Temple University
Department of Economics
Answer Key for Printing Sheet Music

Some descriptive statistics

1.a. Simple linear regression
Coefficient estimates

t-statistics for engraving method
and for typeset method

For use in 5.c. we might also look at

Indicating that the intercepts are different for the two methods.
Goodness of fit for engraving method
Goodness of fit for typeset method

2.  Simple regression with the data in log-log form.

Coefficient estimates

Goodness of fit for engraving method
Goodness of fit for typeset method

3. Do the model in semi-log form

Goodness of fit for engraving method
Goodness of fit for typeset method

4. Do it all again for a model quadratic in Q

Goodness of fit for engraving method
Goodness of fit for typeset method

5.a. Which model do you prefer?
My preference is for the simple linear model. The raw data shows little curvature that needs to be accounted for by exotic nonlinear forms.  As a general rule, the parsimonius model is better since the estimators are more efficient in the statistical sense.  This is especially true when there are so few observations.  Finally, the more exotic forms do not result in appreciably better model statistics; the simple linear model is already very good.
5.b. Fixed Costs for the different processes. Technically there are no fixed costs for the Ln-Ln model. When Q is zero total cost is undefined.
Engraved 
Type set
Simple Model             919.7                1070.

Ln-Ln                           102.6                  128.6

Semi-ln                        1317                   1260

Quadratic                     921.6                  891.7
5.c. To do this construct a confidence interval around each intercept estimate and see if, for a given model, they overlap.  In order to preserve unbiasedness you will have to the 2nd and 3rd models in their original form.  We also need to do some bookkeeping initially so that we are picking out the correct variances.

Simple Linear model
Engraving 

Typeset 

Since the confidence intervals do not overlap we can say that the intercepts for the two processes are different.
Ln-Ln Model

Engraving

Typeset 

Again, the intervals do not overlap.  We conclude that fixed costs are different.
Semi log model
Engraving 

Typeset 

This time the intervals overlap and we conclude that fixed costs are not different for the two processes. Perhaps we are not too surprised since the fitted curve is convex and the data is ever so slightly concave.
 The quadratic form model
Engraved method

Typeset method

For the quadratic specification there is no apparent difference in fixed costs.
5.d. Test the joint hypothesis that fixed costs are 900 pfennige and that marginal costs are 5 pfennige for the engraving method.  Test this hypotheses for the linear and quadratic models.
The Linear model

Do not reject the null.
The quadratic model

Do not reject the null.
5.e. Setup costs for typeset process are 750.  This is a t-test for both the simple and quadratic specifications.

For the simple linear model we reject the null.

The observed t-statistic cuts off 5% in the upper tail.  For such a small sample a 10% level of test is probably appropriate, therefore reject the null.
5.f. In order for the MC to change over the output range Scherer must be referring to the quadratic specification.  We will do this as an F-test.

Although some simple DTC/DQ arithmetic suggests that Scherer is correct, the statistical model tells us something else.
6. a. Total Cost was done in the previous parts.
    b. Derive Marginal Cost for each process for each model
                                            Engraved                                         Type set

Simple Linear                    4.9                                                            2.5

Ln-Ln                                   2.64/Q.429                                               2.29/Q.527
Semi Ln

4.96 + .000022Q
3.396 - .00162Q
Quad