Optimal Taxation and Revenue Maximization - Prof. Singh, Study notes of Economics

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Solution

Manual

DSE & ISI

MA (Economics)

Entrances

1. ISI - 2013

Solved Problems - ISI 2013 - PEB

Exercise 1.1 An agent earns w units of wage while young, and earns nothing while old. The agent lives for two periods and consumes in both the periods. The utility function for the agent is given by u = log c 1 + log c 2 , where ci is the consumption in period i = 1 , 2. The agent faces a constant rate of interest r (net interest rate) at which it can freely lend or borrow, (a) Find out the level of saving of the agent while young. (b) What would be the consequence of a rise in the interest rate, r, on the savings of the agent?

A 1. Agent’s utility maximization problem is the following:

max c 1 ,c 2 log c 1 + log c 2

s.t. c 1 ( 1 + r) + c 2 = w( 1 + r) & c 1 ≥ 0 , c 2 ≥ 0

(a) Solving the above problem we get: (c 1 , c 2 ) =

w 2

w( 1 + r) 2

Hence, Savings = w − c 1 = w 2 (b) Clearly, Savings doesn’t change with change in rate of interest rate, r. 

Exercise 1.2 Consider a city that has a number of fast food stalls selling Masala Dosa (MD). All vendors have a marginal cost of Rs. 15 per MD, and can sell at most 100 MD a day.

A 1.

The production possibility frontier of the two inputs is given by k 1 10

k 2 20

Since final product can be sold at the end of the day at a per unit price of Rs. 1. the firm’s profit maximization problem is

max k 1 ,k 2

k 1 k 2 − 20

s.t.

k 1 10

k 2 20

& k 1 ≥ 0 , k 2 ≥ 0 ,

Thus, firm will hire the worker, produces k 1 = 50 , k 2 = 100 and produces and sells output = 50



Exercise 1.4 A monopolist has contracted with the government to sell as much of its output as it likes to the government at Rs. 100 per unit. Its sales to the government are positive, and it also sells its output to buyers at Rs. 150 per unit. What is the price elasticity of demand for the monopolist’s services in the private market?

A 1. Since monopolist’s sale to the government is positive, his marginal revenue at the point of sale in the private market must be Rs. 100. Now price in the private market is Rs. 150. We can compute the price elasticity of demand in the following way:

TR(x) = p(x) · x

Differentiating TR(x) w.r.t. x, we get,

MR(x) = p(x) + x

d p(x) dx MR(x) = p(x) + p(x)

x p(x)

d p(x) dx

MR(x) = p(x)

η

Now substituting p(x) = 150 and MR(x) = 100 in above we get elasticity, η = − 3 

Exercise 1.5 Consider the following production function with usual notations.

Y = Kα^ L^1 −α^ − β K + θ L with 0 < α < 1 , β > 0 , θ > 0

Examine the validity of the following statements. (a) Production function satisfies constant returns to scale. (b) The demand function for labour is defined for all non-negative wage rates. (c) The demand function for capital is undefined when price of capital service is zero.

8 Chapter 1. ISI - 2013

A 1. (a) Let f (K, L) denotes the production function. f (tK,tL) = (tK)α^ (tL)^1 −α^ − βtK + θtL = t(Kα^ L^1 −α^ − β K + θ L) = t f (K, L) Thus, production function satisfies constant returns to scale. (b) Profit maximization problem of the competitive producer is max K,L Kα^ L^1 −α^ − β K + θ L − wL − rK s.t. L ≥ 0 , K ≥ 0 (b) The above problem is equivalent to max K,L Kα^ L^1 −α^ − (r + β )K − (w − θ )L s.t. L ≥ 0 , K ≥ 0 Clearly, when 0 ≤ w ≤ θ , demand function for labor is not defined. (c) Also, the demand function for capital is defined when price of capital service is zero provided w > θ. 

Exercise 1.6 Suppose that due to technological progress labour requirement per unit of output is halved in a Simple Keynesian model where output is proportional to the level of employment. What happens to the equilibrium level of output and the equilibrium level of employment in this case? Consider a modified Keynesian model where consumption expenditure is proportional to labour income and wage-rate is given. Does technological progress produce a different effect on the equilibrium level of output in this case?

