Supply and Demand and Marginal Cost

Topics: Supply and demand, Economics, Profit maximization Pages: 12 (2732 words) Published: March 12, 2014
1i) Demand function for air travel between the U.S. and Europe has been estimated to be: ln Q = 2.737 - 1.247 ln P +1.905 ln I
where Q denotes number of passengers (in thousands) per year, P the (average) ticket price and I the U.S. national income. Determine the price elasticity and income elasticity of demand (8 points). From Lecture Module 3 Equation 4 we learned the alternative formulation of elasticity. Alternative formulation of elasticity

EP = dQ/dP * P/Q = dlnQ/dlnP

Natural log: ln, uses the base “e”

How?
∂lnQ/∂lnP =(d lnQ/dQ) * (dQ/dP) * (dP/dlnP)
[ Note: dY/dX = 1/(dX/dY)
since, dlnX/dX = 1/X, dX/dlnX = X]

Example:
Q = AP-α
A:Constant>0
lnQ=lnA + ln(P-α)
=lnA – αlnP
EP = dlnQ/dlnP = -α

∝ =∆lnQ/∆lnP
∝ =P/Q* (∆Q/∆K) = Elasticity
The coefficients of double log model are the corresponding elasticity Price elasticity = -1.247
Income elasticity = 1.905

(1ii) It has been estimated that the price elasticity of demand for U.S. manufactured automobiles is -1.2, while the income elasticity of demand is 2.0 and the cross price elasticity of demand with respect to the foreign automobiles is 1.5. The current volume of sales for U. S. manufactured automobiles is 10 million per year. It is expected that over the next year the average income of the consumers in the U.S. will increase by 2.5 percent. It has been determined that the price of the foreign imports will increase by 6% over the next year. By how much should the U.S. automakers adjust the price of their automobiles if they wish to increase the volume of their sales by 9.2% next year (8 points)? Price elasticity = -1.2

Income elasticity = 2
Cross price elasticity = 1.5
Current volume = 10 million
2.5% Average income increase
We know from Module 3 created by Dr. Ghosh that:
EP = %ΔQx / %ΔPx, where only Px changes
%ΔQx = EP * %ΔPx if only Px changes
Exy = %ΔQx / %ΔPy, where only Py changes
%ΔQx = Exy * %ΔPy, if only Py changes
EI = %ΔQx / %ΔI, where only Income changes
%ΔQx = * EI %ΔI, if only Income changes

Total % Δ in Demand,
%ΔQx = EP * %ΔPx % + Exy * %ΔPy + EI * %ΔI

We want the new Total % Δ in Demand (%ΔQx) = 9.2%. Applying this highlighted formula we can calculate the % change in price for US automobiles to obtain a 9.2% increase in demand: 9.2 = -1.2*%ΔPx(what we want to solve for) + 1.5*6 + 2*2.5

X=4%

(1iii) Bright Future, Ltd (BF) is a nonprofit foundation providing medical treatment to emotionally distressed children. BF has hired you as a business consultant to design an employment policy that would be consistent with its goal of providing the maximum possible service given its limited financial resources. You have determined that the service (Z) provided by BF is a function of its medical staff input (M) and sound service input (S) which is given by: Z = M + .5S + .5 MS - S2

BF’s staff budget for the coming year is $1,200,000. Annual employment costs are $30,000 for each social service staff member (S) and $60,000 for each medical staff member (M).

(1iiia) Using the Lagrangean multiplier approach calculate the optimal (i.e., service maximizing) combination of medical and social staff. Determine the optimal amount of service provided by BF (32 points). Objective function: Z = M + .5S + .5 MS – S2

Constraint: 30000S+60000M = 1200000
Lagrangian:
L = M + .5S + .5 MS - S2+λ[1200000-30000S-60000M]
First we need to calculate the first order conditions from the objective function: dL/dS = 0 implies 0.5+0.5M-2S = 30000λ or 0.5+0.5M-2S-30000λ=0 dL/dM = 0 implies 1+0.5S=60000λ or 1+0.5S-60000λ=0

dL/dλ = 0 implies 30000S+60000M = 1,200,000 or 30000S+60000M-1,200,000=0 Next we need to equate dL/dS = dL/dM
60000*(0.5+0.5M-2S) = 30000*(1+0.5S)
30000+30000*M-1,200,000S = 30000+15000S
M = 4.5S

Substituting this value into the last equation of dL/dλ
30000S+60000*4.5S = 1,200,000
S= 4 = Social Staff

M* = 4.5*4 = 18 = Medical Staff
Z = M + .5S + .5 MS - S2
Z=...
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