Bài tập chương 4:
Hàm nhiều biến trong kinh tế (Functions of Two or More Variables and Applications)
1) Tìm tập xác định (domain) các hàm sau: a. 2 2 z = x + y 2 x + y
f. z = x − y d. z = x
b. z = 4x − 3y
g. q = p + 3 p 1 2 3 4x y − x 4x − 3 e. z = c. z =
2x − y
h. q = 5 p − p − p y 1 1 2
2) Tìm giá trị hàm tại các điểm đã chỉ ra. p + 4p f 0,7 . a. tìm q d. ( p , p ) 1 2 ( ) 2 2 , x
f x y = ye + y ( ) = , tìm q 40,35 . 1 1 2 1 ( ) p − p 1 2 ln ( xy) 5 p e. f x, y tìm f 3 − , 4 − . − p ( ) = ( ) 2 2
b. q ( p , p ) 1 2 = , tìm q 50,10 . x + y 1 1 2 1 ( ) p + 3p 1 2
f. z (x, y )= x ln( y)− y ln(x ) tìm z (1,1). c. ( , ) x y z x y xe + = tìm z (3, 3 − ).
3) (Investment) The future value S of an investment earning 6% compounded continuously is a function of the principal
(vốn) P and the length of time t that the principal has been invested. It is given by = ( ) 0.06 , t S f P t = Pe
Find f (2000, 20), and interpret (giải thích ý nghĩa) your answer.
4) (Utility – Hàm thỏa dụng) Suppose that the utility function for two goods X and Y is given by 2 U = xy , and a
consumer purchases (mua) 9 units of X and 6 units of Y.
(a) If the consumer purchases 9 units of Y, how many units of X must be purchased to retain the same level of utility?
(b) If the consumer purchases 81 units of X, how many units of Y must be purchased to retain the same level of utility?
(c) Graph the indifference curve for the utility level found in parts (a) and (b). Use the graph to confirm your answers to parts (a) and (b). 5)
(Production - Hàm sản xuất) Suppose that a company’s production for Q units of its product is given by the 1 3
Cobb-Douglas production function 4 4
Q = 30K L , where K is dollars of capital investment and L is labor hours.
(a) Find Q if K = $10,000 and L = 625 hours. 1
(b) Show that if both K and Lare doubled, then the output is doubled.
(c) If capital investment is held at $10,000, graph Q as a function of L.
6) (Profit) The Kirk Kelly Kandy Company makes two kinds of candy, Kisses and Kreams. The profit, in dollars, for the company is given by P( x y) 2 ,
= 10 x + 6.4 y − 0.001x − 20.025 y
where x is the number of pounds of Kisses sold per week and y is the number of pounds of Kreams. What is the
company’s profit if it sells
(a) 20 pounds of Kisses and 10 pounds of Kreams?
(b) 100 pounds of Kisses and 16 pounds of Kreams?
(c) 10,000 pounds of Kisses and 256 pounds of Kreams?
7) Tìm đạo hàm riêng cấp 1 của các hàm sau: z ∂ z ∂ 2s − 3t ∂Q Q ∂ a. If 4 2 3
z = x − 5x + 6x + 3y −5 y + 7 find and .
f. If Q ( ,st )= and . x find ∂ ∂y 2 2 s + t s ∂ t ∂ x b. If 3 2 2
z = x + 4x y + 6 y g. If 2
z = e + y ln x find z and z .
find z and zy. x x y f ∂ f ∂
h. Find the partial derivative of
c. If f (x y) = (x + y )3 3 2 , 2 find and . x ∂ y ∂ f (x y) 2 ,
= 3x + 4x + 6xy f ∂ f ∂
d. If f (x y ) 2 2 , =
2x − 5y find and . x
with respect to y at x = 2, y = −1. ∂ ∂y C ∂ C ∂
e. If C( x ) 2
, y = 600 − 4 xy+10 x y find and . x ∂ y ∂
8) Tìm các đạo hàm riêng cấp 2 của các hàm sau: a. 2 3
z = x + 4x −5y b. 2 2
z = x y− 4xy c. 2 xy
z = x +e d. 2
z = y − ln xy 2x
e. If f (x y ) 3 4 ,
= x y + 4xy find f (1, 1
− ). f. If f (x, y ) f 1 − , 4 , f 1 − , 4 f 1 − , 4 . xx = yy ( ) 2 2
x + y find xx ( ) and xy ( ) 2 9)
(Cost) Suppose that the cost function (in dollars) of producing a product is C (x y) 2 2 ,
= 25 + 2 x + 3 y , where
x is the cost per pound for material (chi phí mỗi đơn vị nguyên liệu) and y is the cost per hour for labor (chi phí của
mỗi giờ công lao động).