A 1. Suppose the production function has changed from Y = F 0 (L) = aL to Y = F 1 (L) = F 0 ( 2 L) = 2 aL where a > 0. Labor demand curve is, therefore,

L 0 (w) ∈

{ 0 } if a < w R+ if a = w 0 / if a > w

where w is the real wage and it changes to

L 1 (w) ∈

{ 0 } if 2a < w R+ if 2a = w 0 / if 2a > w

Let us assume that Labor supply is exogenously given and is equal to L. Solving for the equilibrium in labor market we get that the real wage has changed from a to 2 a but the equilibrium employment is L in both cases. Therefore, Aggregate Supply curve has shifted from Y = aL to Y = 2 aL. Given any aggregate demand curve, Y = AD(w) it is easy to see that the new equilibrium level of output in the model will be twice as much as it was earlier. 

10 Chapter 1. ISI - 2013

(a) Analyse how wage and rental rate on capital would change over time. (b) Can the economy attain steady state equilibrium?

A 1. (a) In the Solow model, where the entire income is consumed, there will be no capital formation and in the presence of depreciation capital depletes over time. If the population is either fixed or grows over time then there will be fall in capital by labor ratio over time. Therefore wage rate would fall and rental rate would increase over time. (b) Yes, this economy will attain a steady state at k = 0 which is disappointing but steady. 

2. ISI - 2015

Solved Problems - ISI 2015 - PEB

Exercise 2.1 Consider an agent in an economy with two goods X 1 and X 2. Suppose she has income 20. Suppose also that when she consumes amounts x 1 and x 2 of the two goods respectively, she gets utility

u(x 1 , x 2 ) = 2 x 1 + 32 x 2 − 3 x^22

(a) Suppose the prices of X 1 and X 2 are each 1. What is the agent’s optimal consumption bundle? (b) Suppose the price of X 2 increases to 4, all else remaining the same. Which consumption bundle does the agent choose now? (c) How much extra income must the agent be given to compensate her for the increase in price of X 2?

A 2. (a) Check yourself that u(x 1 , x 2 ) is a concave function and hence is also quasi-concave, therefore solution to the above problem can be obtained through the standard slope analysis. MRS 12 =

32 − 6 x 2 and the budget line is x 1 + x 2 = 20. Solution is not at the

corner x 2 = 0 because at this corner, MRS 12 =

p 1 p 2 , therefore consumer will benefit from spending some money on x 2. Solution is not at the other corner x 2 = 20 because MU 2 < 0 at this consumption level, therefore it is beneficial to spend less on x 2. Hence, the solution is in the interior and satisfy MRS 12 = 1 , and we get x 2 = 5 and x 1 = 15.

of the population living within some region of B is simply the proportion of the state’s total land mass contained in that region), with total population normalized to 1. For any resident of B, the cost of travelling a distance d is kd, with k > 0. Every resident of B is endowed with an income of 10, and is willing to spend up to this amount to consume one unit of a good, G, which is imported from outside the state at zero transport cost. The Finance Minister of B has imposed an entry tax at the rate 100 t% on shipments of G brought into B. Thus, a unit of G costs p( 1 + t) inside the borders of B, but can be purchased for just p outside; p( 1 + t) < 10. Individual residents of B have to decide whether to travel beyond its borders to consume the good or to purchase it inside the state. Individuals can travel anywhere to shop and consume, but have to return to their place of origin afterwards. (a) Find the proportion of the population of B which will evade the entry tax by shopping outside the state. (b) Find the social welfare-maximizing tax rate. Also find the necessary and sufficient conditions for it to be identical to the revenue-maximizing tax rate. (c) Assume that the revenue-maximizing tax rate is initially positive. Find the elasticity of tax revenue with respect to the external price of G, supposing that the Finance Minister always chooses the revenue-maximizing tax rate.

