Homework 01
F21 econ 3010-100
Marc Santugini
August 27, 2021
You must provide details to your answers including your diagrams. You may collaborate
freely with other students from F21 econ 3010-100. If you form a group, each homework
group will submit one paper. A few more rules: 1) you cannot hire a tutor to solve the
problems for you; 2) failing to fully collaborate (i.e., free-riding) on homework assignments
is not acceptable; 3) your work is expected to be neat and legible. If the TA …nds your
work too sloppy and/or illegible, it will not be graded, or points will be taken o¤. The
assignment is due at 5:00pm on Wednesday, September 8, 2021 (deadline has been
extended).
1. The original budget line is p1 x1 + p2 x2 = I. The government decides to impose a
lump-sum tax of T 2 (0; I), a quantity tax on good 1 of t > 0 dollars, and a quantity
subsidy on good 2 of s 2 (0; p2 ) dollars.
(a) Write down an equation for the new budget line. (1 point)
(b) Sketch the original and new budget lines. Clearly label your graph. (1 point)
2. Consider an individual who lives only two periods. Let c1 > 0 be consumption in period
1 and c2 > 0 be consumption in period 2. Therefore, (c1 ; c2 ) is the (intertemporal)
consumption bundle. Let p1 be the price of the good in period 1. In period 2, the
price is higher due to in‡ation. Let 2 (0; 1) be the rate of in‡ation. For t = 1; 2,
her (per-period) income is It > 0. In addition to receiving income, she may save or
borrow in period 1. Let s 2 ( I2 =(1 + i); I1 ) be the amount that is saved or borrowed
in period 1. If she saves (i.e., s > 0), then she receives back her savings plus interests
at the beginning of period 2. If she borrows (i.e., s < 0), she pays back her loan plus
interests at the beginning of period 2. Whether she borrows or saves, the interest rate
is i 2 (0; 1).1
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A few comments.
(a) The individual decides to save when she wishes to consume c2 > I2 =p2 in period 2. What is p2 in this
problem?
(b) The individual decides to borrow when she wishes to consume c1 > I1 =p1 in period 1. Note that the
individual is not allowed to borrow more than I2 =(1 + i). In other words, there is no default in this
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(a) For s 2 [ I2 =(1 + i); I1 ], write down an equation for the intertemporal budget
line (i.e., an equation in terms of c1 and c2 ). (2 points)
(b) Sketch the intertemporal budget line and indicate the segment corresponding
to s 2 ( I2 =(1 + i); 0], the segment corresponding to s = 0, and the segment
corresponding to s 2 (0; I1 ]. What do each of these cases represent? Clearly label
your graph. (2 points)
(c) Show graphically the e¤ect of an increase in in‡ation on the budget set. (1 point)
3. Let T = 12 be total available hours, L be hours of leisure, and n be hours of work. Let
w > 0 be the hourly wage and p > 0 be the price of the consumption good. Finally,
let C be consumption.
(a) Suppose that the agent receives a lump-sum amount of B dollars if he works
more than 5 hours. Sketch the corresponding budget constraint, plotting L on
the horizontal axis and C on the vertical axis. Clearly label your graph. (2
points)
(b) Suppose that if the agent decides to work beyond regular hours (say 5 hours),
then he is given an additional dollars for every hour (beyond 5 hours). Sketch
the corresponding budget constraint, plotting L on the horizontal axis and C on
the vertical axis. Clearly label your graph. (2 points)
(c) Suppose that the law forces the agent to work no less than 3 hours and no more
than 5 hours. Sketch the corresponding budget constraint, plotting L on the
horizontal axis and C on the vertical axis. Clearly label your graph. (2 points)
(d) Suppose that the agent receives an amount B > 0 only if the agent is inactive, i.e.,
n = 0. Sketch the corresponding budget constraint, plotting L on the horizontal
axis and C on the vertical axis. Clearly label your graph (2 points)
4. Let the utility function be U (x1 ; x2 ) = x1
2x21 + x22 .
(a) Sketch the indi¤erence curves for utility levels 1 and 2. (2 points)
(b) Calculate the marginal rate of substitution of good 1. (1 point)
p
5. Let the utility function be U (x1 ; x2 ) = maxf x1 ; x2 g. Sketch the indi¤erence curves
for utility levels 2 and 3. (2 points)
problem. Why?
(c) If s = 0, then (c1 ; c2 ) = (I1 =p1 ; I2 =p2 ) is the corresponding consumption bundle on the budget line.
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