Estimate an empirical demand function
1) Running regression analysis on data for 24 cities, Excel Data Analysis output is
Regression Statistics
Multiple R 0.9693
R Square 0.9396
Adjusted R Square 0.9306
Standard Error 188.2038
Observations 24 ANOVA df SS MS F Significance F
Regression 3 11022960 3674320 103.73 2.3E-12
Residual 20 708414 35420.68
Total 23 11731374 Coefficients Standard Error t Stat P-value Lower 95% Upper 95%
Intercept 2308.5 219.9996 10.4933 1.4E-09 1849.618 2767.440
Price(P) -49.06 3.2748 -14.9809 2.46E-12 -55.891 -42.229
Income(M) 0.070380 0.0044 16.0778 6.65E-13 0.061 0.080
Population(N) 0.033636 0.0061 5.4695 2.36E-05 0.021 0.046
As indicated by p-value of coefficients, all of them are significant. Therefore, demand function can be written as
Q = 2308.5 – 49.06*P + 0.07038*M + 0.033636*N
2) Demand function has coefficient of price as -49.06, meaning every increase of $1 in membership price causes demanded quantity to fall by about 49.
Coefficient of average income is 0.07038, meaning a rise of $1000 in average income leads to an increase of about 70 in quantity demanded.
Coefficient of population is 0.033636, meaning for every increase of 1000 in population, demanded quantity increases by about 34.
3) For town D, P = 63, Q = 3263, M = 45000
Point price elasticity of demand = (P/Q) dQ/dP = (63/3263)*(-49.06) = -0.947
Point income elasticity of demand = (M/Q) dQ/dM = (45000/3263)*(0.07038) = 0.971
4) As only costs are the fixed costs, profit is maximized when revenue (=PQ) is maximum. Revenue is maximized when marginal revenue becomes 0. Meaning at, d/dP (PQ) = 0
Or, d/dP (P*(2308.5 – 49.06*P + 0.07038*M + 0.033636*N)) = 0
Or, 2308.5 – 98.12*P + 0.07038*M + 0.033636*N = 0
Or, P = (2308.5 + 0.07038*M + 0.033636*N)/98.12
For town H, M = 41000, N = 28000, giving P = (2308.5+0.07038*41000+0.033636*28000)/98.12 = $62.53
For town W, M = 24000, N= 24000, giving P =
Regression Statistics
Multiple R 0.9693
R Square 0.9396
Adjusted R Square 0.9306
Standard Error 188.2038
Observations 24 ANOVA df SS MS F Significance F
Regression 3 11022960 3674320 103.73 2.3E-12
Residual 20 708414 35420.68
Total 23 11731374 Coefficients Standard Error t Stat P-value Lower 95% Upper 95%
Intercept 2308.5 219.9996 10.4933 1.4E-09 1849.618 2767.440
Price(P) -49.06 3.2748 -14.9809 2.46E-12 -55.891 -42.229
Income(M) 0.070380 0.0044 16.0778 6.65E-13 0.061 0.080
Population(N) 0.033636 0.0061 5.4695 2.36E-05 0.021 0.046
As indicated by p-value of coefficients, all of them are significant. Therefore, demand function can be written as
Q = 2308.5 – 49.06*P + 0.07038*M + 0.033636*N
2) Demand function has coefficient of price as -49.06, meaning every increase of $1 in membership price causes demanded quantity to fall by about 49.
Coefficient of average income is 0.07038, meaning a rise of $1000 in average income leads to an increase of about 70 in quantity demanded.
Coefficient of population is 0.033636, meaning for every increase of 1000 in population, demanded quantity increases by about 34.
3) For town D, P = 63, Q = 3263, M = 45000
Point price elasticity of demand = (P/Q) dQ/dP = (63/3263)*(-49.06) = -0.947
Point income elasticity of demand = (M/Q) dQ/dM = (45000/3263)*(0.07038) = 0.971
4) As only costs are the fixed costs, profit is maximized when revenue (=PQ) is maximum. Revenue is maximized when marginal revenue becomes 0. Meaning at, d/dP (PQ) = 0
Or, d/dP (P*(2308.5 – 49.06*P + 0.07038*M + 0.033636*N)) = 0
Or, 2308.5 – 98.12*P + 0.07038*M + 0.033636*N = 0
Or, P = (2308.5 + 0.07038*M + 0.033636*N)/98.12
For town H, M = 41000, N = 28000, giving P = (2308.5+0.07038*41000+0.033636*28000)/98.12 = $62.53
For town W, M = 24000, N= 24000, giving P =