A 2. (a) An individual located at distance r − dˆ from the center of B will be indifferent between buying G from inside and outside if p( 1 + t) = p + 2 k dˆ or equivalently, consumer located dˆ =

pt 2 k distance inside the boundary is indifferent between buying from inside and outside the region. Therefore, proportion of people who will buy from outside the circular region equals 1 − π(r − dˆ)^2 πr^2

(r − dˆ)^2 r^2

r (^) −^ r dˆ

14 Chapter 2. ISI - 2015

A 2. (b) Given the tax rate t, social welfare of B is given by the sum of welfare of people buying from outside the state plus the welfare of the people buying G from inside the state and the Government tax revenue. ∫ (^) dˆ

x= 0

2 π(r − x) πr^2

( 10 − p − 2 kx)dx + π(r − dˆ)^2 πr^2

( 10 − p) where ˆd = pt 2 k Differentiating it with respect to t, we get the first order condition 2 π(r − dˆ) πr^2 ( 10 − p − 2 k dˆ)

p 2 k

2 π(r − dˆ) πr^2 ( 10 − p)

p 2 k = 0 where ˆd =

pt 2 k The above holds when dˆ = 0 (welfare maximizing) or dˆ = r (welfare minimizing) Therefore, the social welfare maximizing tax rate is t = k dˆ/p = 0. Now to find tax revenue maximizing tax rate, we will first write the expression for Tax-Revenue: pt π(r − dˆ)^2 πr^2

= 2 k dˆ π(r − dˆ)^2 πr^2

[

because ˆd = pt 2 k

]

Finding the revenue maximizing t is equivalent to finding the revenue maximizing dˆ. Maximizing above, we get dˆ = r 3 and hence the revenue maximizing tax rate t = 2 rk 3 p Therefore, necessary and sufficient condition for the the revenue maximizing tax rate to be the same as welfare maximizing tax rate is k = 0. (c) From (b), we know that revenue maximizing tax rate is t =

2 rk 3 p and the revenue is

pt (r − [pt/( 2 k)])^2 r^2 Therefore, optimal revenue is 4 k 9 Therefore, the elasticity of tax revenue with respect to p is 0. 

Exercise 2.4 Suppose there are two firms, 1 and 2, each producing chocolate, at 0 marginal cost. However, one firm’s product is not identical to the product of the other. The inverse demand functions are as follows:

p 1 = A 1 − b 11 q 1 − b 12 q 2 , p 2 = A 2 − b 21 q 1 − b 22 q 2 ;

where p 1 and q 1 are respectively price obtained and quantity produced by firm 1 and p 2 and q 2 are respectively price obtained and quantity produced by firm 2. A 1 , A 2 , b 11 , b 12 , b 21 , b 22 are all positive. Assume the firms choose independently how much to produce. (a) How much do the two firms produce, assuming both produce positive quantities? (b) What conditions on the parameters A 1 , A 2 , b 11 , b 12 , b 21 , b 22 are together both necessary and sufficient to ensure that both firms produce positive quantities?

16 Chapter 2. ISI - 2015

A 2. Firm solves the following profit maximisation problem:

max pQ − wF LF − wM LM s.t. Q = LF + LM p = A −

Q

LF = wε FF LM = wε MM

The firm chooses (p, Q, wF , LF , wM , LM ) since it is a monopolist as well as a monopsonist. By eliminating Q and P using the demand constraints and the production function, the above problem can be rewritten as:

max

A −

LF + LM

(LF + LM ) − wF LF − wM LM

s.t. LF = wε FF LM = wε MM

Now we will use the labor supply equations to write the above problem just in terms of input prices:

max

A −

wε FF + wε MM 2

(wε FF + wε MM ) − w^1 F+ εF− w^1 M+εM

FOCs: ( A − wε FF + wε MM 2

εF wε FF −^1 −

(wε FF + wε MM )εF wε FF −^1 − ( 1 + εF )wε FF = 0 ( A −

wε FF + wε MM 2

εM wε MM −^1 −

(wε FF + wε MM )εM wε MM −^1 − ( 1 + εM )wε MM = 0

Above can be rewritten as ( A − wε FF + wε MM 2

εF −

(wε FF + wε MM )εF − ( 1 + εF )wF = 0 ( A −

wε FF + wε MM 2

εM −

(wε FF + wε MM )εM − ( 1 + εM )wM = 0

Again, above can be rewritten as ( A − wε FF + wε MM 2

εF −

(wε FF + wε MM )εF = ( 1 + εF )wF ( A −

wε FF + wε MM 2

εM −

(wε FF + wε MM )εM = ( 1 + εM )wM

A 2.

