Midterm Exam - Answer Key

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1 Pofesso Salye, Economics 37, Fall 00 Midtem Exam - Answe Key Diections: Answe all questions; the questions ae weighted equally Fo full cedit, you must povide complete explanations fo you answes Geece is cuently expeiencing a sevee public nance cisis In paticula, some analysts ague that the combination of high inteest ates on Geek debt combined with a low gowth ate of GDP fo Geece imply that the path of Geek debt is not sustainable Using the govenment budget constaint (and abstacting fom money and in ation) povide an agument suppoting this foecast ANSWER: To analyze the sustainability of Geece s debt we use the govenment s ow budget constaint: P t G t + R t P t T t + ( ) + (M t M t ), () whee P t is the pice level at time t, G t is govenment spending (in eal tems) at time t, epesent the dolla amount in outstanding govenment bonds at time t, T t denotes the tax evenue (in eal tems) at time t, and R t is the nominal inteest ate on teasuy bonds issued in peiod t We will ignoe in ation and money and can theefoe set P t and M t M t fo all time peiods, t 0; ; ; ::: With these simplifying assumptions equation () educes to G t + t T t + ( ) (G t T t ) + ( t ) D t + ( t ) () whee D t denotes Geece s pimay de cit and t R t t R t is the eal inteest ate on govenment debt issued in peiod t (Remembe, we ae abstacting fom in ation and hence t 0) In ode to elate Geece s path of debt to its economic gowth we divide both sides of equation () by Geece s GDP, D t + ( t ) D t + ( t ) D t t Bt +, (3) + n t whee n t Yt denotes Geece s GDP gowth ate Assuming that the pimay de cit as a faction of GDP, d t Dt, the inteest ate, as well as the ate of GDP gowth stay constant, then the path of Geece s debt to GDP atio, b t Bt, evolves accoding to the di eence equation b t d + + n b t (4) To see how the inteest ate,, and GDP gowth, n, detemine Geece s path of debt let s assume that its cuent debt to GDP atio was 00% of GDP, ie b 0 Futhe, let s assume that Geece manages to keep a zeo pimay de cit fom now on, ie d 0 Fom equation (4) it follows that the path of Geece s debt is b 0 b + n b + n b t t + n (5)

2 Hence, Geece s debt as a faction of GDP in the vey distant futue is given by the limit lim b t t! 0 if < n, and if > n (6) In othe wods, if inteest ates ae highe than GDP gowth ates, then even if Geece issues no futhe debt the inteest payments on the cuent 00% of GDP debt buden keep accumulating faste than the Geek economy gows This is why some analysts believe that Geece s cuent debt is not sustainable given the pojections of slow GDP gowth not even if Geece wee to maintain a zeo pimay de cit In his model of optimal taxes without a commitment mechanism, S Fische solved the economy backwads in time That is, he st solved fo optimal choices in the last peiod of the economy and then solved fo optimal choices in the st peiod Fom this pespective, answe the following: (a) In the second peiod of the economy, what detemined the optimal tax ates on capital and labo Explain (b) The time consistent equilibium was chaacteized by the intesection of two functions that summaized the backwads solution pocedue Explain what these functions descibe and why thei intesection implies a time-consistent equilibium (It is not necessay to wite down the functions - simply give a vebal desciption of the elationships epesented by the two functions) ANSWER: (a) The optimal tax ate in the second peiod is no tax on labo The eason is that capital is inelastically supplied in the second peiod so thee is no deadweight welfae loss to taxing capital (b) The two functions ae: () the optimal tax ate on capital (chosen by the govenment in peiod ) and () the desied amount of savings (ie capital) chosen by households in peiod The function in () is detemined by the govenment taking as exogenous the capital stock chosen by households in the st peiod of the economy That is k f (k) The function in () is detemined by households taking as given the tax ate they expect in peiod (in paticula, that the tax ate on labo will be zeo) and choosing capital That is, k g ( e k ) A time consistent solution equies the tax ate to be consistent with these two functions That is, k f (g ( k )) : 3 Assume that a household lives fo two peiods and has pefeences given by U (c 0 ; c ) c0 c whee c 0 denotes st peiod consumption, c is second peiod consumption, and 0 < < epesents the time discount facto The household eceives no income in the st peiod and an income of in the second peiod Consumption in the st peiod must be nanced by boowing - the inteest ate is () : Assume that () (a) Set up the household s maximization poblem as a Lagangian and show that the intetempoal maginal ate of substitution is equal to () Intepet this esult (b) Deive the household s demand function fo c How esponsive is c to a change in? esult (c) Solve fo the household s indiect utility function Intepet you ANSWER: Since the household doesn t eceive any income in peiod 0 the intetempoal budget constaint is given by c 0 + c and we can epesent the maximization poblem as the following Lagangian max L(c 0; c ; ) c0 c Y + c 0;c ; The st ode conditions ae then c 0 c (7) 0 ; c ; ) c0 0, 0 0 ; c ; ) 0 0 ; c ; ) c c 0 ()

