Answer Key Practice Final Exam

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1 Answer Key Practice Final Exam E. Gugl Econ400 December, (0 points)consider the consumer choice problem in the two commodity model with xed budget of x: Suppose the government imposes a price of good 1 in the amount of t; so that the new price of good 1 is equal to p t: (a) Suppose that the amount of good 1 that the consumer buys before the implementation of the tax and after the implementation of the tax stays the same. Is good 1 a normal good? Prove your answer. Good 1 is not a normal good. If a good is normal, then its Marshallian demand decreases with an increase in its price; it cannot stay the same. To prove this, we use the Slutsky decomposition of the price e 1 (x; p) 1 (u; 1 (x; p) g 1 (x; p) This equation tells us that the change in Marshallian demand due to an increase in the good s price is decomposed into an own-substitution e ect and a income e ect. is one of the elements of the main diagonal of the substitution matrix and the substitution matrix is negative semide nite, we know 0: Moreover the de nition of a normal good is that its demand increases with increase in x, i.e. > 0: At any g 1 (x; p) > 0; we must thus < > 0: It s also not possible to have no subtitution e ect in combination with no income e ect, because no subtitution as prices increase implies that the old consumption bundle is no longer feasible. = 0 is possible only if have an inferior good, i.e. < 0; = 0, the negative own-substitution e ect must cancel out the positive income e ect. (b) Given that the amount of good 1 that the consumer buys before the implementation of the tax and after the implementation of the tax stays the same, does this imply that there is no DWL of the commodity tax? Prove your answer by means of a well drawn graph. Any tax that creates a substitution e ect creates deadweight loss. Just because people consume the same amount of the taxed good as 1

2 before does not mean that there is no deadweight loss. The reason why the demand of the taxed good is unchanged is that the substitution e ect and the income e ect of the increase in good 1 s price due to the tax cancel each other out. There clearly is a substitution e ect and thus ine ciency. The below diagram illustrates the deadweight loss associated with the situation described in the question. EV is the negative of the vertical distance between the budget line tangent to A and the dashed line tangent to B, TR is the vertical distance between A and C, DWL is the vertical distance between C and D. Good A u 0 C D B BC w/ tax on good 1 BC w/lump sum tax Good 1 u 1 Original choice: A; Taxed choice: C; EV: vertical distance b/w A and D; Revenue from per unit tax: vertical distance b/w A and C; DWL: vertical distance b/w C and D. (0 points) Explain very carefully the role of the expenditure minimization problem in developing restrictions on the Marshallian demand that can be empirically tested. In your answer sketch the path from the expenditure minimization problem to these restrictions on the Marshallian demand. The Marshallian demand g(x; p) is the solution to the consumer s utility maximization problem. The UMP is given by (I focus on the two good

3 case, here) max u (q 1; q ) + (x p 1 q 1 p q ) q 1;q ; It tells us how much of all the goods a consumer will buy given prices and his total outlay x in order to maximize his utility. Based on the utility maximization problem, we can only say that the Marshallian demand is (1) homogeneous of degree zero in prices and x, and () that the Marshallian demand satis es Walras Law (or "adding up"): p 1 g 1 (x; p 1 ; p ) + p g (x; p 1 ; p ) = x: The expenditure minimization problem, although a thought experiment only, allows us to place more restrictions on the observed behaviour of consumers. The idea of the expenditure minimization problem is to minimize expenditure subject to a utility constraint. That is max p 1q 1 + p q + (u u (q 1 ; q )) q 1;q ; It is the dual problem of the utility maximization problem, that is, it is the constrained optimization problem in which the objective function and the constraint of the UMP switch places. The solution to the EMP is the Hicksian demand, h (u; p) :It tells us the expenditure minimizing consumption bundle for given prices p in order to reach utility level u. The value function of the EMP is the expenditure or cost function, c(u; p) and it has the following properties: (1) increasing in u; () concave in p; (3) homogenous of degree 1 in p; (4) non-decreasing in p and increasing in at least one p i ; (5) Shepard s Lemma. Since c(u; p) is concave in prices, its second order derivative matrix with respect to prices (the Hessian) is negative semide nite. By Shepard s Lemma this matrix is also the rst derivative matrix with respect to prices of the Hicksian demand. As such, each element of the matrix tells us how a change in a particular price changes the expenditure minimizing amount of a particular good when keeping the person all the while on the same utility level. The nd element in the rst ; for example, tells us how the Hicksian demand for good 1 changes as the price of good increases. That is, keeping a person s utility the same, how would we have to change the amount of good 1 as the price of good increases in order to expenditure minimize. The elements along the main diagonal are the own substitution e ects. This matrix is therefore also known as the substitution matrix. Another important insight is that elements o the main diagonal with the same indices, e.g. the nd element in the rst row and the nd element in the rst column, must be the same, because the Hessian is always symmetric. Fourth, because the Hicksian demand is homogenous of degree zero in prices, each row evaluated at a given price vector and multiplied with the same price vector is equal to zero (by Euler s Theorem). To sum up, (1) negative semi de niteness (due to concavity of c (u; p)), () By Shepard s Lemma D pp c(u; p) = D p h (u; p) ; (3) 3

