econometrics assignment no 1

MPH 511: ECONOMIC ANALYSIS FORMPH 511: ECONOMIC ANALYSIS FOR BUSINESSBUSINESS

ASSIGNMENT ONEASSIGNMENT ONE

SUBMITTED TO:Prof. Rajendra Prasad Shrestha, Ph.D

Master of Philosophy in ManagementFaculty of Management

Tribhuvan University

Submitted by:

Nischal ThapaRoll No 14/012

MPhil in Management, TU

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April, 2013

PROBLEM 1Suppose, A B C Book Seller located at the university complex is selling Econometric Book. The cost per book of the seller (including transport and other related costs) is 76. The firm has other overhead and marketing cost which is considered to be fixed cost. The marginal cost is 76 assuming it is constant. There is another book seller located in the same complex. So, the manager of ABC Book Seller estimates very high price elasticity of demand for econometric books. The estimated price elasticity of book is 2.5. Therefore, the manager of ABC Book Seller decided to put 67% mark-up on the cost per book. Do you think the price decided by the manager is optimal?

ANSWER 1Given in the question, Cost per book (including transport and other related cost) = Rs. 76Marginal Cost (MC) = Rs. 76

Estimated price elasticity of book (ep) = - 2.5

Mark-up = 67% of costs per bookPrice per book (P) = Cost per book (1+ Mark-up) = Rs. 76(1 + 0.67) = Rs. 126.92Here, According to Arc elasticity demand Marginal Revenue (MR) = P (1 – )Or, 76 = P (1– )Or, P = 76/0.6 P = 126.67 P = Rs. 127 (roundoff)Conclusions: Even though in the absolute figures, the price might not be absolutely optimal, but in roundoff figure, the price is optimal.

Nischal Thapa, MPhil-TU

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PROBLEM 2An American company provides fertilizers and weed control lawn services to residential customers of Lagankhel. Its seasonal service package, regularly priced at $250, includes several chemical spray treatments. As part of an effort to expand its customer’s base, American company offered $50 off its regular price to customers in the Lagankhel area. Response was enthusiastic, with sales rising to 5750 units (Package) from the 3250 units sold in the same period last year.

a) Calculate the arc price elasticity of demand for American company service.b) Assume the arc price elasticity (from part A) is the best available estimates

of the point price elasticity of demand. If marginal cost is $ 135 per unit for labour and materials. Calculate company’s optional markup on price and its optimal price.

ANSWER 2Given in the question,Price of each service package (P1) = $ 250Change in Price ( P) = $ 50Price of new service package (P2) = 250–50 = $ 200Quantity demand before new service package (Q1) = 3250 unitsQuantity demand after new service package (Q2) = 5750 unitsChange in Quantity demand ( Q) = 2500 unitsNow, Calculation of Arc Price Elasticity (ep);

We know that, ep = ×

Or, ep = × = = 2.5Arc Price Elasticity (ep)= 2.5

Now, calculation the optimal price and markupHere, Marginal cost (MC) = $135We know that, at the level of optimal profit, MC is equal to MR


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MR = MC = $ 135Again,We know that, MR = P (1– )Or, 135 = P (1– )Optimal price (P) = $ 225Therefore, Price per book (P) = Cost per book (1+ Mark-up)or $225 = MC (1+Mark-up%)or, $ 225 = $ 135(1 + Mark-up%)Mark up percentage = – 1Mark up percentage = 66.67%Conclusion:Hence, the company’s optimal price is $ 225 and its optimal markup is 66.67%.

PROBLEM 3


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ABC Company is operating for the last 10 years in Kathmandu Valley. The company wanted to hire you as a consultant to determine profit maximizing output and price. From the ten years data on the sales and price supplied by the Accountant of the company you run regression program and came with the following demand and cost functions P=90-Q and a cost function as C=10+0.5Q2 . The General Manager of the company told you that the company was producing 160 units as an optimal quantity of production. How do you suggest the General Manager of the company?

ANSWER 3Given in the question,Demand function; P = 90–2Q Cost function; C = 10+0.5 Q2

General Manager's assumption of optimal quantity of production (Q) = 120 unitHere;Total Revenue (TR) = P × Q

Or, TR = (90–2Q) × QRevenue function; R = 90Q –2Q2

Again, we know that, Profit (П) = TR – TC П = (90Q –2Q2) – (10+0.5 Q2) Profit Function; П= –2.5Q2 +90Q –10 Now, MR = == 90– 4Qand, MC = == 0+1Q = QAccording to the profit maximization principle,MC = MROr, Q = 90 – 4QOptimal quantity (Q) = 18 unitsTesting of reliability of above calculation:Condition 1: At the profit maximization level of output; Marginal profit or 1st order derivative of profit must be equal with zero, Taking 1st order derivative of profit, Marginal Profit = = = –5Q+90


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Then, substituting the value of Q = 18,

Marginal profit = = –5 ×18 + 90 = 0 = 0Condition 2: Again, at the profit maximization level of output; 2nd order of derivative of profit must be less than zero (i.e. negative), Taking 2nd order derivative of profit, < 0Or, = – < 0Or, = – < 0Or, = – < 0Or, = –4 – 1 < 0Or, = –5 < 0Thus, the first derivative or marginal profit is zero and 2nd order derivative of the firm is found less than zero (i.e. – 5). Hence, the optimal quantity of production that has profit maximization is 18 units.

Conclusion: At ABC Company, since the General Manager's assumption of optimal quantity of production (i.e. 120 units) is not equal to the calculated optimal quantity (i.e. 18 units); hence, the assumption of optimal quantity of production is wrong. Therefore the general manager should produce 18 units as an optimal quantity of production for profit maximization.

PROBLEM 4A M.Phil Graduate wanted to started a business. He estimated cost and revenue of the business. Suppose his cost ( C ) and Revenue( R ) functions are given as C= 100 + 5 Q2 and R = 150 Q – 2.5 Q2 . find:

a) Profit maximizing output and price/output combinationb) Maximum profitc) Optimal price/output combination if marginal cost of production is 10Q.