Dividing them we get εF εM

( 1 + εF )wF ( 1 + εM )wM

Using εM εF = 1 and wM = 2 wF , we get εM = 2 , εF =



Exercise 2.6 An economy comprises of a consolidated household sector, a firm sector and the government. The household supplies labour (L) to the firm. The firm produces a single good (L) by means of a production function Y = F(L); F′^ > 0 , F′′^ < 0 , and maximizes profits Π = PY −W L, where P is the price of Y and W is the wage rate.The household, besides earning wages, is also entitled to the profits of the firm. The household maximizes utility (U), given by:

U =

lnC +

ln

M

P

− d(L)

where C is consumption of the good and

M

P

is real balance holding. The term d(L) denotes

the disutility from supplying labour; with d′^ > 0 , d′′^ > 0. The household’s budget constraint is given by:

PC + M = W L + Π + M − PT ;

where M is the money holding the household begins with, M is the holding they end up with and T is the real taxes levied by the government. The government’s demand for the good is given by G. The government’s budget constraint is given by:

M − M = PG − PT ;

Goods market clearing implies: Y = C + G.

(a) Prove that dY dG

∈ ( 0 , 1 ), and that government expenditure crowds out private consumption

(i.e.,

dC dG

(b) Show that everything else remaining the same, a rise in M leads to an equiproportionate rise in P.

A 2.

Now we will will write the conditions that the equilibrium prices (W, P) and the equilibrium vector (Y,C, M, T, Π, L) must satisfy: From firm’s profit maximisation problem:

Y = F(L)

F′(L) =

W

P

Π = PY −W L

From household’s utility maximisation problem:

PC + M = W L + Π + M − PT M = PC d′(L) =

W

P

×

2 C

And we have the government’s budget constraint:

M − M = PG − PT

Finally, the market clearing condition

Y = C + G

Market clearing conditions for the money market and labor market are implicit in above since we denoted labor demand and labor supply by the same variable L and money demand and money supply by the same variable M. We will reduce the above system of conditions by using the household’s optimisation condition and substituting M = PC everywhere in the system:

Y = F(L) F′(L) =

W

P

Π = PY −W L

PC + PC = W L + Π + M − PT

d′(L) =

W

P

×

2 C

PC − M = PG − PT

Y = C + G

Now we will eliminate T from the system by substituting PT = −PC + M + PG (using the government’s budget constraint)

Y = F(L) F′(L) =

W

P

Π = PY −W L

PC = W L + Π − PG

d′(L) =

W

P

×

2 C

Y = C + G

20 Chapter 2. ISI - 2015

A 2. Next, we will eliminate Π by substituting it with Π = PY −W L everywhere, we will then reduce the system to

Y = F(L) F′(L) =

W

P

d′(L) =

W

P

×

2 C

Y = C + G

Now we eliminate

W

P

by substituting

W

P

= F′(L) everywhere,

Y = F(L) d′(L) = F′(L) ×

2 C

Y = C + G

Differentiating the above system with respect to G, dY dG

= F′(L)

dL dG d′′(L) dL dG

F′′(L)

2 C

×

dL dG

F′(L)

2 C^2

×

dC dG dY dG

dC dG

Eliminating

dC dG , we get

dY dG

= F′(L)

dL dG d′′(L) dL dG

F′′(L)

2 C

×

dL dG

F′(L)

2 C^2

×

dY dG

Now solving for dY dG , we get

dY dG

(F′(L))^2

(F′(L))^2 + 2 C^2 d′′(L) − F′′(L)

The above follows from F′(L) > 0 , F′′(L) < 0 and d′′(L) > 0. Since

dC dG

dY dG − 1 , we get dC dG



Exercise 2.7 Consider the Solow growth model in continuous time, where the exogenous rate of technological progress, g, is zero. Consider an intensive form production function given by:

f (k) = k^4 − 6 k^3 + 11 k^2 − 6 k ( 1 )