5. Consider a monocentric model with fixed housing unit h and the utility of residents u=c, where c is the consumption of the composite goods. The budget constraint of each resident living at distance d from the CBD is c + ph = w-dt, where w is wage, p is housing rental rate, t is the unit commuting cost in distance. The city is open and free migration in and out. The external utility they obtain out of the city is ū= 10. The rental rate on agricultural land is pA. (a) Illustrate why the consumption of residents living in all parts of the city will be equalized. (b) Derive the housing rental rate (p) as a function of d, distance from the CBD, What is the slope? Draw a figure with p at the y-axis and d at the x axis. (c) Discuss how the boundary of the city is determined in this model, how would the boundary of the city respond to wage w and commuting cost t?
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- a) Consider a couple whose behaviour follows the unitary household model. Their preferences can be represented by the utility function: U(CM; CH) = min (CM, CH), where CM, denotes market goods and CH denotes home production. Each spouse can work up to 50 hours per week, and those 50 hours can be divided between market work and home production. Joe and Anna are each paid £20 per hour for market work. Joe produces £20 of home production per hour, while Anna produces £30 per hour of home production. How many hours are each of the spouses allocating to home production and market work? b) Suppose that Anna is offered a pay raise, so that her hourly market wage increases to £25, and nothing else changes. Will that change the identity of the spouse who works more hours on the market? Explain your answer. [Hint there is no need to calculate the full solution to this case]Consider a couple (a husband and a wife) that jointly represents their collective preferences between combinations of household production time (X) and purchased goods and services (Y) according to the formula W = X2Y, where W represents the level of welfare. Suppose the maximum time available in a day is 16 hours and currently the wife devotes 6 hours to market work (H) at a wage of $16 per hour. a. What is the level of welfare associated with the wife’s current situation? b. How much additional purchasing power would the wife contribute if her market work hours increased to 8? c. How much of an increase in purchased goods and services would be necessary to compensate for the additional 2 hours of lost household production? d. Should this couple choose to have the wife increase her market work by 2 hours? e. If this couple is raising a child, suppose that combinations of household production and purchased goods are now ranked according to the formula U = X3Y. Would the additional 2…Consider a one period model in which a representative agent maximises the utility function: u(c,l) = lnc + 5lnl subject to the budget constraints: c = (1-t)w(1-l) + v where c is consumption and l is the amount of leisure, they enjoy out of a total of one unit of time available, t is the tax on wage earnings which pays for v in government transfer payments. A. Solve for first order conditions of the representative agent. B. Write down the market clearing condition (resource constraint) for the aggregate economy. C. Solve for equilibrium consumption and labour choices.
- In the two good specific factor model with diminishing returns to the mobile factor the utility possibility frontier is a straight line. True FalseHow 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.Consider a couple whose behaviour follows the unitary household model. Their preferences can be represented by the utility function: U(CM; CH) = min (CM, CH), where CM, denotes market goods and CH denotes home production. Each spouse can work up to 50 hours per week, and those 50 hours can be divided between market work and home production. Joe and Anna are each paid £20 per hour for market work. Joe produces £20 of home production per hour, while Anna produces £30 per hour of home production. (a)How many hours are each of the spouses allocating to home production and market work? and Suppose that Anna is offered a pay raise, so that her hourly market wage increases to £25, and nothing else changes. Will that change the identity of the spouse who works more hours on the market? Explain your answer.
- Anna and Bob are the only residents of a small town. The town currently funds its fire department solely from the individual contributions of these two residents. Each of the two residents has a utility function over private goods ? and total number of firemen ?, of the form: ?(?,?)=2ln?+ln?. The total provision of firemen hired, ?, is the sum of the number hired by each of the two persons: ?=??+??. Ann and Bob both have income of 200 each, and the price of both the private good and a fireman is 1. They are limited to providing between 0 and 200 firemen. For the purposes of this problem, you can treat the number of firemen as a continuous variable (it could be man-years). Consider the setup from Question 2. Suppose that the government recruits additional ?N firemen and taxes Ann and Bob equally to cover the cost. Therefore, the total number of firemen is ??+??+?MA+MB+N, where ??,??MA,MB are appropriate individually-optimal contributions of ?A and ?B(i.e., the agents behave optimally,…Consider a two-person exchange economy in which initial endowments for both individuals are such that (e1 = e1) = (1,1). Suppose the two individuals have the following indirect utility functions: V1 (x, y) = ln M1 - a ln Px - (1-a) ln Py V2 (x, y) = ln M2 -b ln Px - (1-b) ln Py Where Mi is the income level of person i and Px and Py are the prices for goods x and goods y, respectively. a) Calculate the market clearing prices.Consider two towns with 4 million individuals each. The figure below shows the initial locational equilibrium for these two towns where the utility per worker is $80 as shown by point I. Use this figure to answer parts a and b. a. If one town creates a growth boundary which restricts the population to 3 million people, graphically show the adjustment process to a new equilibrium and describe how you reached this new level. (Hint: you should talk about 3 new points and draw a new curve) b. What is the final utility level in each of the two cities following the growth boundary implementation? Are residents better or worse off than before the growth boundary?
- 3. A firm that is located in country H, where price levels are p = (1,1), needs to send one of its two employees to its branch in country F. However, in country F price levels are p′ ≫ p, so the firm will have to pay additional salary to ensure that its employee is equally well-off in country F as she was in country H. Suppose the utility functions of the two employees are u1(x1, x2) = x1 + x2 and u2(x1, x2) = min {x1, x2}. The two employees are otherwise identical, including current salary. If the firm wants to minimize the additional salary it needs to pay, which employee should it send? Explain.Person 1 and person 2 are the only two residents of an economy. Person i (where i is either 1 or 2) has the utility function Here, Yi is person i’s income and β is parameter between 0 and 1. Assume that the social welfare function is where α is a parameter between 0 and 1. Initially, person 1’s income is 1 and person 2’s income is 2. a) Express W in terms of Y1 and Y2. If a social indifference curve shows all the pairs (Y1, Y2) that yield the same value of W, what would a social indifference curve look like if it were drawn in the (Y1, Y2) quadrant? Find an algebraic expression for the slope of a social indifference curve. b) Imagine that income redistribution is costless, in the sense that the economy can reach any pair (Y1, Y2) that satisfies the condition Draw a graph of the attainable pairs in the (Y1, Y2) quadrant. This set is called the “utility possibility frontier.” Using this frontier and the social indifference curves,…Susan obtains utility by consuming carrots C and enjoying leisure L. Suppose that she has a daily non-wage income Y of £100 and is paid a fixed hourly wage rate of £10 for every hour she works in a local coffee shop. Assume that Susan is a utility maximiser and is free to choose x hours of work per day where 0 ≤ x ≤ 10. Assume also that the unit price of C is £1. a) Suppose that L is measured on the horizontal axis and C on the vertical axis. Use these axes to draw the set of all C and L combinations that Susan can choose from. Write down Susan’s budget equation. b) Suppose that Susan’s preferences over carrots and leisure are expressed by the following utility function: U(C,L) = min{C, 10L}. Calculate Susan’s optimal consumption bundle, both algebraically and graphically. Calculate the value of MRS at the optimal choice. c) Suppose instead that Susan’s preferences are such that indifference curves in the L-C space are strictly convex to the origin, and that she chooses to work 5…