ANSWER 4


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Given in the question, Cost function; C = 100 + 5 Q2

Revenue function; R = 150Q –2.5 Q2

We know that, Revenue (R) = Price (P) × Quantity (Q)

P × Q = Q (150 – 2.5 Q)Price function; P = 150 – 2.5QProfit (П) = TR – TC

П = (150Q –2.5 Q2)- (100 + 5 Q2)Profit function; = –7.5Q2+150Q – 100 We have,

Marginal Revenue (MR) = = 150– 5Q Marginal Cost (MC) = = =0+10Q = 10QAccording to profit maximization principle, Marginal revenue must be equal to Marginal cost.Therefore, MC = MR

10Q = 150 – 5QOptimal quantity (Q) = 10 units

Condition 1:At profit maximizing level of output; Marginal profit or 1st order derivative of profit must be equal with zero, Taking 1st order derivative of profit,

Marginal Profit (MP) = = = –15Q+150Substituting the value of Q (= 10) then we get,

= –15 × 10 + 150

= 0

Condition 2:Again, at profit maximizing level of output; 2nd order of derivative of profit must be less than zero (i.e. negative), Taking 2nd order derivative of profit,

< 0


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Or, = <0Or, = < 0Or, = –15< 0Thus, the first derivative or marginal profit and 2nd order derivatives of the firm are found zero and < 0 i.e. – 15 respectively. Therefore, the profit maximizing optimal quantity of production is 10 units.

Now, substituting the value of Q (= 10 units) in price function, revenue function and cost function, we get P = 150 – 2.5 ×10Optimal price (P) = Rs. 125Now, Total Revenue (TR) = P × Q = Rs. 125 ×10 = Rs. 1250 Total Cost (TC) = 100 + 5 × 10 2 =Rs.600

Now, Maximum profit = TR –TC Maximum profit = Rs.1250 –Rs. 600 = Rs. 650Conclusion: Therefore, the maximum profit that can be earned is Rs. 650 by producing 10 units of output.

PROBLEM 5 With the launching of M.Phil Program of Faculty of Management, a term of graduate students wanted to start a Book Stall close to M.Phil building. There are only 25 students in the M.Phil Program. The estimated demand of book at different levels of price is given below:

Books (in No.)

5 10 15 17 18 20

Price (Rs.) 1500 1200 1000 800 500 400The cost function is assumed to be TC= C = 100 + 5 Q2. Suppose the Book Stall is only selling the books related to Economic Analysis for Business. At what level of


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book sale the Book Stall maximizes its profit and what is the price output combination to the Stall. (Note: Formulas for obtaining the value α(constant) and β (coefficient) are

β =

respectively or it can be solved by using two simultaneous equations as ∑Yi = nα + β ∑Xi and ∑Xi ∑ Yi = α ∑Xi + β (∑Xi)2

What will happen if cost function Changed to TC = 100 + 10 Q2. ANSWER 5

Given in the questionHere,

Books(X) Price(Y) XY X2

5 1,500 7,500 2510 1,200 12,000 10015 1,000 15,000 22517 800 13,600 28918 500 9,000 32420 400 8,000 400

X= 85 Y= 5,400 XY=65,100

X2=1,363

= = 14.17 = = 900 = = = –71.77Again, = – = 900– (-71.77) 14.17 = 1917We have regression equation of

Y = α +βXPrice function; P = 1917 – 71.77QRevenue Function; R = P × Q = 1917 Q –71.77Q2

Marginal revenue, MR = = 1917– 143.54Q


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Cost function; C = 100 + 5Q2

Marginal cost, MC = = 0+10QП = TR –TC = (1917 Q –71.77 Q2) – (100 + 5Q2) = 1917 Q – 76.77 Q2 – 100

According to profit maximization principle, Marginal revenue must be equal to Marginal cost.Therefore, MC = MR

10 Q = 1917 – 143.54 QOptimal quantities (Q) = 12.49 units

= 13 unit (approximately )Testing of reliability of above calculation:Condition 1At profit maximizing level of output; Marginal profit or 1st order derivative of profit must be equal with zero, Taking 1st order derivative of profit, Marginal Profit = = =1917 – 153.55QSubstituting the value of Q = 12.48 then we get,

= 1917 – 153.55 × 12.48 = 0

Condition 2:Again, at profit maximizing level of output; 2nd order of derivative of profit must be less than zero (i.e. negative), Taking 2nd order derivative of profit, Or, = – Or, = –143.55 – 10 Or, = –153.55 < 0Thus, the first order derivative or marginal profit and 2nd order derivatives of the firm are found zero and < 0 (i.e. – 153) respectively. Hence, the profit maximizing optimal quantity of production is 13 units.Now substituting the value of Q (=12.48) in price, revenue and costs and profit function, we getP = 1917 – 71.77 × 10Optimal price (P) = Rs. 1200Total Revenue (TR) = P × Q = 12.48 × 1200 =Rs. 14976Total cost (TC) = 100 + 5Q2 = 100 + 5 × (12.48) 2 = Rs. 878.75


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Total profit (П) = TR –TC = Rs. 14976 – Rs. 878.75 = Rs. 14097.25 Conclusion: Here profit can be maximized up to Rs. 14097.25 by producing 13 units of output.

PROBLEM 6Let the demand facing a firm for its product be expressed by the following functions Q = 25 - 0.5 P (Where Q = quantity and P = Price), and cost function as C = 25 – 2 Q + 4 Q2. Compute (a) profit maximizing output, (b) Justify profit maximizing output.

