Okay, lets apply these tools to an actual economics problem. Suppose we want to relate consumer demand for coffee Q_C to four variables: the price of coffee P_C, the price of tea P_T, the price of non-dairy coffee creamer P_N and consumer income I. We can specify and test a linear demand model

1. What signs would you hypothesize for each of the slope coefficients b₁, b₂, b₃ and b₄? Why?
2. A graduate research assistant interviewed 22 households in different regions of the US and obtained data on coffee consumption, local prices of coffee, tea and non-dairy creamer, and household income. Copy these data into 6 columns of an Excel spreadsheet.

   
   obs  Q_C     P_C     P_T     P_N        I
     1   3  $1.04  $1.77  $0.29  $11,300
     2   6  $0.95  $2.15  $0.26  $10,500
     3   5  $0.88  $1.66  $0.23  $14,700
     4   4  $0.95  $1.58  $0.22  $26,600
     5   6  $0.90  $2.07  $0.24  $13,200
     6   4  $1.16  $1.99  $0.28  $80,700
     7   4  $0.95  $1.49  $0.20  $45,600
     8   5  $1.04  $2.35  $0.28  $28,100
     9   8  $0.87  $2.17  $0.28  $18,600
    10   4  $0.97  $1.63  $0.26  $44,400
    11  10  $1.08  $1.80  $0.27  $77,100
    12   6  $0.93  $1.96  $0.28  $50,800
    13   2  $1.15  $1.88  $0.28  $13,400
    14   0  $1.11  $1.48  $0.25  $12,800
    15   3  $1.08  $1.99  $0.20  $32,900
    16   7  $1.00  $2.20  $0.26  $32,500
    17   3  $1.11  $1.52  $0.23  $60,600
    18   3  $1.15  $1.74  $0.20  $13,900
    19   2  $1.11  $1.68  $0.25  $47,500
    20   8  $0.96  $1.88  $0.26  $54,100
    21  10  $0.91  $2.16  $0.26  $61,000
    22   2  $1.16  $1.64  $0.26  $24,900

3. Regress Q_C (the Y-range) against the data in all four columns P_C, P_T, P_N and I (the X-range).

4. Analyze your results.  Which of your regression coefficients are significant at the 95% level.  Which are significant at the 99% level?

5. Is the own-price demand for coffee negatively-sloped?  Is tea really a substitute for coffee?  Is non-dairy creamer really a complement?  Is coffee a normal good?

6. Calculate the own-price elasticity of demand for coffee at the mean value of P_C.
   Calculate the cross-price elasticity of demand for coffee with respect to P_T at the mean value of P_T.
   Calculate the cross-price elasticity of demand for coffee with respect to P_N at the mean value of P_N.
   Calculate the income elasticity of demand for coffee at the mean value of I.

7. Graph actual and predicted consumption of coffee versus P_C.  Why don't the predicted Q's trace out a straight line?