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Hello, long time no see!Can you help me do these 21 questions. You can only insert the answers next to the equations.You do not need to write the process how you solved it, just answers please like last time.Thank you

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Problem Set 4

Find all first-order partial derivatives ππ₯ , ππ¦ for each of the following:

1. π(π₯, π¦) = 4π₯ 2 + 2π₯π¦ β 8π¦ 2

2. π(π₯, π¦) = 3π₯ 3 π¦ 3 β 2π₯ 2 + 3π¦ + 40

3. π(π₯, π¦) = 6π₯ 2 β 5π¦ 3 + 2π₯ 2 π¦ 3 + 100

Building off of 1-3, find second-order partial derivatives ππ₯π₯ , ππ¦π¦ for each of the following:

4. π(π₯, π¦) = 4π₯ 2 + 2π₯π¦ β 8π¦ 2

5. π(π₯, π¦) = 3π₯ 3 π¦ 3 β 2π₯ 2 + 3π¦ + 40

6. π(π₯, π¦) = 6π₯ 2 β 5π¦ 3 + 2π₯ 2 π¦ 3 + 100

7. Use the product rule to find f x and f y .

a. For π(π₯, π¦) = (3π₯ 2 β 3π¦)(2π₯π¦ + 4π¦ 2 ),

8. What is the general rule to find the values that optimize a two variable function?

9. What is the general rule to show that optimized values are a relative maximum?

10. What is the general rule to show that optimized values are a relative minimum?

11. A retail store estimated the following market response function giving sales (s) as a function

of the number of advertisements in circulars (x) and the number of advertisements in

newspapers (y):

π (π₯) = 420π₯ β 3π₯ 2 β 2π₯π¦ β 4π¦ 2 + 570π¦ + 1725

Find the optimal x and y that will maximize the sales (show that it is indeed a maximum)

12.

Find the profit-maximizing prices and outputs for the multi-product monopolist with the

following demand and cost functions. Also find maximum profit.

ππ₯ = 90 β 2π₯ ππ¦ = 180 β 4π¦

πΆ(π₯, π¦) = 3π₯ 2 + π₯π¦ + 2π¦ 2 + 424

13. Assume that a firm produces sidewalk slabs at a constant average and marginal cost of $3

each (assume no fixed costs). The weekly demand for slabs facing the firm is given by

Q=200 – 20P

Find the monopolistβs optimal price and quantity that would maximize profits.

14. Use Solver to find solutions to the following constrained optimization problems (include

printouts of the Solver models):

πππππππ§π ππΆ(π₯, π¦) = 6π₯ 2 + 14π¦ 2

π π’πβ π‘βππ‘ π₯ + π¦ β₯ 90

15. A retail store estimated the following market response function giving sales (s) as a function

of the number of advertisements in circulars (x) and the number of advertisements in

newspapers (y):

s = 420 x β 2 x 2 β 3xy β 5 y 2 + 640 y + 1725

Find the optimal advertising plan given an advertising budget of $180 per period,

and assuming the cost is $1 per ad for circulars and $4 per ad for newspapers. Also,

use the Lagrange multiplier to estimate the effect on sales of a $1 increase in the

advertising budget.

16. Suppose a firm has a production function (called a βCobb-Douglasβ production function)

given by

π = 2.5πΎ 0.6 πΏ0.4

Assume the budget is $960, the price of capital (K) is ππΎ = $6, and the price of

labor (L) is ππΏ = $2. Determine the amounts labor and capital that maximize

output. Estimate the benefit of having an additional dollar in the budget.

17. A consulting firm for a manufacturing company arrived at the following Cobb-Douglas

production function for a particular product:

π(π₯, π¦) = 40π₯ 0.7 π¦ 0.3

where x is the number of units of labor and y is the number of units of capital required

to produce Q units of the product. Each unit of labor costs $40 and each unit of

capital costs $80. If $400,000 is budgeted for production of the product, determine

how this amount should be allocated to maximize production, and find the maximum

production.

18. A store sells two brands of color print film. The store pays $2 for each roll of brand A film

and $3 for each roll of brand B film. A consulting firm has estimated the following daily

demand equations for these two competitive products:

ππ΄ = 75 β 40ππ΄ + 25ππ΅

ππ΅ = 80 + 15ππ΄ β 30ππ΅

where PA is the selling price for brand A and PB is the selling price for brand B. Use

calculus to determine how the store should price each brand of film to maximize daily

profits. What is the maximum daily profit from these two products? (Show your work.)

(Hint: write an expression that gives profit as a function of the two prices, and maximize

the function.)

19. Use calculus to find the first order conditions. Use solver then to solve the first order

conditions. Show your work for the first order conditions and include the relevant portion of

excel.

π(π₯, π¦) = β22π₯ 2 + 22π₯π¦ β 11π¦ 2 + 110π₯ β 40π¦ β 23

20. Use calculus to find the profit maximizing level of (a) output, (b) price, and (c) profit for a

two-product firm facing the following demand and cost functions (show your work):

π1 = 475 β 6π1

π2 = 450 β 8π_2

ππΆ(π1 , π2 ) = 4π12 + 5π1 π2 + 3π22 + 700

where quantities are expressed on a weekly basis.

21. Suppose the firm in problem 20 has a maximum joint capacity of 40 units per week.

a. Use Solver to find the profit maximizing level of output, price, and profit given this

production constraint.

b. Use the Lagrange multiplier to estimate the effect of expanding capacity by one more

unit per week.

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Tags:

product rule

partial derivatives

critical point

optimized values

market response

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