(a) If material costs are held constant, at what rate will the total cost increase for each $1-per-hour increase in labor?
(b) If the labor costs are held constant, at what rate will the total cost increase for each increase of $1 in material cost? 10)
(Profit) Suppose that the profit (in dollars) from the sale of Kisses and Kreams is given by P( x ) 2 2
, y = 10x + 6.4 y − 0.001x − 0.025 y P ∂
where x is the number of pounds of Kisses and y is the number of pounds of Kreams. Find , and give the y ∂
approximate rate of change of profit with respect to the number of pounds of Kreams that are sold if 100 pounds of
Kisses and 16 pounds of Kreams are currently being sold. What does this mean? P ∂ 11)
(Utility) If U = f (x, y) is the utility function for goods X and Y, the marginal utilityof X is and the x ∂ P
marginal utility of Y is ∂ . If 2 2
U = x y , find the marginal utility of (a) X. (b) Y. y ∂ 1 2 12)
(Production) Suppose that the output Q(in units) of a certain company is 3 3
Q = 75K L , where K is the ∂ Q ∂Q
capital expenditures in thousands of dollars and L is the number of labor hours. Find and K ∂ L ∂ when capital
expenditures are $729,000 and the labor hours total 5832. Interpret each answer. 13)
The cost (in dollars) of manufacturing one item is given by C (x, y )= 30+ 3x+ 5y, where x is the cost of 1
hour of labor and y is the cost of 1 pound of material.
(a) If the hourly cost of labor is $20, and the material costs $3 per pound, what is the cost of manufacturing one of these items?
(b) Find and interpret the partial derivative of C with respect to x. 14)
The total cost of producing 1 unit of a product is 3 xy
C( x, y) = 30 + 2x + 4y + 50 dollars
where x is the cost per pound of raw materials and y is the cost per hour of labor.
(a) If labor costs are held constant, at what rate will the total cost increase for each increase of $1 per pound in material cost?
(b) If material costs are held constant, at what rate will the total cost increase for each $1 per hour increase in labor costs? 2 x y 15)
The total cost of producing an item is C ( ,
x y) = 40+ 4x + 6y +100 dollars, where x is the cost per pound
of raw materials and y is the cost per hour for labor. How will an increase of
(a) $1 per pound of raw materials affect the total cost?
(b) $1 per hour in labor costs affect the total cost? 16)
The joint cost (in dollars) for two products is given by C( x y ) 2 ,
= 30+ x + 3y + 2xy , where x represents the
quantity of product X produced and y represents the quantity of product Y produced.
(a) Find and interpret the marginal cost with respect to x if 8 units of product X and 10 units of product Y are produced.
(b) Find and interpret the marginal cost with respect to y if 8 units of product X and 10 units of product Y are produced. 17)
Suppose that the production function for a product is z = 4xy, where x represents the number of
workhours per month and y is the number of available machines. Determine the marginal productivity of (a) x. (b) y. 2 3 18)
Suppose the production function for a product is 5 5
z =60 x y , where x is the capital expenditures and y is
the number of work-hours. Find the marginal productivity of (a) x. (b) y. 3 2 19)
Suppose the Cobb-Douglas production function for a company is given by 5 5
z = 400x y , where x is the
company’s capital investment and y is the size of the labor force (in work-hours).
(a) Find the marginal productivity of x. 4
(b) If the current labor force is 1024 work-hours, substitute y = 1024 in your answer to part (a) and graph the result.
(c) Find the marginal productivity of y.