3 Combining equations (9) and (0) yields c c 0 c0 c 0 () The left hand side of equation () is the intetempoal maginal ate of substitution (IMRS) since 0;c 0;c c 0 c (3) Theefoe, equation () implies that, at an optimum, the IMRS is equal to This means that, at an optimum, the ate at which the consumes ae willing to shift consumption fom tomoow to today has to exactly equal the ate at which they can boow that additional unit of consumption Gaphically, this means that, at the optimal consumption po le (c 0; c ), the consume s indi eence cuve is exactly tangent to the budget line (b) Fom equation () and the assumption that () it follows that the optimal consumption po le is at and households choose to pefectly smooth thei consumption: c0 c () c0 c c 0 c c (4) Consequently, the budget constaint (7) o, equivalently, the FOC with espect to the Lagange multiplie () implies that c c + + c + c c + This is the annuity value of the consume s lifetime income It is the constant payment whose pesent value is equal to the pesent value of the household s lifetime income To bette see this we can also ewite equation (5) using the same notation as in class and in chapte 4 of Doepke, Lehnet and Sellgen s text c 0 + Y Y + + (5) W (6) Also, since the household eceives all he income in the second peiod of life she can only achieve he goal of pefect consumption smoothing by boowing at the inteest ate If this inteest ate ises, and hence boowing becomes moe expensive, the optimal amount of consumption in both peiods will fall Fomally, we can easily show that dc d ( + ) < 0 (7) (c) The household s indiect utility function is the utility function evaluated at the optimal consumption choices: V ( ; ) c 0 c + + ( + )( + ) (8) 3

4 A quick inspection of the indiect utility funciton tells us that highe inteest ates imply lowe utility and highe income implies highe utility Fomally, we have ( ( ; ( + ) < 0, and ( + )( + ) Y Again, this is because the household can only smooth consumption by boowing at the inteest ate The highe this ate, the hade it is fo the household to achieve he consumption smoothing objective Obvioulsy, moe lifetime income will allow he to consume moe in eithe peiod and hence enjoy moe lifetime utility 4 The govenment of Finlandia is faced with the Ramsey poblem of choosing a sequence of excise taxes so that the utility of its citizens is maximized It uses this evenue to pay fo a wa in the st peiod of the economy The govenment knows that households optimal consumption is given by: c t P t0 > 0 (+) t + t (9) denotes income and it is assumed that 0 in evey peiod The govenment has expenditues of G 0 in the vey st peiod (this is the wa expenditues) and nothing theeafte; that is, G t 0 fo t Given this envionment, answe the following (a) Find the optimal excise tax, (Hint: Use the conditions that the tax smoothing hypothesis implies taxes ae constant, t, and the govenment s intetempoal budget constaint implies that the pesent discounted value of tax evenues (given by c t t in evey peiod) equals the pesent discounted value of govenment expenditues You do not have to solve fo the indiect utility function) (b) Find the implied path of govenment debt Does this satisfy the tansvesality condition? ANSWER: Thee is a typo in this question Consumption should have been given as: c t + P t0 + t (+) t whee the numeato is the annuity value of household s consumption Since they eceive 0 units evey peiod, and we know that consumption and taxes will be constant, we have c ( + ) 0 We also know that the pesent discounted value of tax evenues must equal the pesent discounted value of govenment expenditues: X c () t 0 (0) + t0 Solving fo yields: Denote tax evenues (also constant) as T c + 0: But note fom eq(0) that 5 + : So we have T To nd the path of debt, use the govenment budget constaint: G t + () T In peiod 0 (assuming B 0) we have: B 0 4 +

5 In peiod we have: B 0 + () This implies that the debt stays constant at () : This clealy satis es the tansvesality condition: lim () t lim t!() t+ 0 t! 5