Practice Problems: ECON 2220A
Prof. Simona Cociuba
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
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 : Briey 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
(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
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 planners 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 reects 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
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