(a) Formulate the agent's dynamic problem of maximizing her life-time utility with a discounting factor p. (b) Write the Hamiltonian with a co-state variable and the maximum principle conditions. (c) Using the maximum principle conditions from part (b) above, explain the evolution of the co-state variable (i.e.,). 77
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- Suppose that the first P300,000 income earned is not taxed at all, the next P500,000 earned is taxed at a rate of 20%, the next P1,500,000 is taxed at a rate of 30%, and any additional income is taxed at the rate of 40%. Let T(y) represent the income tax schedule. a.Find and graph the tax function T(y), defined on y 2 0.Suppose that the government imposes a proportional income tax on the representative consumer’s wage income. That is, the consumer’s wage income is w(1 − τ )(h − l) where τ is the tax rate. (a) Determine effect of a decrease in income tax on consumption and labor supply. Explain your results in terms of income and substitution effects. Show the income and substitution effect in a graph. (b) Is your answer in (1) consistent with the observation that hours of work in European countries is about 30% less than in the U.S, while the average income tax rate is about 60% in European countries and 40% those in the U.S.? Under what condition? Please show all your work. Please do all parts of eth question. Thank you for yourhelp.The representative consumer has the following utility function: U(c, I) = In c + In I where c is the consumption good, and I is the leisure hours. She has h hours of time available which she can allocate between work and leisure. For each hour of work, she earns w units of consumption good. She also earns t as dividend (profit) income which is measured in consumption goods. The government collects revenues using a consumption tax; in particular, the representative consumer pays t units of goods to the government for each unit of consumption good she buys. Note that there is no lump-sum tax. (a) Determine the consumer's budget constraint. (b) Solve for the c and I that maximize the utility of the consumer. (c) Suppose that the tax ratet increases. What is the impact of this increase in the tax rate, t, on the consumer's choice for c and l? What is the impact of this on the labour hours of the consumer?
- 1. For what kind of preferences will the consumer be just as well off facing a quantity tax as an income tax? 2. Someone is willing to maximize his/her utility. Assume he/she knows his/her u(x,y) = %3D x0.Sy0$ (Cobb-Douglas) (a) Find his/her optimal consumption given the income = 20, Px=2, Py=4! %3D (b) Depict his/her optimal consumption graph! 3. Suppose that x and y (from your answer in number 2) are now perfect complement with u(x,y) = min(x,y). (a) Find his/her optimal consumption (given the same income and prices)! (b) Depict his/her optimal consumption graph! Best of Luck!1. Consider a static economy. There are neither government spending nor taxes. The total amount of time available to consumers h for simplicity is set equal to 1. Suppose the utility function of the representative consumer is given by U(C, 1) = ylog C+ (1 - 7) log l where 0 0 is total factor productivity, K is the exogenously given capital stock and N is labour. (a) Write down the social planner's problem. State the variables with respect to which the social planner maximises. (b) Find the conditions characterising the Pareto efficient allocation. Interpret them. (c) Find the real wage w and the firm profits that decentralise the Pareto efficient allocation as a competitive equilibrium. (d) Suppose the economy experiences a fall in the capital stock. Analyse the equilibrium effects of this shock on consumption, leisure, hours worked, profits and the real wage. Use a graphical analysis to decompose these effects into income and substitution effects. Explain your answer intuitively. (e)…A closed economy one-period macroeconomic model with a government taxing a lump-sum tax Consider the following utility function for the representative consumer U(c,l) = 2Vc - a(1–1)1.5 s.t.c = wN-T 1.5 Where /is leisure, c es consume, N is the total time dedicated to work, l=1-N, T is the lump-sum tax, so government purchases equal taxes G=T, w is wage, and it has the following production function with K fixed and equal to 1 Y = zN°,5 zN0,5 Find the Labor supply curve Ns, labor demand curve Nd, and the optimum wage
- Consider a government that raises money in a two-good economy by taxing good 1 at a rate of t per unit. The government is considering replacing these taxes with a lump-sum tax to the consumer that raises the same revenue. Thus, if the consumer consumes x units of good 1 before the change in taxes, she must pay the government a lump sum of tx after the change. Suppose, moreover, that prices change only by the amount of the tax; i.e., if prices are (p₁+t, p2) before the change, then they become (P₁, P2) after. Let x = (x1, x₂) be the consumer's demand before the change, and x' (x1,x2) the consumer's demand after. Suppose that x = x'. = (a) Is one of x or x' revealed preferred to the other (and if so, which)? (b) Assuming that the consumer's demand satisfies the Weak Axiom of Revealed Preference, does the change in taxes lead to an increase or decrease in consumption of Good 1, or is it impossible to determine? What about Good 2?An agent consumes quantity (x1,x2) of goods 1 and 2. Here is his utility function: U(x1, x2) = 2 * x1 * x2 + 1, p1 = p2 = $1, his income m = 20. (a) Calculate the agent's Marshallian demand (x*, x*,). (b) If the government put a $1 tax on x1, which increase pl to $2, assume p2 and m do not change, what is the demand for x1? (c) If the government collect tax on the agent's income, the amount of tax is the same as the tax revenue collected in (b), what is the agent's utility? Compare it with the agent's utility in (b).Consider the following consumer problem's facing a proportional income tax. Utility function over consumption (C) and leisure (L) U(C,L)= 3 ln(C)+ 2 ln(L) Total hours : H=40 Labor hours :NS =H-L Proportional income tax rate = 0.2 Production function: Y =zND Total factor productivity : z =4 The representative consumer maximizes utility, the representative firm maximizes profit, and the government balances budget . What is the tax revenue at equilibrium.?
- Consider 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 pathSuppose that a consumer cannot vary hours of work as he or she chooses. In particular, he or she must choose between working q hours and not working at all, where q > 0. Suppose that dividend income is zero, and that the consumer pays a tax T if he or she works, and receives a benefit b when not working, interpreted as an unemployment insurance payment. a) If the wage rate increases, how does this affect the consumer’s hours of work? What does this have to say about what we would observe about the behavior of actual consumers when wages change? b) Suppose that the unemployment insurance benefit increases. How will this affect hours of work? Explain the implications of this for unemployment insurance programs.Consider an economy in which every person’s utility function takes the form Here, c is consumption and h is hours of work. The parameter θ varies across individuals. Within the economy, θ is uniformly distributed on the unit interval. Each unit of consumption goods costs $1, and the wage rate is $2. a) Assume that there are no taxes or subsidies within the economy. How much work does a person of any type θ do? What is his consumption and what is his utility? If a lump-sum tax were imposed upon him, or a lump-sum subsidy given to him, how would that tax or subsidy affect his hours of work, consumption and utility? b) People with low values of θ find work difficult. (A low θ might be interpreted as a physical disability which hampers a person’s ability to earn an income.) Suppose that the government decides to assist the people for whom θ is less than 1/4. Unfortunately, it cannot directly observe each person’s θ, so it instead…