5. Consider a representative household in the static consumption-leisure model with prefer- ences given by u(c, l) = Ac¹/² + 1¹/2. The household has a unit time endowment given by 1 = l +n, faces a price of P on consumption, and earns nominal wages at rate W on their labor supply, n. In addition, the household faces a proportional tax on wage income of T.
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- Suppose that a person has 2000 hours to allocate each year between leisure and work. a. Derive the equation of his budget constraint given an hourly wage of $(15)/hour. b) Graph his budget constraint line based on the equation you derived in part a. (Consumption (C) on the vertical axis and leisure (L) on the horizontal axis). Please make sure to include the value for the vertical and horizontal intercepts. c) Now suppose that the local government introduces an income guarantee program for single parents in which the income transfer is $10,000 per year if an individual does not work during that year (this dollar amount represents the benefit guarantee). If the individual decides to work, this transfer program imposes a 100% benefit reduction rate (e. g.. each additional hourly wage earned is reduced by 100%). Derive the new budget constraint equation that corresponds to this scenario. d) Draw the budget line that corresponds to the new scenario on a new graph. (Consumption (C) on the…For this question, assume that indifference curves are strictly convex, consumption andleisure are normal goods, and the optimal amounts of consumption, leisure, and labor arealways positive. A wage increase ______. (SE = substitution effect; IE = income effect)(a) increases labor supply via the SE and decreases labor supply via the IE(b) decreases labor supply via the SE and decreases labor supply via the IE(c) increases labor supply via the SE and increases labor supply via the IE(d) decreases labor supply via the SE and increases labor supply via the IE(e) Can’t tell without knowing the utility functionConsider a utility function defined over hours of leisure and consumption expenditure: U(c, h) ch. The after- tax wage rate is given by w and the amount of nonwage income, by N. Assume the number of hours of leisure in a year is 8000. Potential annual income, I, is given by 1-8000wN. It is allocated over hours of leisure, which cost wh, and consumption expenditure, c. The budget constraint is thus 8000wN-cwh 0. 1. Find the demand functions for leisure and consumption. a. Write the Lagrangean function for this problem. b. Find the first order conditions. c. Solve the first order conditions to obtain the condition that the MRS wage rate. Show your steps in the solution. d. Use the results from c to obtain the expansion path
- 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…I Consider a consumer who has the following utility function of consumption ( C 1. -- ) and leisure ( L ), U = CL?. The wage rate is $20 per hour, the price of consumption is $10 per unit, and the consumer's non-labor income is $60. The time endowment of the consumer is 24 hours. (1).( ) What's the utility-maximizing choice of consumption and leisure? (2).( Show that at that choice, the marginal rate of substitution between consumption and leisure is equal to the real wage.How would you demonstrate part b) diagramatically 6. Assume you can work as many hours you wish at £12 per hour (net of tax). If you do not work, you have no income. You have no ability to borrow or lend, so your consumption, c, is simply equal to your income. a) Derive and plot the feasible set, between daily values of consumption c, and “leisure”, l. Label the values at the intercepts (the points where the feasible frontier cuts the two axes). b) Assume that your optimal choice of consumption and leisure is to work 8 hours per day. Illustrate this choice diagrammatically using the feasible set and indifference curves.
- 4. A consumer’s utility function over leisure and consumption is given by u(L, Y) =LY. Wage rate is w and the price of the composite consumption good is p = 1. (a) Suppose w = 10. Find the optimal leisure - consumption combination. (b) Suppose the overtime wage law is passed so that the firm must pay 1.5 times the normal wage for hours worked beyond the first 8 hours. Find the effect on the hours worked. Decompose the effect into substitution effect and income effect2. Consider the labor leisure decision with utility of In(c)+ aln(1). This utility function is characterized by an elasticity of substitution equal to one. a. What observation leads us to use this utility function? In answering this question, be sure to both list this observation and explain how it relates to the unit elasticity of substitution property. b. Explain why this utility function leads to the following claim: "To predict the effect of a change in the tax rate on work hours, one needs to know how the tax revenues are spent." In your explanation, you must demonstrate with the indifference curve/budget constraint diagram(s) two cases: one where the use of the tax revenues has no effect on the equilibrium work hours and another use of tax revenues where it does affect equilibrium work hours. .- Find the marginal revenue function for the total revenue function given by 1 TR(Q) = 500Q Q3 - - 3 Find the marginal cost function for the total cost function given by TC(Q) = Q3 – 90Q² + 7,500Q
- no chagpt answer urgent. The marginal rate of substitution of current consumption for future consumption is A) the slope of the indifference curve. B) minus the slope of the difference curve. C) the downward slope of the budget constraint. D) the endowment point. E) the slope of the lifetime budget constraint.An individual derives utility from consumption spending C and leisurel according to the following utility function: U(C,1)=C"1¹-a where 0>a>1. Leisure time in hours is given by: 1=T-H where T is hours of total time available and H is hours of work. The consumer's real income is given by: C=w (T-1)+N where w is real wage and N is real non-labour income. d) Verify that the second-order conditions for a constrained maximum are met. Reduce your answer (which should only be function a, w, N, and T) to the lowest terms.An individual derives utility from consumption spending C and leisurel according to the following utility function: U(C,1)=C"1¹-a where 0>x>1. Leisure time in hours is given by: 1=T-H where I is hours of total time available and H is hours of work. The consumer's real income is given by: C=w (T-1)+N where w is real wage and N is real non-labour income. c) What does the Lagrangian multiplier represent in this context? I