Income and sales taxes
.
1. Consider a world with no income and sales taxes. The utility function of a representative
consumer is given by u =
p
x1x2.
Her budget constraint is given by 7x1 + 10x2 = 3000.
Solve for the optimal consumption bundle that the consumer will choose.
Now suppose the government is debating between a 9:75% áat sales tax on both the
commodities or a 5% income tax.
The government is going to have either the sales tax or the income tax but NOT both.
Solve for the optimal consumption bundles under sales tax and under income tax
regimes separately.
Which regime (sales tax or income tax) raises higher revenue for the government?
Calculate the precise revenue collection in each scenario.
Which regime (sales tax or income tax) results in higher consumption for our representative
consumer?
Which regime (sales tax or income tax) results in higher utility for our representative
consumer?
If there are one million consumers like our representative consumer, how much of the
two commodities will be demanded in the economy? (Compute the total demand with
NO taxes and then with sales and income taxes. Now compares these three sets
of numbers and reáect on their di§erences.)
2. Consider three consumption bundles X; Y; Z.
Prove that if X is preferred to Y and Y is preferred to Z then X is preferred to Z:
Use the insight from this proof to show that two indi§erence curves may not intersect
one another.
3. Suppose two commodities are perfect complements and the consumerís utility function
is given by U = minfX1; X2g.
Both the commodities are priced the same way and they cost $20 per unit. The
consumer makes $2000.
Write down the budget constraint of the consumer.
How much amount for each of the commodities will the consumer consume and why?
What will be the utility level of the of the consumer if she is maximizing her utility
subject to the budget constraint?
4. Argue in EACH case below as what returns to scale the production function exhibits
(Show reason behind your conclusion):
f(x1; x2) = x1 + x2;
f(x1; x2) = (x1 + x2)
2
;
f(x1; x2) = (x1 + x2)
1=2
5. Critically assess if the following statements are true or false:
(You MUST provide 3
4 linesíof explanations for EACH answer and draw graphs
wherever you can. Providing crisp mathematical explanation wherever possible will be
good.)
i) MC intersects the AV C where AV C is minimum.
ii) MC intersects the AT C where AV C is minimum.
iii) Break-even point refers to the point where MR = MC
6. The inverse demand curve faced by a monopolist is given by P = 200010Q.
Monopolistís
marginal cost is constant is given by c = 25.
Assume that the monopolist maximizes its proÖts.
What is the amount that the monopolist would produce and what will be the market
price?
Compute the total proÖt of the monopolist and also the value of the consumer surplus.
7. Consider Question #6 and assume that the market is served by two Örms playing a
Cournot game. Their constant marginals costs of production are given by c1 = 20 and
c2 = 25. Assume that both Örms are proÖt maximizers.
Compute the output produced by each of the Örms. Also compute the aggregate supply
in the market.
What will be the prevailing price in the market?
Compute the proÖt for each of the Örms.
Compare the consumer surpluses between Question #6 and Question #7.
8. ProÖt function of a representative Örm in a competitive market is given by = P Q
wL
rK
F.
Here, P is the market price, Q is the output produced (sold), w is the wage rate, L is
the labor employed, K is the capital employed, r is the rental rate and F is the Öxed
cost that does not depend on the amount of quantity produced.
The Örm uses a production function Q = AKL
.
Algebraically solve for the proÖt maximizing labor (L
) and capital (K
) that the Örm
should be using.
What is the total output produced by the Örm? What is the proÖt earned by the Örm?
Now assume that w = 15; = 0:4; = 0:5; A = 10; and P = 15.
Graph the proÖt maximizing labor (L
) and capital (K
) amounts when r increases
continuously from 1% to 5%.
(Use Excel to simulate as smooth an increase as possible. You may get quite a smooth
path if you plot the changes in r at one Öfth of one percent intervals).
Suppose this economy has 100 such Örms. How will the total employment change as
the rate of interest inches up from 1% to 5%?
9. Find the best strategies for EACH player and carefully Önd the Nash equilibrium of
EACH of the following games:
(Be careful about identifying pure strategy Nash equilibrium (equilibria). If
no pure strategy Nash equilibrium exists then check for the mixed strategy
Nash equilibrium)
(i)
Prisoner 2 !
Confess Donít Confess
Prisoner 1# Confess 18;
18
2;
20
Donít Confess 20;
(ii)
Country 2 !
Trade Donít Trade
Country 1# Trade 1000; 1000 600; 500
Donít Trade 500; 600 500; 500
(iii)
Intel !
cooperate Donít cooperate
Apple# cooperate 20; 20 13; 17
Donít cooperate 17; 13 18; 18
(iv)
Tax Payer Joe !
Cheat Donít Cheat
IRS# Audit (with Probability 0:5) 450; 500
150; 200
Donít Audit (with Probability 0:5) 0; 0 200; 200
(v)
Applicant Russell !
Negotiate Donít Negotiate
Company X# O§er a job 50; 40 60; 30
Donít O§er a job 0; 10
0; 0
(vi)
Player B !
Go to Opera Go to Movie
Player A# Go to Opera 1; 1
1;
1
Go to Movie 1;
1 1; 1
10. Prove that all Gi§en goods are inferior goods but all inferior goods are NOT Gi§en
goods. (A graphical proof will be su¢ cient).
11. When prices are (P1; P2) = (2; 4) a consumer demands (x1; x2) = (4; 2): When the
prices are (4; 2) the consumer demands (2; 4). Argue if the behavior of the consumer
is consistent with the utility maximizing behavior.
12. The total cost of a Örm is given by C = 2 + 3Q + 4Q2 where Q is the total amount of
output.
What is the cost of producing zero units? What do you call this amount?
What is the total cost of producing four units? What is the marginal cost of producing
the Öfth unit? What is the marginal cost of producing the sixth unit?
Draw a graph depicting the following as the output rises from 0 to 100: MC, AVC,
ATC, AFC. (Tip: Use Microsoft Excel)
Looking at your answers, what can you say about the change in marginal cost as the
output level keeps increasing?
13. Suppose that the inverse market demand is given by P = 200
5Q. Assume now that
the market is served by two Örms (Cournot Model).
Marginal cost of production of Örm 1, c1 is 5 and marginal cost of production of Örm
2, c2 is 7. In other words, Örm 1 is the low cost Örm and Örm 2 is the high cost Örm.
Derive the following for this m
(i) equilibrium price (P
)
(ii) equilibrium output of the Örst Örm (q
1
)
(iii) equilibrium output of the second Örm (q
2
).
(iv) equilibrium total supply in the market (Q = q
1 + q
2
)
(v) equilibrium output of the Örst Örm (
1
)
(vi) equilibrium output of the Örst Örm (
2
)
14. Consider the demand function of Question #13. Suppose the two Örms are competing
in a leader-follower fashion (Stackelberg Model) where Firm 1 is the Leader and Firm
2 is the follower. Marginal cost of production of Örm 1, c1 is 5 and marginal cost of
production of Örm 2, c2 is 7. In other words, Örm 1 is the low cost Örm and Örm 2 is
the high cost Örm.
Derive the following for this market:
(i) equilibrium price (P
)
(ii) equilibrium output of the Örst Örm (q
1
)
(iii) equilibrium output of the second Örm (q
2
).
(iv) equilibrium total supply in the market (Q = q
1 + q
2
)
(v) equilibrium output of the Örst Örm (
1
)
(vi) equilibrium output of the Örst Örm (
2
)