(d) If the current capital investment is $59,049, substitute x = 59, 049 in your answer to part (c) and graph the result. 20)
(Profit) Suppose that the quarterly profit from the sale of Kisses and Kreams is given by P( x ) 2 2
, y = 10x + 6.4y − 0.001x − 0.025y dollars
where x is the number of pounds of Kisses and y is the number of pounds of Kreams. Selling how many pounds of
Kisses and Kreams will maximize profit? What is the maximum profit? 21)
(Profit) The profit from the sales of two products is given by P( x ) 2 2
, y = 20 x+70 y− x − y dollars
where x is the number of units of product 1 sold and y is the number of units of product 2. Selling how much of
each product will maximize profit? What is the maximum profit? 22)
(Production) Suppose that 2 2 3 3
P = 3.78 x +1.5 y − 0.09 x − 0.01 y tons is the production function for a
product with x units of one input and y units of a second input. Find the values of x and y that will maximize
production. What is the maximum production? 23)
(Profit) Suppose that a manufacturer produces two brands of a product, brand 1 and brand 2. Suppose the
demand for brand 1 is x = 70− p y = 80 − p
1 thousand units and the demand for brand 2 is
2 thousand units, where
p1and p 2 are prices in dollars. If the joint cost function is C = xy , in thousands of dollars, how many of each brand
should be produced to maximize profit? What is the maximum profit? 24)
(Profit) A company manufactures two products, A and B. If x is the number of thousands of units of A and
y is the number of thousands of units of B, then the cost and revenue in thousands of dollars are C( x ) 2 2
, y = 2 x − 2 xy+ y − 7 x−10 y+11, R (x, y) = 5x +8y.
Find the number of each type of product that should be manufactured to maximize profit. What is the maximum profit? 5 25)
Một công ty sản xuất độc quyền hai loại mặt hàng A và B với số lượng và giá bán tương ứng lần lượt là x,
y (sản phẩm) và 580− 5x (ngàn đồng/sản phẩm), 740 − 8y (ngàn đồng/sản phẩm). Cho biết hàm chi phí để sản
xuất sản phẩm là C (x, y ) = 2xy + 4(ngàn đồng). Công ty nên sản xuất bao nhiêu sản phẩm mỗi mặt hàng để lợi
nhuận thu được là lớn nhất. 26)
Giả sử công ty sản suất hai loại sản phẩm có sản lượng , x y với mức giá và hàm chi p = 60, p = 75 1 2 phí là C (x y) 2 2 ,
= x + xy + y .Tìm mức sản lượng ,
x y để công ty đạt lợi nhuận lớn nhất. Tìm lợi nhuận lớn nhất đó. 27)
Công ty A sản xuất hai mặt hàng X và Y. Thông tin cho trong bảng sau Sản lượng Giá bán X x p = 50.000 (đ/kg) 1 Y y p = 80.000 (đ/kg) 2
Hàm chi phí cho hai sản phẩm là C (x y) 2 2 ,
= 5x + 3xy + 4y .
a. Tìm hàm lợi nhuận P (x,y).
b. Công ty A phải chọn mức sản lượng (x,y) như thế nào đạt mức lợi nhuận lớn nhất. Tìm lợi nhuận lớn nhất đó. 28)
A multiplant monopoly operates two plants whose total cost schedules are 3 3
C = 36 + 0, 003Q , C = 45 + 0, 005Q 1 1 2 2
If its total output is sold in a market where the demand schedule is p = 320 − 0,1Q , where Q = Q +Q , how 1 2
much should it produce in each plant to maximize total profits? 29)
Một công ty sản xuất và và bán độc quyền hai loại mặt hàng A và B. Giả sử nếu bán một sản phẩm của mặt
hàng loại A, B lần lượt với giá x, y (đvtt) thì sẽ bán được 120 − 6x+ 4 y sản phẩm loại A và 80 + 4x − 4y sản
phẩm loại B. Biết rằng, chi phí để sản xuất một sản phẩm của mặt hàng loại A, B lần lượt là 40, 50 (đvtt). Xác
định giá bán của mỗi loại mặt hàng để tổng lợi nhuận thu được lớn nhất. Tính lợi nhuận thu được tương ứng. Khi
đó, mỗi mặt hàng sẽ bán được bao nhiêu sản phẩm? 30)
Một công ty sản xuất và và bán độc quyền hai mặt hàng A và B. Giả sử nếu bán một sản phẩm của mặt
hàng loại A, B lần lượt với giá x, y (đvtt) thì sẽ bán được 70− 5x+ 4 y sản phẩm loại A và 80 +6x −7y sản
phẩm loại B. Biết rằng, chi phí để sản xuất một sản phẩm của mặt hàng loại A, B lần lượt là 30, 40 (đvtt). Xác
định giá bán của mỗi mặt hàng để lợi nhuận thu được lớn nhất. Tính lợi nhuận thu được tương ứng. Khi đó, mỗi
mặt hàng sẽ bán được bao nhiêu sản phẩm? 6