ANSWER 6Given in the question, Demand function; Q = 25–0.5 POr, Demand function; P = 50 – 2QCost function; C = 25 – 2Q + 4Q2

Revenue function; R = P × Q = (50 – 2Q) × Q = 50Q – 2Q2

Again, Marginal Revenue (MR) = = = 50 – 4QAgain, Marginal Cost (MC) = = = – 2 + 8QProfit (П) = TR – TC Or, П = 50Q –2Q2 –(25 – 2Q +4Q2)Profit, П = 52Q –6Q2–25


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According to profit maximization principle, Marginal revenue must be equal to Marginal cost.Therefore, MC = MROr, –2 +8Q = 50 –4Q Q = unit = 4.33 units = 5 units (aprox)Testing of reliability of above calculation:Condition 1:At profit maximizing level of output; Marginal profit or 1st order derivative of profit must be equal with zero, Taking 1st order derivative of profit,

MP = = = 52 – 12 Q

If Q = then,

= 52 – 12 × = 0

Condition 2:At profit maximizing level of output; 2nd order of derivative of profit must be less than zero, Taking 2nd order derivative of profit, Or, = – Or, = – Or, = –4 – 8 Or, = –12 < 0Conclusion: The first order derivative or marginal profit and 2nd order derivatives of profit are found zero and < 0 i.e. – 12 respectively. Hence, the profit maximizing optimal quantity of production is 13/3 units.

Now substituting the value of Q (=13/3) in Profit function, we getOptimal profit, П = 52Q – 6Q2 – 25Or, П = 52 × – 6 × × –25П = Rs. 87.66Under the given condition, optimal profit is Rs. 87.66.


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PROBLEM 7A manufacturing company is operating in Kathmandu valley with the demand function given as P = 40 – Q, and total cost function as C = Q2 + 8 Q + 2. If the company wanted to maximize profit what is the price combination and the total profit and revenue? The management of the company realizes the need for capturing market. Therefore, it started to promote its product with the strategy of sales revenue maximization instead of profit maximization. What will be the price output combination and total profit under the condition of sales revenue maximization? The shareholders of the company did not like market share capture strategy (sales revenue maximization) followed by the management. The shareholders shoed strong dissatisfaction against the management in its Annual General Meeting (AGM). They argued that management should not be given opportunities for free play in the company. The shareholder’ meeting consensually decided to put restriction with minimum profit of Rs. 10. Under this condition, what is the optimum price (P) output (Q) combination and total revenue?

ANSWER 7Given in the question,Demand function; P = 40 – Q


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Cost function; C = Q2 +8Q +2Revenue Function; R = P × Q = (40 – Q ) × Q = 40Q – Q2

We have, Marginal Revenue (MR) = = = 40 – 2QMarginal Cost (MC) = = = 2Q + 8

П = TR – TC Or, П = 40Q – Q2 – (Q2 + 8Q + 2) П = 32 Q – 2Q2 – 8Case 01: Profit maximizationAccording to profit maximization principle, Marginal revenue must be equal to Marginal cost.Therefore, MC = MR 40 – 2Q = 2Q + 8

Q = 8 unitsCondition 1: At profit maximizing level of output; Marginal profit or 1st order derivative of profit must be equal to zero, Taking 1st order derivative of profit,

Marginal Profit (MP) = = =32 – 4Q

If Q = 8 unit, then,

= 32 – 4 × 8= 0

Condition 2:At profit maximizing level of output; 2nd order of derivative of profit must be less than zero (i.e. negative), Taking 2nd order derivative of profit, = – = – = –2 – 8 = –10 < 0Conclusion: The first order derivative or marginal profit and 2nd order derivatives of the profit are found zero and < 0 i.e. – 10 respectively. Hence, the profit maximizing optimal quantity


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of production is 8 units. Now, substituting the value of Q (= 8 units) in Price Function, Revenue Function, Cost function and Profit Function, we get P = 40–8 Optimal price (P) = Rs. 32Total Revenue (TR) = P × Q = Rs. 32 × 8 = Rs. 256 Total Cost (TC) = Q2 + 8Q + 2TC = 82 +8x8 +2Total Cost = Rs. 130And, Optimal profit = TR –TC Optimal profit = Rs.256 –Rs. 130 = Rs. 126.00

Case 02: Sales Revenue maximizationAccording principle of sales revenue maximization; Marginal Revenue (MR) = 0MR = 40 – 2Q = 0Output (Q) = 20 unitsNow, Price (P) = 40 – QOr, P = 40 –20 = 20Price at sales maximizing output (P) = Rs.20Total revenue (TR) = P × QTR = 20 × 20TR = Rs.400

Total Cost (TC) = Q2 + 8Q + 2TC = 202 + 8 × 20 + 2TC = Rs. 62

Total Profit (П) = TR – TCП = 400 – 562 = – Rs. 162

Case 03: Sales Revenue Maximization with Constraint of Profit Amount of Rs. 10Profit (П) = TR – TC = 10 П = 32Q –2Q2–8=10Or, -2Q2 +32 Q -18 = 0


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Using quadric equation,

a = -2, b = 32 and c = -18

or, Q = or, Q = or, Q = Taking positive value, or, Q = or, Q = 0.58 unitsTaking negative value Q = Q = 15.42 units = 16 unitsConclusion: Sales maximizing output with profit constraint of Rs. 10 is 15.42 unitsNow, Substitute the value of Q (= 15.42) in price equation to calculate price,Price (P) = 40 – Q = 40 –15.42 = Rs.24.58Total revenue (TR) = P × QTR = 24.58 × 15.42TR = Rs.379.02

Total Cost (TC) = Q2 + 8Q + 2TC = (15.42) 2 + 8 × 15.42 + 2TC = Rs. 363.14

Total Profit () = TR – TCП = 379.02 –363.14 =Rs. 15.88


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PROBLEM 8A company developed be a group of M. Phil Graduates is facing with following demand and cost functions: P = 50 - 0.5Q (Demand function) C = 1.5 Q2 +5 Q + 10 (Cost function), where, Q = quantity in thousands of unit, and P = Price in Rupees, C = Total cost. The graduates wanted to set the objective of the company. Therefore, they conducted a brainstorming session. In the session, certain group of graduates put forwarded profit maximization as an objective of the company while other group argued in favour of revenue maximization as objective of the company. Similarly, agroup of graduates wanted to have revenue maximization with a minimum profit of Rs. 20. In this situation, they wanted to use their knowledge gained from the course entiled “Economic Analysis for Business”. What is the process they will use for determining price output combination and what is the price output combination including total profit and revenue based on the above demand and cost functions under different conditions?

