1Econ 2316 Microeconomic Theory HOMEWORK SET #5(Due Friday, November 9, 2012) Name:...

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1Econ 2316 Microeconomic Theory HOMEWORK SET #5(Due Friday, November 9, 2012) Name: _________________________________________________________ Please print this sheet and staple it in front of your answer set. Homework is due at the beginning of class (on Friday, November 9) or leave it in my mailbox before class (Do not write your answers on this sheet)2HOMEWORK SET #5Your completed homework is due in class at the beginning of class or in my mailbox before class in the Economics Department 301 Lake Hall. No credit will be given for late homework sets.1.A firm has the following production function: q = LK. The firm wants to produce a target output level of q=200. The costs of the two inputs are r=$2 (for capital K) and w=$1(for labor L). a.What is the Marginal Rate of Technical Substitution between labor and capital? b.What is the least cost method of producing the target level of output (i.e., what are the optimal choices of K and L to minimize costs)? c.Suppose now a payroll tax is imposed on the firm, so that the firm must pay $3 per unit of labor to the government. How does this payroll tax affect the cost minimizing combinations of L and K if the firm still wants to produce q=200? Explain your answer. d.How much revenue does the government raise? Do the costs to the firm rise? If so, by how much? Explain your answer. e.How would your answer to part (a)-(d) be affected if the production function changed to q = L + K? 2.Consider the market for coffee at Northeastern University. The market supply is given by: QS = -20 + PS, where QSis the quantity supplied and PSis the price per cup of coffee received by coffee firms. The market demand curve (for students) is given by: QD = 100 – PD, where QDis the quantity demanded and PDis the price per cup of coffee paid by students. Prices are in cents. Initially, suppose that there are no taxes or subsidies on coffee, so that PD = PS . Call this common price P.
	
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HOMEWORK SET #5 
 
(Due Friday, November 9, 2012)

Name:  _________________________________________________________

Please print this sheet and staple it in front of your answer set.  Homework is due at 
the beginning of class (on Friday, November 9) or leave it in my mailbox before class 
 
(Do not write your answers on this sheet)

1HOMEWORK SET #5

Your completed homework is due in class at the beginning of class or in my mailbox before 
class in the Economics Department 301 Lake Hall.  No credit will be given for late 
homework sets.

1. A firm has the following production function: 
 
q = LK. 
 
The firm wants to produce a target output level of q=200.  The costs of the two inputs 
are r=$2 (for capital K) and w=$1(for labor L). 
 
a. What is the Marginal Rate of Technical Substitution between labor and 
capital? 
b. What is the least cost method of producing the target level of output (i.e., what 
are the optimal choices of K and L to minimize costs)? 
c. Suppose now a payroll tax is imposed on the firm, so that the firm must pay 
$3 per unit of labor to the government.  How does this payroll tax affect the 
cost minimizing combinations of L and K if the firm still wants to produce 
q=200?  Explain your answer. 
d. How much revenue does the government raise?  Do the costs to the firm rise?  
If so, by how much?  Explain your answer. 
e. How would your answer to part (a)-(d) be affected if the production function 
changed to 
q = L + K? 
2. Consider the market for coffee at Northeastern University. The market supply is 
given by: 
 
Q = -20 + P , 
S S
 
where Q is the quantity supplied and P is the price per cup of coffee received by 
S S
coffee firms. The market demand curve (for students) is given by: 
 
Q = 100 – P , 
D D
 
where Q is the quantity demanded and P is the price per cup of coffee paid by 
D D
students.  Prices are in cents....

David · Accepted Answer

Answer to 1: 
q = LK 
Firm want q = 200 
Cost of inputs r = $2 and w = $1 
a. 
Marginal rate of technical substitution between labor and capital = MPL/MPK 
MPL = dq/dL = K 
MPK = dq/dK = L 
So Marginal rate of technical substitution between labor and capital (MRTS) = K/L 
b. 
At the cost minimization point,  
MRTS = w/r i.e. 
K/L = 1/2, implies K = (1/2)L ……………………. (1) 
We want output q = 200 i.e. 
KL = 200 ………….. (2) 
Substituting (1) and (2) 
(1/2)L2 = 200, implies cost minimizing value of L = 400^(1/2) = 20 
And cost minimizing value of K = (1/2)*20 = 10 
c. 
Now government pays $3 per unit of labor to the government, so labor cost (w) now becomes;  
w = 4 
In such a case, the condition (MRTS = w/r) implies 
K/L = 4/2 = 2 
So we get K = 2L  ……………….. (2) 
Substituting this in required output constraint i.e. in (2),  
2L2 = 200, implies cost minimizing value of L = 10 
And the cost minimizing value of K = 2*10 = 20 
d. 
Revenue received by government = 10*3 = $30 
Total cost before payroll tax = 20*1+10*2 = $40 
Total cost after payroll tax = 10*4+20*2 = $80 
So we note that after payroll tax, the cost to the firm has risen by $40. 
e. 
If production function changed to q = L + K, then the two inputs would be taken as perfect 
substitutes. So firm would employ only that input which is cheaper in cost. 
Before payroll tax, labor was cheaper, so firm would have employed only labor, where labor 
employed (L) = 200. So total cost to the firm = 200*1 = $200 
After payroll tax, capital was the cheaper, so firm would employ only capital, where capital 
employed (K) = 200. So total cost to the firm = 200*2 = $400 
We note that after the imposition of tax, labor is not hired, so there is no government revenue in 
this case. 
Before the payroll tax, total cost to the firm = $200 and after that it became $400. So after the 
imposition of tax, the cost to the firm has increased by $200.
Answer to 2: 
Market supply; Qs = -20 + Ps 
Market demand; Qd = 100 – Pd 
a. 
At the equilibrium, Qs = Qd i.e. 
-20 + Ps = 100 – Pd, implies equilibrium price P = 60 
b. 
Equilibrium amount of cups of coffee Q = -20+60 = 40
c. 
In this case, price paid by consumers (students) Pd is 60 cent less than price received by seller (Ps) 
i.e. 
Pd = Ps – 0.

1Econ 2316 Microeconomic Theory HOMEWORK SET #5(Due Friday, November 9, 2012) Name: _________________________________________________________ Please print this sheet and staple it in front of your...

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