4 symmetry (Young s Theorem), (4) Euler s equation (due to h (u; p) being homogenous of degree zero.): D pp c(u; p)p = 0. While all these properties are remarkable, how are they related to the Marshallian demand? Since the expenditure minimization problem and the utility maximization problem are dual to each other, their solution will coincide if we pick the constraints the right way. That is, if we choose total outlay in the budget constraint to be equal to c(u; p), then the utility maximizing consumption bundle is the same as h(u; p) and the utility level reached through utility maximization is equal to the set utility level in the expenditure minimization problem. This fact is exploited by the Slutsky equation that allows us to express the slope of the Hicksian demand in terms of changes in the Marshallian i (u; p) i (x; p) i (x; p) g j (x; p) This means every element of the substitution matrix can be expressed in terms of changes of the Marshallian demand with respect to prices and money income. For example, symmetry leads to the following restriction of the Marshallian i (x; p) i (x; p) g j (x; p) j (x; p) j (x; p) g i (x; j These are restrictions on the Marshallian demand that we would have never guessed without going through the expenditure minimization problem. Since they place more restrictions on how the Marshallian demand can change with a change in prices and money income, they make our theory stronger. If we nd these restrictions to hold in our empirical work, this lends enormous support to the theory of utility maximization. In fact, if all the restrictions hold, we must conclude that people behave as if they are maximizing a utility function subject to a budget constraint. 3. Consider our labour supply model. Suppose a person s preferences over leisure L and a consumption good y are represented by the utility function u (L; y) = ln L + ln y; (a) (10 points) Find the uncompensated labour supply z(w; p; M): There are several ways to do this. The quickest is to point out that we have a Cobb-Douglas utility function and since utility is ordinal, we can rewrite the utility function as Then using full income (F ), u (L; y) = L 1= y 1= : L = F w 4

5 Substituting for F L p; w; M = wt + M w : Uncompensated labour supply is given by z p; w; M = T L p; w; M = T M w : The alternative is the long way through setting up the Lagrange function. I do this below using v (y; z) = (T z)y: $ = (T z)y + M + wz py First order conditions for an @y From these equations we nd = y + w = 0 = T z + p = 0 y = w p (T z) Substituting this into the budget constraint p w p (T z) = wz + M Making z explicit z p; w; M = T M w : (b) (10 points) Write down the Slutsky equation for a change in labour supply as the wage rate increases. What can you say about the magnitude of the substitution e ect and the magnitude of the income e ect if non-wage income M is equal to zero? We can decompose the total e ect of w on labour supply into a substitution and an income e = z Setting M = 0; z w; p; M = wt 0 w = T : This function does not depend on w: Hence an increase in w; leaves the uncompensated labour supply unchanged. Thus = 0:

