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Economics 2152A/B

Practice Problems: ECON 2220A Prof. Simona Cociuba Note These problems are last years second midterm. These are meant to give you some practice as you prepare for midterm 2. Your solutions to these problems will not be graded. Lecture notes 15 (November 5) will have detail on the material you need to cover for midterm 2. 1. Long-Run Changes in Productivity (20 pts) Consider the following static model. The representative consumer has preferences de…ned over bundles of consumption and leisure (C;l) and chooses a bundle that is a⁄ordable and that maximizes utility. The consumer earns labor income and receives dividend payments from the …rm. The …rm uses capital (K) and labor (N) to produce goods according to the a 1▯a Cobb-Douglas production function Y = zK N ; where z is total factor productivity and a 2 (0;1): There is no government taxation and government spending is zero. (a). (2 points) Give a de…nition of the production possibilities frontier (PPF). (b). (2 points) Write down the expression for the PPF in this economy. (c). (2 points) Plot the production possibilities frontier and label it1PPF . Make sure you label the two goods produced in this economy on the x-axis and the y-axis of your graph. Do not mark an optimal bundle on the graph. (d). (6 points) On the same graph, illustrate how the production possibilities frontier changes when productivity (z) increases. Label the new curve as PP2 : Brie‡y explain why the production possibilities frontier changes the way that you illustrated in you graph. Note: For every change that you make to the PPF write at most 1 or 2 sentences explaining why that change was necessary. If no change is necessary, explain that as well in at most 1 or 2 sentences. (e). (8 points) Canadian data from 1950 to present shows increases in real GDP, real private consumption and the real wage, but practically no change in hours worked by Canadians. Can long-run increases in productivity explain these observations? To answer 1 this question, explain what happens to the model’ s output (Y ), private consumption (C), wage (w) and hours worked (N), as a result of a long-run increase in total factor productivity (z): 2. Pareto Optimal Allocation (20 pts) Consider a benevolent social planner who maximizes the utility of a representative consumer: U (C;l) = C 0:l 0; where C denotes consumption and l denotes leisure and where C > 0 and l > 0: The social planner knows that the consumer has h hours of available time that can be split between work and leisure. Goods in the economy are produced using the technology Y = zN; where z denotes the level of total factor productivity and N denotes the total amount of time the consumer works. Of the goods produced, the social planner distributes amount G of goods to the government and the rest is given to the consumer. Recall: There are no prices in the social planners problem! (a). (2 points) Write down the social planner’ s problem in this economy. (b). (8 points) Solve for the social planner’s allocation: (C;l;N;Y ) given the following exogenous parameters: h = 100, z = 2 and G = 20: (c). (2 points) Illustrate the optimal bundle (C;l) on a graph that has leisure (l) on the x-axis and consumption (C) on the y-axis. Make sure your graph re‡ects the functional forms given to you in the economy. (d). (8 points) How does the optimal allocation (C;l;N;Y ) change when government consumption increases to G = 40? Provide a brief explaination for your results. Note: You are not required to recompute equilibrium here. In addition, mark the new optimal bundle on the graph you drew in part (c). Make sure you illustrate any changes in the production possibilities frontier that occur due to the change in G: 2 3. Competitive Equilibrium in One-Period Model (30 pts) Consider a representative consumer who has preferences over consumption of a …nal good (C) and leisure time (l) given by the following utility function: U (C;l) = lnC + b ▯ l where b > 0 is a▯par▯meter and where C > 0 and l > 0: The consumer splits available time (h) into work time N S and leisure time (l): The consumer’ s labor income is taxed at the proportional tax rate t > 0; so that after-tax labor income of the consumer is (1 ▯ t)wN . The tax revenues are used to …nance government expenditures, G > 0: The consumer also receives dividend income (▯): The representative …rm produces a …nal good which is sold to the consumer at the price ▯ D▯ of 1: The …rm hires labor N and produces the …nal good using the following technology of production: ▯ ▯ Y = z N D a where z is the total factor productivity and a is a parameter which satis…es 0 < a < 1: (a): (4 points) Compute GDP according to the income approach. Be speci…c about what are all the components of income. Label each component using words, and also write the expression of each component of GDP using model variables. (b): (6 points) De…ne an equilibrium in this economy. Be speci…c about which variables are endogenous and exogenous; write down everyone’ s problem expli
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