A management consulting firm produces reports for clients using highly skilled c

Question

A management consulting firm produces reports for clients using highly skilled consultants (H) and moderately skilled analysts (M). Its production function is f(H,M)=H0.5M 0.5. The wage rate for consultants is $360/hour and the wage rate for analysts is $90/hour.

a) What returns to scale property does this production function exhibit? Show this mathematically.

b) Show that the isoquant associated with producing 6 reports is given by H=36/M (or H=36M-1) . Using simple calculus, provide an expression for the marginal rate of technical substitution (the slope of the isoquant).

c) What is the marginal rate of technical substitution when 3 analysts are employed (and 6 reports are produced)? What does this imply about how many consultants could be substituted for 1 analyst and still produce the same number of reports?

d) State the cost minimization problem of the firm mathematically. How many consultants and analysts will the firm employ to produce q reports if its objective is to minimize costs? In other words, solve for H*(q) and M*(q).

e) Derive an expression for the firm’s long-run cost function.

f) The management consulting industry in New York City is very competitive as it easy for experienced executives to start their own boutique consultancies. Given that firms will enter the industry until economic profits are driven to zero, what will the price of a report be in the longrun?

I really need to understand this so if you could be as clear as possible that would be great. Thanks!!

Explanation / Answer

Production function, Q = H0.5M0.5

Total Cost, TC = 360H + 90M

(a)

We can find out the returns to scale by doubling both the inputs such that, new production function becomes:

Q1 = (2H)0.5(2M)0.5 = 2 x H0.5M0.5 = 2Q

So, doubling of both inputs exactly doubles the output, signifying it displays Constant Returns to Scale (CRS).

(b)

Q = 6, therefore:

6 = H0.5M0.5

Squaring both sides,

36 = HM

So,

H = 36 / M

Equivalently, M = 360 / H

Sope of isoquant, MRTS = MPH / MPM

MPH = dQ / dH = 0.5 x (M / H)0.5

MPM = dQ / dM = 0.5 x (H / M)0.5

So, MRTS = 0.5 x (M / H)0.5 / 0.5 x (H / M)0.5 = M / H

(c)

M = 3, Q = 6

From part (b), when Q = 6, H = 36 / M

H = 36 / 3 = 12

This implies that 12 analysis can be substituted by 1 analyst, or (1/12) = 0.0833 consultants can be substituted by 1 analyst.

(d)

Cost minimization problem:

Minimize Total cost, TC = H x PH + M x PM (= 360H + 90M)

Subject to Q = H0.5M0.5

In optimal condition, slope of isoquant (MRTS) equals the slope of total cost line (PH / PM).

Since MRTS = M / H, we have

M / H = 360 / 90 = 1 / 4

H = 4M

Substituting in Q:

Q = (4M)0.5 (M)0.5 = 2 x (M)0.5 (M)0.5 = 2 x M

So, M = Q / 2 [Demand function for M]

H = 4M = 4 x (Q / 2) = 2Q [Demand function for H]

NOTE: Out of 6 sub-questions, the first 4 are answered.

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.