ANSWER 8Given in the question, Demand function; P = 50 – 0.50 QCost Function; C = 1.5 Q2 + 5Q + 10Revenue Function; R = P × Q = 50Q – 0.50Q2

Total Profit (П) = TR – TCOr, П = – 2Q2 + 45 Q – 10Minimum Profit = Rs.20Marginal Revenue (MR) = = = 50 – QMarginal Revenue (MC) = = = 3 Q + 5Case 01: Profit maximizationAccording to profit maximization principle, Marginal revenue must be equal to Marginal cost.Therefore, MC = MR

MR– MC = 0


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Or, 50 – Q – (3 Q +5) = 0 Q = 11.25 unitCondition 1: At profit maximizing level of output; Marginal profit or 1st order derivative of profit must be equal to zero, Taking 1st order derivative of profit,

Marginal Profit = = = 45 – 4Q

If Q = 11.25 then,

= 45 – 4 × 11.25 = 0

Condition 2:At profit maximizing level of output; 2nd order of derivative of profit must be less than zero (i.e. negative), Taking 2nd order derivative of profit,

= 45 – 4Q

= = - 4 >0Conclusion: The first order derivative or marginal profit and 2nd order derivatives of the firm are found zero and < 0 i.e. – 4 respectively. Hence, the profit maximizing optimal quantity of production is 11.25 units.

Now, substituting the value of Q (= 11.25 units) in Price Function, Revenue Function Cost function and Profit Function, we get Price (P) = 50 – 0.50 Q = 50 – 0.50 × 11.25 =Rs. 44.37Total revenue (TR) = 50 Q – 0.50 Q2

TR = 50 × 11.25 – 0.50 × (11.25) 2

TR = Rs.562.5 – Rs. 63.28 = Rs. 499.22

Total Cost (TC) = 1.5 Q2 + 5Q + 10TC = 1.5 × (11.25)2 + 5 × 11.25 +10TC = 189.84 + 56.25 + 10 = Rs. 256.09


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Total Profit (П) = TR – TCП = 499.22 – 256.09 = Rs. 243.13Case 02: Sales Revenue MaximizationAccording principle of sales revenue maximization; Marginal Revenue (MR) = 0MR = 50 – Q = 0Output (Q) = 50 unitsNow, Price (P) = P = 50 – 0.50 Q or, P = 50 – 0.5 × 50 = 25Price at sales maximization output (P) = Rs.25

Total revenue (TR) = P × QTR = 25 × 50Total Revenue = Rs.1250

Total Cost (TC) = 1.5 Q2 + 5Q + 10TC = 1.5 × 502 + 5 × 50 + 10TC = Rs. 4010

Total Profit (П) = TR – TCП = 1250 – 4010 = –Rs. 2760Case 03: Sales Revenue Maximization with Constraint of Profit Amount of Rs. 20At minimum level of profit of Rs.20; Total Profit (П) = TR – TC = Rs. 20Or, 50 Q – 0.50 Q2 – (1.5 Q2 + 5 Q + 10) = 20Or, – 2Q2 + 45 Q – 30 = 0Using quadric equation formula,a = -2, b = 45 and c = -30Or, Q = Or, Q = Or, Q = Taking positive sign, Or, Q = Or, Q = 0.6875 unitsTaking negative sign, Nischal Thapa, MPhil-TU

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Or, Q = Or, Q = 21.81 units = 22 units (aprox)Conclusions: The required quantity of output is 21.81 unitsNow, Price (P) = P = 50 – 0.50 Q or, P = 50 – 0.5 × 21.81= 39.09Price at sales maximization output (P) = Rs.39.09Total revenue (TR) = P × QTR = 39.09 × 21.81Total Revenue = Rs.852.63Total Cost (TC) = 1.5 Q2 + 5Q + 10TC = 1.5 × 21.812 +5 × 21.81+ 10TC = Rs. 832.56Total Profit () = TR – TCП = 852.63 – 832.56.00 = Rs. 20.06

From the above computation, the 1st case of profit maximization has an optimum output of 11.25 units and optimal price of Rs.44.37 that generate the total revenue, total cost and profit of Rs. 499.22, Rs. 256.09 and Rs. 243.13 respectively. The 2nd condition of sales revenue maximization has an optimal output of 50 units and optimal price of Rs. 25 that generate the Total Revenue and Total Cost are Rs. 1250 and Rs. 4010 respectively. But there is loss of Rs.2760.The 3rd condition of sales revenue maximization with constraint of profit of Rs.20, the optimum output is 21.81 units and optimal price is Rs. 39.09 that should generate the total revenue and the total cost at the level of output are Rs.852.63 and Rs.832.56 respectively.

As, the profit maximization condition generates highest profit with lowest sales, the best approach is profit maximization approach.

PROBLEM 9


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A group of students started a business with a view to create employment for themselves. Their demand and cost functions of their product are as follows: P = 75 – 1.5 Q (Demand function) C = 3Q2 + 15 Q + 30 (Cost function)Where, Q = quantity in thousand of unit, and P = Price in Rupees, C = Total cost. Find the price, total revenue, total cost and total profit of the group under the objective of:

(a)Profit maximization, (b)Sales revenue maximization, and (c) Sales revenue maximization with a minimum profit of Rs. 30.

ANSWER 9Given in the question,Demand function; P = 75 – 1.50 QCost Function; C = 3 Q2 + 15Q + 30Revenue Function; R = P x Q = 75Q -1.50Q2

Total Profit (П) = TR – TCOr, П = (75Q –1.50Q2) – ( 3 Q2 + 15Q + 30) Or, П = -4.5Q2 +60 Q -30Minimum Profit = Rs 30Marginal Revenue (MR) = = = 75 – 3QMarginal Revenue (MC) = = = 6 Q +15Case 01: Profit maximizationAccording to profit maximization principle, Marginal revenue must be equal to Marginal cost.Therefore, MC = MR

MR– MC = 0Or, 75 – 3Q – (6 Q +15) = 0 Optimal output (Q) = 6.67 unitCondition 1: At profit maximizing level of output; Marginal profit or 1st order derivative of profit must be equal to zero, Taking 1st order derivative of profit,


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Marginal Profit = = = – 9 Q +60

If Q = 6.67 then,

= –9 x 6.67 + 60= 0

Condition 2:At profit maximizing level of output; 2nd order of derivative of profit must be less than zero (i.e. negative), Taking 2nd order derivative of profit,

Or, = – 9 Q +60

Or, = = - 9

Conclusion: The first order derivative or marginal profit and 2nd order derivatives of the profit are found zero and < 0 i.e. – 9 respectively. Hence, the profit maximizing optimal quantity of production is 6.67 units.

