Consider an earnings function with the dependent variable ymonthly usual earnings and as independent variables years ofeducation x1, gender x2 coded as 1 if female and 0 if male, andwork experience in years x3. We are interested in the partialeffect of years of education on earnings. We consider the followingpossible relations (that are assumed to be exact) y =β0 + β1x1 +β2x2 + β3x3 (1) y =β0 + β1x1 + β2x2 + β3x3 + β4x 2 1 (2) y =β0 +β1x1 + β2x2 + β3x3 + β4x1x2 (3) We are interested in the partial,i.e. ceteris paribus, effect of x1 on earnings y. (i) Use partialdifferentiation to find the partial effect in the threespecifications above. (ii) For which specifications are the partialeffects constant, i.e. independent of the level of x1, x2, x3? Ifnot constant how does the partial effect change with x1, x2, x3?(iii) If we have data that allow us to estimate the regressioncoefficients, how would you report the partial effects if they arenot constant and you still want to report a single number? (iv) Canyou use partial differentiation to find the partial effect of x2?Why (not)? (v) Often work experience is not directly observed, butmeasured as AGE YEARS OF EDUCATION - 6. Does this change youranswers to (i) and (ii)?

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