Question

A production company has observed that there are changes in revenue every time it tries to manipulate the price. The concerns of management were brought to your attention needing your expertise advice on how price influences quantity and subsequently revenue. The market survey revealed that the quantity demanded of the product follows a Cobb-Douglas function as presented below:

Q"dx"= αP"x^β"+ε^u ........... of the form Q=AL^αK^β (Cobb Douglas production fuction)

Where Qdx is the quantity demanded of commodity X and Px is the price of commodity X. Further, the company observed that from January to October, the Quantity demanded at a given price level was given as in the table below.

Month	Jan	Feb	Mar	April	May	Jun	Jul	Aug	Sep	Oct
Qty	583	580	618	695	724	812	887	991	1186	1940
Price	61	54	50	43	38	36	28	23	19	10

As the only company’s economist, management has asked you help them to be able to predict and forecast the sales

a). Estimate the model and interpret the results (15Mks)

b). calculate the price elasticity of demand for this model (5Mks)

c). How much variations in quantity demanded are accounted for by the variations in the price of the commodity (5Mks)

Answer 1

a.)

The Cobb Douglas model is of the form,
Y=aP^x, where a is constant, P is independent variable (Price) and x is price elasticity
So, in order to find the value we need to convert the multiplicative to additive form using log

log(Y) = log(aP^x)
log(Y) = loga+log(P^x)
log(Y) = a+xlog(P)
Running regression in excel we get

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.9938
R Square	0.9876
Adjusted R Square	0.9861
Standard Error	0.0193
Observations	10
ANOVA
	df	SS	MS	F	Significance F
Regression	1.0000	0.2373	0.2373	637.4880	0.0000
Residual	8.0000	0.0030	0.0004
Total	9.0000	0.2403
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	3.9394	0.0407	96.8249	0.0000	3.8455	4.0332
Price	-0.6738	0.0267	-25.2485	0.0000	-0.7353	-0.6122

b). The price elasticity is the coefficient of Price = -0.67

c) From the above table, R-squared tells about the variations in quantity demanded accounted by variations in the price by 98.76 percent