Calculate the optimal leisure and consumption. b) Explain why we usually do not use this type of utility function to model the labour supply.Consider an economy with 100 identical households and 100identical firms. Each household is endowed with one unit of time. Half of the households are endowed with equal shares of firm, while the rest areendowed with no firm shares. A householdís utility is u (x; r) = 2 ln(x*r^2) ,where x is consumption of goods, r = 1- l is leisure time and l is thehouseholdís labour supply. Each firm hires households to produce goodsaccording to technology y = L^1/2, where L denotes the labour input. Goodsprice p, wage rate w, and dividend income are all taken as given. Normalizew = 1. Do the following: (a) Derive the Marshallian demands, x (p, D) andr (p,D), where D is dividend. (b) Derive the firmís input demand functionL (p) and profit function pi(p). (c) List all the market-clearing conditions.(d) Calculate the equilibrium price p
- 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.Provide the functional form of the income effect from a marginal decrease in income and provide the functional form of the substitution and total income effects of a marginal increase in wage.Problem 4 Consider the leisure demand/labor supply model studied in class, and let the consumer have a spccific utility function U(N, Y) = N2/3y!/3. As in lecture, let the price of consumption be normalized to 1, and let w denote the wage. (a) Say that w = 10. What are the optimal N* and Y*? How many hours does the agent work? Draw a sketch to illustrate this situation. (b) Solve for the general demand functions N(w) and Y(w) as a function of the wage, as well as the labor supply function H(w). Calculate the elasticity of labor supply with respect to the wage, w. Do you notice anything special about this particular example? (c) Say the wage rises to some w' > w. What is the change in leisure demand N(w)? Carefully draw a sketch that decomposes this into an income effect and a substitution effect. (d) Sketch the labor supply function. (e) Let H(w) be the labor supply as a function of the (take-home) wage w. Say now that the government imposes an income tax of a. Let T(a) denote the…This question will analyze the impact on a person's labour supply from a shock to their partner's job. Assume leisure is a normal good. Let's assume Vanessa has a wage rate of $20 per hour. Recently her partner, Bill, had to take a wage cut at work, with his wage falling from $45 per hour to $30 per hour, but allowed them to continue working 40 hours per week. Analyze the decision of the household over choice consumption and Vanessa's leisure, taking Bill's hours as given (constant).
- Consider 5 workers who care about their consumption and continuous job satisfaction J.Their preferences are described by the utility function U(C,J) = 2C + J. There are 5 firms thatare producing the output using the production function Q(J,L) = L√20 − J1. What are the marginal rate of substitution between consumption and job satisfaction andthe marginal rate of transformation between wages and job satisfaction?2. What are the equilibrium levels of wage and job satisfaction?3. What is the slope of the wage-job satisfaction locus?√N. The 3. Suppose that the production function of an economy is given by Y = (representative) firm hires workers (N) at a wage rate w. Assume that the firm maximizes profits by choosing how much to produce and how many workers (hours) to hire. (Remember that the profit function is Y - WN.) = The representative consumer has preferences given by the utility function U(-) = log (c) + log (l), with a marginal rate of substitution given by c/l, where c is consumption and I is the leisure time. The endowment of time of an individual is 24, which has to be allocated between labor and leisure. (Recall that N + 1 = 24.) a. Derive an expression for the labor demand function. b. Derive an expression for the labor supply function. c. Compute the equilibrium wage rate and the equilibrium quantity of labor employed. d. Consider a technical improvement of the production process that determines a new production function Y = 2√N. Compute the new equilibrium quantity of labor and wage rate. e. Draw a…Consider 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.