Now, substituting the value of Q (= 6.67 units) in Price function, Revenue function and Cost function, we get

Price (P) = 75 – 1.5 × 6.67 = Rs. 65

Total revenue (TR) = 75Q -1.50Q2

TR = 75 × 6.67 – 1.50 × (6.67) 2

TR = Rs. 433.52

Total Cost (TC) = 3 Q2 + 15Q + 30TC = 3 × (6.67) 2 +15 × 6.67 +30TC = 263.52

Total Profit (П) = TR – TCП = 433.52 – 263.52 = 170.00


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From the above computation, at the level of the profit maximization, the firm has an optimum output is 6.67 units and optimal price is Rs.65 that should generate the total revenue, total cost and amount of profit are Rs. 433.52, Rs. 263.52 and Rs. 170.00 respectively.

Case 02: Sales maximizationAccording principle of sales revenue maximization; Marginal Revenue (MR) = 0MR = 75 – 3Q = 0Output (Q) = 25 units

Price (P) = 75 – 1.5 × 25 =37.5

Total revenue (TR) = P × QTR = 37.5 × 25TR = Rs.937.50

Total Cost (TC) = 3 Q2 + 15Q + 30TC = 3 × (25) 2 +15 × 25 +30TC = 2280

Total Profit (П) = TR – TCП = 937.50 – 2280 = – 1342.50

At the level of sales revenue maximization, the firm has an optimal output of 25 units and optimal price of Rs. 37.5 that generate the total revenue and total cost of Rs. 937.50 and Rs. 2280 respectively. But there is loss of Rs.1342.50.

Case 03: Sales maximization with constraint of profit amount of Rs. 30At minimum level of profit of Rs.30; Total Profit (П) = TR – TC = Rs. 30Or, - 4.5Q2 + 60 Q – 30 = 30Or, - 4.5Q2 + 60 Q – 60 = 0Or, 0.3Q2 – 4 Q + 4 = 0Using quadric equation formula,a = 0.3, b = – 4 and c = 4Or, Q = Nischal Thapa, MPhil-TU

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Or, Q = Or, Q = Taking positive sign, Or, Q = Or, Q = 12.25 unitsTaking negative sign, Or, Q = Or, Q = 1.08 units Hence, the required quantity of output is 12.25 units

Price (P) = 75 – 1.5 Q = 75 –1.5 × 12.25 = 56.625 = Rs. 56.63

Total revenue (TR) = P × QTR = 56.625 × 12.25TR = Rs.693.656 =Rs. 693.66

Total Cost (TC) = 3 Q2 + 15Q + 30TC = 3 × (12.25) 2 + 15 × 12.25 +30TC = 663.938 = Rs.663.94

Total Profit (П) = TR – TCП = 693.66 – 663.94 = Rs.29.72

Conclusion: At the level of sales revenue maximization with profit constraint of Rs.20, the firm has the optimum output of 12.25 units and optimal price is Rs. 56.63 that generate the Total Revenue and the Total Cost are Rs.693.66 and Rs.663.94 respectively.

PROBLEM 10The past records of price and sales of shirt at different points of time are as follows:

Price (Rs.) 100 120 130 120 160 150Sales (No.) 40 38 43 45 20 30


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From the above records of prices and sales of shirts, calculate profit maximizing number of shirt and price including total revenue and profit, given the cost function of C = 3Q2 + 15 Q + 30. What will happen in quantity demand when price increased at Rs. 180 and elasticity of demand? Estimate new function relationship and its price output combination? Interpret your results.

ANSWER 10Given in the question,Here,

Sales Nos. (X) Price (Y) XY X2

40 100 4000 160038 120 4560 144443 130 5590 184945 120 5400 202537 160 5920 136943 150 6450 1849

X= 246 Y= 780 XY= 31920 X2= 10132

= = 41= = = 130XY = 31920X2 = 10132 =

= = – 1.3043Again, = – = 130– (-1.3043) 41 = 183.4763 = 183.48Now, Price function; P = 183.48 – 1.3043 QRevenue Function; R = P × Q = 183.48 Q –1.3043 Q2

Again, MR = = 183.48– 2.6086 QCost function; C = 3Q2 +15Q +30


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Again, MC = = 6Q +15П = TR – TC П = (183.48 Q –1.3043Q2) – (3Q2 +15Q +30) П = – 4.3043 Q2 + 168.48Q – 30 According to profit maximization principle, Marginal revenue must be equal to Marginal cost.Therefore, MC = MR6Q +15 = 183.48– 2.6086 Q8.6086 Q = 168.48Optimal quantity (Q) = 19.57 units = 20 units (approx)

Condition 1: At profit maximizing level of output; Marginal profit or 1st order derivative of profit must be equal to zero, Taking 1st order derivative of profit,

Marginal Profit = = = – 8.6086 Q + 168.48

If Q = 19.57 then,

= – 8.6086 x 19.57+168.48 = 0

Condition 2:At profit maximizing level of output; 2nd order of derivative of profit must be less than zero (i.e. negative), Taking 2nd order derivative of profit, 2nd order derivative, < 0

= – 8.6086 Q + 168.48

= = – 8.6086

Thus, the first order derivative or marginal profit and 2nd order derivatives of the firm are found zero and < 0 i.e. – 8.6086 respectively. Hence, the profit maximizing optimal quantity of production is 19.57 units.Now, substituting the value of Q (= 19.57 units) in Price function, Revenue function and Cost function, we get


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Again, P = 183.48 – 1.3043 × 19.57Optimal price (P) = Rs. 157.95Total revenue (TR) = P × QTR = 157.95 × 19.57TR = Rs. 3091.08

Total Cost (TC) = 3Q2 +15Q + 30TC = 3 × (19.57) 2 + 15 × 19.57 + 30TC = Rs. 1472.50

Total Profit (П) = TR – TCП = 3091.08 –1472.50 = Rs.1618.58

Explanation: At the profit maximizing objective, the firm's optimum output is 19.57 units and optimal price is Rs. 157.95 that generates the total revenue, the total cost and optimal profit of Rs.3091.08, Rs.1472.50 and 1618.58 respectively.


