   Chapter 14.3, Problem 30E Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

Solutions

Chapter
Section Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
Textbook Problem

In Problems 27-30, the demand functions for q A and q B units of two related products, A and B, are given. Complete parts (a)-(e) for each problem. Assume that p A and p B are in dollars.(a) Find the marginal demand of q A with respect to p A (b) Find the marginal demand of q A with respect to p B (c) Find the marginal demand of q B with respect to p B (d) Find the marginal demand of q B with respect to p A (e) Are the two goods competitive or complementary? 30. { q A = 2500 + 600 p A + 2 40 p B q B = 300 − 100 p A + 400 p B + 5

(a)

To determine

To calculate: The marginal demand of qA with respect to pA. The demand functions for qA and qB units of two related products, A and B, are given by qA=2500+600pA+240pB and qB=3000100pA+400pB+5. Assume that pA and pB are in dollars.

Explanation

Given Information:

The demand functions for qA and qB units of two related products, A and B, are given by qA=2500+600pA+240pB and qB=3000100pA+400pB+5.

Formula used:

For a demand function, of the form q=f(p1,p2), the marginal demand function is the partial derivative of the function q. Thus, the marginal demand of q with respect to the price p1 is given by qp1 and the marginal demand of q with respect to the price p2 is given by qp2.

For a function f(x,y), the partial derivative of f with respect to x is calculated by taking the derivative of f(x,y) with respect to x and keeping the other variable y constant. The partial derivative of f with respect to x is denoted by fx.

Power of x rule for a real number n is such that, if f(x)=xn then f(x)=nxn1.

Quotient rule for function f(x)=u(x)v(x), where u and v are differentiable functions of x, then f(x)=v(x)u(x)u(x)v(x)(v(x))2.

Constant function rule for a constant c is such that, if f(x)=c then f(x)=0

(b)

To determine

To calculate: The marginal demand of qA with respect to pB. The demand functions for qA and qB units of two related products, A and B, are given by qA=2500+600pA+240pB and qB=3000100pA+400pB+5. Assume that pA and pB are in dollars.

(c)

To determine

To calculate: The marginal demand of qB with respect to pB. The demand functions for qA and qB units of two related products, A and B, are given by qA=2500+600pA+240pB and qB=3000100pA+400pB+5. Assume that pA and pB are in dollars.

(d)

To determine

To calculate: The marginal demand of qB with respect to pA. The demand functions for qA and qB units of two related products, A and B, are given by qA=2500+600pA+240pB and qB=3000100pA+400pB+5. Assume that pA and pB are in dollars.

(e)

To determine

Whether the two goods, A and B, are competitive or complementary. The demand functions for qA and qB units of two related products, A and B, are given by qA=2500+600pA+240pB and qB=3000100pA+400pB+5. Assume that pA and pB are in dollars.

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