6 On the other hand, the income e ect = 1 w : = 0; the substitution e ect and income e ect have the same magnitude but are of opposite sign. Moreover, the substitution e ect is z = T 4w : (c) (5 points) Suppose the government imposes a tax on wages. Would such a tax cause a deadweight loss? Proof your answer. You can do this graphically or analytically. In case of a graph, you use the information that there is a subtitution e ect and then you show in a grpah similar to the one in question one that there must be DWL. A tax of wages decreases the amount of money a person receives for each unit of labour while leaving the price of the consumption good unchanged. This lowers the real wage ratio after taxes and, given the utility function from above, leads to a negative substitution e ect. Uncompensated labour supply is una ected by a change in the aftertax wage rate, but this fact is irelevant for determing DWL. What counts is that the compensated labour supply decreases as expressed by the substitution e ect and therefore leads to EV >actual tax revenue. Alternatively, you can calculate EV and TR and then show that DW L = EV T R > 0: For EV, rst nd minimum non-wage function bm (u; p) v M; w; p = wt + M 1 p wp and hence bm (u; w; p) = p wpu The new utility level is given by wt p w (1 t)t v (0; w (1 t) ; p) = p p The non-wage income necessary to reach the new utility level with the old wage rate is bm u 1 ; p p = wt (1 t) 1 = EV This amount is negative and given that M = 0; this amount is equal to EV. Hence the goverment could have taken bm u 1 ; p away at 6

7 the old wage rate and left the person as well of as with the wage tax in place. Actual government revenue is given by T R = twz (p; w (1 t) ; 0) = tw T p t DW L = wt 1 (1 t) t p = wt (1 t) = wt 1 + (1 t) p (1 t)! Note that 1 + (1 t) p (1 t) = 1 p (1 t) Since 1 > p (1 t) as long as 0 < t < 1, DW L > 0: 4. (0 points)suppose the prices in the second period of our model of intertemporal choice increase by an amount i due to in ation. Analyze the impact of in ation on the intertemporal consumption decision by deriving the Slutsky equation for an increase in in ation. (Hint: Which role does in ation play in the intertemporal choice model? Is there an analogue change in the two-commodity model?) Without loss of generality set p 1 = 1: Then p = 1+i: Note that in ation is changing the price of second period consumption. To see more clearly the implications of this price change, consider the change in budget constraint, before deriving the Slutsky equation. c 1 = y 1 + A c = y A The intertemporal budget constraint is thus De ne c r c = y 1 + y W 1 = y 1 + y From above analysis, we see that in ation makes consumption tomorrow more expensive, but has no impact on the intertemporal wealth as measured by W 1 : For that reason, a change in in ation is like a change in the price of one of our goods in the standard two-good model in consumer 7

8 theory. Hence we can use the same technique as in the standard model to derive the Slutsky equation. h t u; = c t W 1 ; Replacing Let W 1 = c u; p = and di erentiation with respect to i we t 1 + r = t By Shepard s Lemma and r = r u; (1+i) 1 1 t c 1 5. (0 points) A rich man is facing the risk of being ruined. He is an expected utility maximizer. His initial wealth is $10 million. The risk of ruin is 1/3. The man has the option to buy any amount of insurance for 80 cents per dollar of insurance and v (y) = ln y: (a) Would the man choose to be fully insured? Prove your answer. The man has initial wealth 10 million but faces the prospect of incurring a loss of 10 million with probability 1 3 :. Insurance for coverage q is available at a premium cost of :8q. Once he buys coverage in the amount of q; his wealth in the good state is 10 :8q and in the bad state (i.e. when loss occurs but insurance kicks in) 0 + q :8q Hence his expected utility maximization problem max q First order condition is given by 3 ln (10 :8q) + 1 ln (0 + q :8q) 3 : :8q + 1 : 3 :q = 0 8

9 Rearranging 1:6 q = (10 :8q) q = 10 :4 = 4:1667 As we can see, the man chooses insurance coverage well below 10 million. (b) Suppose that the man s wealth is $1 million instead of $10 million, and he risks losing 10 million at a probability of 1/3. Does the man s optimal amount of coverage change with this increase in wealth? Prove your answer. The man has initial wealth 1 million but faces the prospect of incurring a loss of 10 million with probability 1 3 :. Insurance for coverage q is available at a premium cost of :8q. Once he buys coverage in the amount of q; his wealth in the good state is 1 :8q and in the bad state (i.e. when loss occurs but insurance kicks in) + q :8q Hence his expected utility maximization problem max q First order condition is given by Rearranging 3 ln (1 :8q) + 1 ln ( + q :8q) 3 :8 3 1 :8q + 1 : 3 + :q = 0 1:6 ( + :q) = : (1 :8q) q = 1: The man cannot take out a negative amount of coverage, so we have a corner solution given by q = 0; the man is better o without insurance than buying any positive amount of coverage at p = :8: 9