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A Case of Wry Island[I] Initial Equilibrium Initial Equilibrium (for 9 workers from Farm A, B and C)Table 1.1Total Physical Product (TPP), Marginal Physical Product (MPP), Marginal Revenue Product (MPR) of

Wry Island Farm

Unit of labour

Farm A Farm B Farm C

TPP Rye

MPP Rye

MRP Rye @ $4

RPP Rye

MPP Rye

MRP Rye @ $4

TPP Rye

MPP Rye

MRP Rye @

$4

0 0 0 $ -0- 0 0 $ -0- 0 0 $ -0-1 400 400 $ 1,600 675 675 $ 2,700 350 350 $ 1,4002 725 325 $ 1,300 1250 575 $ 2,300 675 325 $ 1,3003 1000 275 $ 1,100 1725 475 $ 1,900 975 300 $ 1,2004 1225 225 $ 900 2125 400 $ 1,600 1250 275 $ 1,1005 1425 200 $ 800 2450 325 $ 1,300 1500 250 $ 1,0006 1600 175 $ 700 2725 275 $ 1,100 1725 225 $ 9007 1750 150 $ 600 2950 225 $ 900 1925 200 $ 8008 1875 125 $ 500 3150 200 $ 800 2100 175 $ 7009 1975 100 $ 400 3325 175 $ 700 2250 150 $ 600

10 2050 75 $ 300 3475 150 $ 600 2375 125 $ 500Table 1.2Allocation of Resources and Distribution of Income

Units ComputationRye only @ $4 per bu. 9

workerWages rate $ $ 1,300Farm A:

Workers No. 2Rent $ 725×$4 – 2×$1300 $ 300

Farm B:Workers No. 5Rent $ 2450×$4 –

5×$1300$ 3,300

Farm C:Workers No. 2Rent $ 675×$4 – 2×$1300 $ 100


29

Farm D:Workers No. –Rent @ $ –

Total wages bill (A + B + C + D)

9 × $ 1300 $ 11,700

Total rent $ 300 + $ 3300 + $ 100

$ 3,700

Total Farm income (wages & rent)

$ 15,400

Total value of Farm (P × Q) $ 4 × 3850 $ 15,400Conclusions:Out of 9 labourers, 2 work in Farm ‘A’, 5 work in Farm ‘B’ and 2 work in Farm ‘C’. The equilibrium wages rate of all Farms is $ 1300. The rent earned by Farm ‘A’, Farm ‘B’ and Farm ‘C’ are $ 300, $ 3,300 and $100 respectively being total of $ 3,700.[II] InflationIn case of inflation, the price of rye rises to $40 per bushel. Therefore, at the new equilibrium, MRP (= the wage rate) will be $13000. The number of laborers and rent earned on various farms are as follows:Table 2.1Total Physical Product (TPP), Marginal Physical Product (MPP), Marginal Revenue Product (MPR) of

Wry Island Farm

Unit of labour


TPP Rye

MPP Rye

MRP Rye @ $40

RPP Rye

MPP Rye

MRP Rye @ $40

TPP Rye

MPP Rye

MRP Rye @ $40

1 400 400 $ 16,000 675 675 $ 27,000 350 350 $ 14,0002 725 325 $ 13,000 1250 575 $ 23,000 675 325 $ 13,0003 1000 275 $ 11,000 1725 475 $ 19,000 975 300 $ 12,0004 1225 225 $ 9,000 2125 400 $ 16,000 1250 275 $ 11,0005 1425 200 $ 8,000 2450 325 $

13,0001500 250 $ 10,000

6 1600 175 $ 7,000 2725 275 $ 11,000 1725 225 $ 9,0007 1750 150 $ 6,000 2950 225 $ 9,000 1925 200 $ 8,0008 1875 125 $ 5,000 3150 200 $ 8,000 2100 175 $ 7,0009 1975 100 $ 4,000 3325 175 $ 7,000 2250 150 $ 6,000

10 2050 75 $ 3,000 3475 150 $ 6,000 2375 125 $ 5,000


30

Table 2.2Allocation of Resources and Distribution of Income

Units ComputationsRye only @ $40 per bu. 9


Workers No. 2Rent $ 725×$40 – 2×$1300 $ 3,000

Farm B:Workers No. 5Rent $ 2450×$40 – 5×$1300 $ 33,000

Farm C:Workers No. 2Rent $ $ 1,000

Farm D:Workers No. –Rent $ –


9 × $1300 $ 117,000

Total rent $ 3000 + $ 33000 + $ 1000

$ 37,000


$ 154,000

Total value of Farm (P × Q) 3850 × $ 40 $ 154,000Conclusions: Because of inflation, the price of rye and wage rate increased simultaneously. In the same way,

the marginal revenue to the Farm and wages to labourers also increased. With the rise in prices, the purchasing power of consumers decreases. But the inflation has also

increased the revenue to the farmers and wages to the labourers, their standard of living will be unchanged.

If the prices of the goods bought by Wry Islanders rise eleven fold (or nine fold), the revenue and wage rate also change proportionately. In the same way, the standard of living will be unchanged.

In general inflation, with increase in price level, revenue does not increase proportionately. As such, the living standard will be decreased.

Therefore uneven rise in price affect the wage rate and rent earned on the farms.


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[III] Growth of Factor Inputs (a)In case of adding new labourers in Wry Island, the total number of labourers will increase causing increase in supply of labour. With 20 labourers, the new equilibrium wage rate will be $800.

