Problem Set #6



Problem Set #8

Suggested Answers to Problems 6.9, 7.1; 7.3, 7.5 and 7.9

6.9 A utility function is termed separable if it can be written as

U(X, Y) = U1(X) + U2(Y)

Where Ui’ > 0 and Ui”< 0, and U1 and U2 need not be the same function.

a. What does separability assume about the cross partial derivative UXY? Give an intuitive discussion of what word this condition means and in what situations it might be plausible.

The cross partial derivative must equal zero. Thus, increases in the quantity of one good available do not affect the marginal utility of another. Such an assumption is plausible when goods are essentially unrelated (neither substitutes nor complements). For example, the marginal utility a consumer may derive from making charitable contributions likely does not affect their marginal utility for a private good, such as an automobile.

b. Show that if utility is separable, neither good can be inferior.

In any equilibrium the optimal consumption bundle will be Ux/Px = Uy/Py. If income increases, consumers must consume more of one of the goods. But given Uii50, Q1 = 0.

Dick’s demand curve is given by Q2 – 160 – 4P for P40, Q2 = 0.

Harry’s demand curve is given by Q3 – 150 – 5P for P30, Q3 = 0.

Using this information, answer the following:

a. How much scrod is demanded by each person at P = 50? At P = 35? At P = 25? At P = 10? At P =0?

|Price |Tom (1) |Dick (2) |Harry (3) |Market |

|50 |0 |0 |0 |0 |

|35 |30 |20 |0 |50 |

|25 |50 |60 |25 |135 |

|10 |80 |120 |100 |300 |

|0 |100 |160 |150 |410 |

b. What is the total market demand for scrod at each fo the prices specified in part (a)?

See the rightmost column in the table above.

c. Graph each individual’s demand curve (n.b. the right most two curves should not have kinks)

[pic]

d. Use the individual demand curves and the results of part (b) to construct the total market demand curve for scrod. Summing horizontally,

[pic]

7.5 For this linear demand, show that the price elasticity of demand at any given point (say point E) is given by minus the ratio of distance X to distance Y in the figure. How might you apply this result to a nonlinear demand curve?

Our task is to show that

[(Q/(P] [(P/(Q] = -X/Y

= P*/(Po-P*)

where Po is the price intercept.

Notice that -Y/Q* is the slope dP/dQ Observing that X is P, we immediately get

( = -(Q*/Y)(X/Q*) = -X/Y

Given a linear demand curve, we may express demand as

Q = a – bP. Thus

← = -bP/Q

= -P/[Q/b]

Now at point E, P is P*.

Solving the demand curve for P yields P = a/b – Q/b. P0 = a/b. Thus

← = dQ/dP[P/Q]

= P/[Q(dP/dQ)]

More generally what does this imply about nonlinear demand curves?

It implies that for any curve at a point E, elasticity may be illustrated as the ratio –X/Y for the ray extending from E with the slope dP/dQ.

7.9 In Example 7.2 we showed that with 2 goods the price elasticity of demand of a compensated demand curve is given by

esX PX = -(1-sx)(

where sx is the share of income spent on good X and ( us the substitution elasticity. Use this result together with the elasticity interpretation of the Slutsky equation to show that:

a. if (=1 (the Cobb-Douglas case),

eX PX + eY PY = -2

b. (>1 implies eX PX + eY PY < -2 and ( -2. These results can easily be generalized to cases of more than two goods.

Both a and b are answered similarly. The Slutsky equation implies for X and Y that

eX PX = esX PX + sx eX I

and

eY PY = esY PY + sY eY I

Adding the two Slutsky equations together

eX PX + eY PY = esX PX + esY PY - sx eX I - sY eY I

Now, by Engel’s law,

sx eX I + sY eY I = 1.

Thus

eX PX + eY PY = esX PX + esY PY - 1

Finally, for either X or Y compensated demand may be written esX PX = -(1-sx)( or esY PY = -(1-sY)(. Inserting

eX PX + eY PY = -(1-sx)( -(1-sY)( - 1

= - ( - 1

(The latter expression since the sum of shares equals 1)

Thus ( = 1 implies the sum of own price elasticities equals -2, and ( ................
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