ECO2066 Mathematics for Economics 2019-2020 Assignment Questions

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1. Given a monopoly firm’s demand function PQ 25,050 −= , and its cost function

QQQC 20)( 2 +=

a. Find the level of output and price, which maximize profit. Calculate the

elasticity ε and mark-up coefficient µ at that point. Show the profit rectangle on

following plane.

b. If the government decide to impose a per unit quantity tax a s amount of t in

order to maximize its tax revenue what would be the best value of t. Use the

value that you found and calculate the tax revenue. Sketch the graph of

demand (inverse), MR, MC (before and after the tax) determine the deadweight

loss area and calculate the deadweight loss caused by taxation.

c. Instead of taxation, if the monopoly is forced to accept marginal cost pricing by

government, what will be the profit maximize price - quantity combinations.

Sketch the graph, determine the deadweight loss area and calculate the

deadweight loss of mark-up pricing.

2. According to given national income model when the tax ratio rises up how would change the equilibrium value of income and corresponding tax revenue. Use partial derivatives to answer this question.

𝑌 = 𝐶 + 𝐼 + 𝐺 𝐶 = 𝑎 + 𝑏 𝑌 − 𝑇 𝑎 > 0, 0 < 𝑏 < 1

𝑇 = 𝑡𝑌 0 < 𝑡 < 1 𝐼 = 𝐼 𝐺 = 𝐺

3. Take first and second derivatives of given function and sketch its graph with its all

details. Find the inflection point.

𝑦 = !!!!!!!,!!

4. What is the growth rate of 𝑦 . ( refers to time, is a constant parameter)

𝑦 = !!.!!"#

!! !

t r

5. Solve the problem, according to given IS – LM Model.

𝐶 = 200 + 0,8 𝑌 − 𝑇

𝑇 = 0,25𝑌

𝐼 = 0,2𝑌 − 2000𝑖

𝐺 = 120

𝐵 = 𝑇 − 𝐺

𝑀! = 0,3𝑌 − 4000𝑖

𝑀! = 60

a. What is the equilibrium income 𝒀 and interest rate 𝒊 ofeconomy? Calculate the corresponding government budget.Sketchthegraphandshowtheequilibriumpoint.

b. If government prefer to keep its budget at balance (withoutanysurplusordeficit)whatwouldbetaxrate?

c. Ifthegovernmentexpendituresareincreasedfrom120to200and tax rate is decreased from 0,25 to 0,2while the CentralBank decide to increasemoney supply from 60 to 130whatwouldbenewequilibriumlevelsofincome𝒀andinterestrate𝒊?Sketchthegraphandshowchanginginequilibriumpoint.

6. Answer the following questions for the given function.

𝑧 =𝑦!

𝑥− 𝑦!𝑥 + 2𝑥𝑦 +

𝑥!

𝑦!

a. Determine whether or not the function is homogeneous.

b. Verify Euler's theorem for that function if it is homogeneous. c. At the point (2, 2) hence estimate the change in z when x increases from 2 to 2.1

and y decreases from 2 to 1,8

d. If the z is fixed at 12 find the equation of tangent line at point (2,2) on xy plane.

7. According to given Cobb – Douglas production Q K, L = 8K!L!!

a. Find the marginal product of capital 𝑀𝑃! =!"!"

and sketch the projection of

the production function onto the Q–K plane (use the first and second

derivatives).

b. Find the marginal rate of technical substitution 𝑀𝑅𝑇𝑆 = !"!"

and sketch the projection of the production function onto the K–L plane (use the

first and second derivatives).

c. Is the production function homogenous? Verify Euler's Theorem for that function if it is homogeneous. Determine the type of returns to scale. When the inputs are doubled how much does output change? Assume that cost function is given like this 𝐶 = 𝑟𝐾 + 𝑤𝐿 with constant coefficients of r and w. How will average cost 𝐴𝐶 change?

d. Obtain the total differential of output ∆𝑄 = !"!"

∆𝐾 + !"!"∆𝐿 and suppose that

initially capital equals 20 and labor equals 16 units. If capital increases from 20

to 22 and labor decreases from 16 to 14 how much will output change?

∆𝑄 =𝜕𝑄𝜕𝐾

∆𝐾 +𝜕𝑄𝜕𝐿

∆𝐿

e. Determine the capital elasticity of output 𝜀!,! =!"!"

!!

f. Suppose that capital and labor may change in time.

Q K(𝑡), L(𝑡) = 8 K(𝑡) ! L(𝑡)!!

If capital increases from 2000 to 2400 and growth rate of labor 24%.

( ) 24,0ln==

dtLdgrwL Calculate the growth rate of Q (output). grwQ =

d lnQ( )dt

?