Table 3.1(a)Total Physical Product (TPP), Marginal Physical Product (MPP), Marginal Revenue Product (MPR) of

Wry Island Farm

Unit of labour


TPP Rye

MPP Rye

MRP Rye @ $4

RPP Rye

MPP Rye

MRP Rye @ $4

TPP Rye

MPP Rye

MRP Rye @

$4

0 0 0 $ -0- 0 0 $ -0- 0 0 $ -0-1 400 400 $ 1,600 675 675 $ 2,700 350 350 $ 1,4002 725 325 $ 1,300 1250 575 $ 2,300 675 325 $ 1,3003 1000 275 $ 1,100 1725 475 $ 1,900 975 300 $ 1,2004 1225 225 $ 900 2125 400 $ 1,600 1250 275 $ 1,1005 1425 200 $ 800 2450 325 $ 1,300 1500 250 $ 1,0006 1600 175 $ 700 2725 275 $ 1,100 1725 225 $ 9007 1750 150 $ 600 2950 225 $ 900 1925 200 $ 8008 1875 125 $ 500 3150 200 $ 800 2100 175 $ 7009 1975 100 $ 400 3325 175 $ 700 2250 150 $ 600

10 2050 75 $ 300 3475 150 $ 600 2375 125 $ 500

Table 3.2(a)Allocation of Resources and Distribution of Income

Units ComputationsRye only @ $4.00 per bu. 20

workerWages rate $ $ 800Farm A:

Workers No. 5Rent $ 1425×$4 – 5×$800 $ 1,700

Farm B:Workers No. 8Rent $ 3150×$4 – 8×$800 $ 6,200

Farm C:Workers No. 7Rent $ 1925×$4 – 7×$800 $ 2,100


32

Farm D:Workers No. –Rent $ $ -0-


20 × $ 800 $ 16,000

Total rent $ 1700 + $ 6200 + $ 2100

$ 10,000


$ 26,000

Total value of Farm (P × Q) 6500 × $ 4 $ 26,000

Conclusions: With increase in supply of labour, the wage rate has decreased to $800. However, the total wage income has increased from $ 11700 to $ 16,000. The total rent income also increased from $ 3700 to $ 10,000, labourers' share of income is 61.54% ($ 16,000/$ 26,000), and total income is $ 26,000.

[III] Growth of Factor Inputs (b)When one of the sailors, named Dalta, turned into gentle wo(men), the total number of laborers decreased to 19. The new equilibrium wage rate of the 19 laborers is $ 1100.

Table 3.1(b)Total Physical Product (TPP), Marginal Physical Product (MPP), Marginal Revenue Product (MPR) of

Wry Island FarmUnit of

labour

Farm A Farm B Farm C Farm D

TPP Rye

MPP Rye

MRP Rye @

$4

RPP Rye

MPP Rye

MRP Rye @ $4

TPP Rye

MPP Rye

MRP Rye @

$4

TPP Rye

MPP Rye

MRP Rye @ $4

0 0 0 $ -0- 0 0 $ -0- 0 0 $ -0- 0 0 $ -0-1 400 400 1,600 675 675 2,700 350 350 1,400 675 675 2,7002 725 325 1,300 1250 575 2,300 675 325 1,300 1250 575 2,3003 1000 275 1,100 1725 475 1,900 975 300 1,200 1725 175 1,9004 1225 225 900 2125 400 1,600 1250 275 1,100 2125 400 1,6005 1425 200 800 2450 325 1,300 1500 250 1,000 2450 325 1,3006 1600 175 700 2725 275 1,100 1725 225 900 2725 275 1,1007 1750 150 600 2950 225 900 1925 200 800 2950 225 9008 1875 125 500 3150 200 800 2100 175 700 3150 200 800


33

9 1975 100 400 3325 175 700 2250 150 600 3325 175 70010 2050 75 300 3475 150 600 2375 125 500 3475 150 600

Table 3.2(b)Allocation of Resources and Distribution of Income

Units ComputationRye only @ $4 per bu. 19


Workers No. 3Rent $ (1000 × $ 4 – 3 × $

1100)$ 700

Farm B:Workers No. 6Rent $ (2725 × $ 4 – 6 × $

1100)$ 4,300

Farm C:Workers No. 4Rent $ (1250 × $ 4 – 4 × $

1100)$ 600

Farm D:Workers No. 6Rent $ (2725 × $ 4 – 6 × $

1100)$ 4,300


19 × $ 1100 $ 20,900

Total rent $ 700 + $ 4300 + $ 600

$ 9,900


$ 30,800


Conclusions:

With 19 laborers, the total value of output increased from $ 26,000 to $30,800, wage rate also increased from $800 to $1,100. Similarly, the total wages also increased from $16,000 to @20,900. However, the total rent decreased from $10,000 to $9,900.

[III] Growth of Factor Inputs (c)


34

According to Malthus and Ricardo’s Theory of Rent, with the population growth, the supply in labor also increases. But the other factor of production – technology and capital stock–remains constant. Though, increase in labour increases production, but it decreases at some level of production. Therefore, the over supply of labour will ultimately decreases the wage rate. Hence, increased production will increase the rent to the landlord but increase of labur will decrease the wage to the labourers.

[IV] Technology Change (a)

Here, new equilibrium wage rate will be set $900

Table 4.1(a)Total Physical Product (TPP), Marginal Physical Product (MPP), Marginal Revenue Product (MPR) of

Wry Island Farm

Unit of

labour

Farm A Farm B Farm C Farm BTPP

Rye

MPP Rye

MRP Rye @

$4

RPP

Rye

MPP Rye

MRP Rye @

$4

RPP

Rye

MPP Rye

MRP Rice @

$1

TPP

Rye

MPP Rye

MRP Rye @

$40 0 0 $ -0- 0 0 $ -0- 0 0 $ -0- 0 0 $ -0-1 400 400 $ 1,600 675 675 $ 2,700 200

02000 $ 2,000 675 675 $ 2,700

2 725 325 $ 1,300 1250

575 $ 2,300 2900

900 $ 900 1250

575 $ 2,300

3 1000

275 $ 1,100 1725

475 $ 1,900 3200

300 $ 300 1725

475 $ 1,900

4 1225

225 $ 900 2125

400 $ 1,600 3425

225 $ 225 2125

400 $ 1,600

5 1425

200 $ 800 2450

325 $ 1,300 3625

200 $ 200 2450

325 $ 1,300

6 1600

175 $ 700 2725

275 $ 1,100 3800

175 $ 175 2725

275 $ 1,100

7 1750

150 $ 600 2950

225 $ 900 3950

150 $ 150 2950

225 $ 900

8 1875

125 $ 500 3150

200 $ 800 4075

125 $ 125 3150

200 $ 800

9 1975

100 $ 400 3325

175 $ 700 4175

100 $ 100 3325

175 $ 700

10 2050

75 $ 300 3475

150 $ 600 4250

75 $ 75 3475

150 $ 600


35

Table 4.3 (a)Allocation of resources and distribution of income

Units ComputationsRye @ $4/bu. And Rice @ $ 1/bu. 20

workerWages rate $ $ 900Farm A:

Workers No. 4Rent $ (1225 × $ 4 – 4 × $

900)$ 1,300


900)$ 5,500

Farm C:Workers No. 2Rent $ (2900 × $ 1 – 2 × $

900)$ 1,100


900)$ 5,500


$ 18,000

Total rent $ 13,400Total Farm income (wages & rent)

$ 31,400

Total value of Farm (P × Q) $ 31,400

[IV] Technology Change (b)When Price of rice is $ 5.50Table 4.1(b)Total Physical Product (TPP), Marginal Physical Product (MPP), Marginal Revenue Product (MPR) of

Wry Island Farm

Unit of

labour


TPP Rye

MPP Rye

MRP Rye @

$4

RPP Rye

MPP Rye

MRP Rye @

$4

RPP Rye

MPP Rye

MRP Rye @ $5.50

TPP Rye

MPP Rye

MRP Rye @

$4


36

0 0 0 $ -0- 0 0 $ -0- 0 0 $ -0- 0 0 $ -0-1 400 400 $

1,600675 675 $ 2,700 2000 2000 $

11000.00675 675 $ 2,700

2 725 325 $ 1,300

1250 575 $ 2,300 2900 900 $ 4950.00

1250 575 $ 2,300

3 1000 275 $ 1,100

1725 475 $ 1,900 3200 300 $ 1650 1725 475 $ 1,900

4 1225 225 $ 900 2125 400 $ 1,600 3425 225 $ 1237.50

2125 400 $ 1,600

5 1425 200 $ 800 2450 325 $ 1,300 3625 200 $ 1100 2450 325 $ 1,3006 1600 175 $ 700 2725 275 $

1,1003800 175 $ 962.50 2725 275 $

1,1007 1750 150 $ 600 2950 225 $ 900 3950 150 $ 825.00 2950 225 $ 9008 1875 125 $ 500 3150 200 $ 800 4075 125 $ 687.50 3150 200 $ 8009 1975 100 $ 400 3325 175 $ 700 4175 100 $ 550.00 3325 175 $ 700

10 2050 75 $ 300 3475 150 $ 600 4250 75 $ 412.50 3475 150 $ 600

Table 4.2 (b) Allocation of resources and distribution of income

Units ComputationsRye @ $4/bu. and Rice $

5.5/bu.Wages rate $ $ 1,100Farm A:

Workers No.Rent $ (1000 × $ 4 – 3 × $

1100)$ 700


1100)$ 4,300

Farm C:Workers No. –Rent $ (3625 × $ 4 – 5 × $

1100)14,438

Farm D:Workers No. 3Rent $ (2725 × $ 5.5 – 6 × $

1100)$ 4,300


37


$ 22,000

Total rent $ 23,738Total Farm income (wages & rent)

$ 45,738

Total value of Farm (P × Q) $ 45,738

Conclusions:

With increase in price of rice to $ 5.50, the total rent earned by producing rice increased to $ 14,438 from $ 1100. This will definitely encourages producing rice instead of rye.

[V] Industrialization

New equilibrium wage = $ 1900Table 5.1Total Physical Product (TPP), Marginal Physical Product (MPP), Marginal Revenue Product (MPR) of

Wry Island Farm

Unit of

labour


TPP Rye

MPP Rye

MRP Rye @

$4

RPP Rye

MPP Rye

MRP Rye @

$4

RPP Rye

MPP Rye

MRP Rye @

$4

TPP Rye

MPP Rye

MRP Rye @

$40 0 0 $ -0- 0 0 $ -0- 0 0 $ -0- 0 0 $ -0-1 400 400 $ 1,600 675 675 $ 2,700 350 350 $ 1,400 675 675 $ 2,7002 725 325 $ 1,300 1250 575 $ 2,300 675 325 $ 1,300 1250 575 $ 2,3003 1000 275 $ 1,100 1725 475 $ 1,900 975 300 $ 1,200 1725 475 $ 1,9004 1225 225 $ 900 2125 400 $ 1,600 1250 275 $ 1,100 2125 400 $ 1,6005 1425 200 $ 800 2450 325 $ 1,300 1500 250 $ 1,000 2450 325 $ 1,3006 1600 175 $ 700 2725 275 $ 1,100 1725 225 $ 900 2725 275 $ 1,1007 1750 150 $ 600 2950 225 $ 900 1925 200 $ 800 2950 225 $ 9008 1875 125 $ 500 3150 200 $ 800 2100 175 $ 700 3150 200 $ 8009 1975 100 $ 400 3325 175 $ 700 2250 150 $ 600 3325 175 $ 700

10 2050 75 $ 300 3475 150 $ 600 2375 125 $ 500 3475 150 $ 600

Table 5.2Allocation of resources and distribution of income

Units Computations @ $4 per bu. Wages rate $ $ 1,900Farm A:

Workers No.


38

Rent $ –Farm B:

Workers No. 3Rent $ (1725 × $ 4 – 3 × $

1900)$ 1,200

Farm C:Workers No. –Rent $ –


1900)$ 1,200


6 × 1900 $ 11,400

Total rent $ 1200 + $ 1200 $ 2,400Total Farm income (wages & rent)

$ 13,800


Conclusions:

When the wage rate will be $ 1900, Farm A and Farm C can not hire workers at that rate. In such a condition, they have to close their production.

☻☻☻


econometrics assignment no 1

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