The assignment goes live tommorow at 9am EST. It involves optimizing and solving models. It is 4 questions. 1 or 2 of them will be qualitative surrounding monetary / fiscal policy changes. The other 2 are pretty robust math problems involving simple calculus & algebra.
I can send example problem sets from the course and their solutions for a reference.
Econ 2221 - Problem Set 1 Sergio Ocampo Díaz 1 [20 points] Static Equilibrium Consider an economy with representative consumer whose utility is given by: u (C, Ns) = ( C− (N s)1+φ 1+φ )1−σ 1− σ This utility function is note defined over consumption and leisure, but instead over consumption and labor. The labor supply is given by Ns. The most important char- acteristic of it is that there is no income (or wealth) effect on the labor supply. The consumer gets income from firm profits π and from working wNS. Her la- bor income is taxed proportionately at a rate τ, so that after tax labor income is: (1− τ)wNs. There is also a representative firm that uses only labor to produce (there is no capital in this economy). The production function is zNd, where z is productivity and Nd is labor demand. The government must finance an exogenously given amount of spending G with la- bor taxes. Hint: The marginal utility of consumption and capital are given by: uc (C, Ns) = ( C− (N s)1+φ 1 + φ )−σ uN (C, Ns) = − ( C− (N s)1+φ 1 + φ )−σ (Ns)φ 1. Define an equilibrium for this economy. 1 2. What is the firm’s optimal labor demand? That is Nd as a function of the market wage. 3. What are profits (π) in equilibrium? 4. What is labor (N) in equilibrium? 5. From the government’s budget (that you needed for part a), what is the equi- librium government’s revenue (Laffer curve)? For which value of taxes (τ) is the government revenue maximized? 6. Is the value of taxes increasing or decreasing in φ? Interpret your result. Hint: The larger φ is the less sensitive the labor supply to a change in the (ef- fective) wage. φ measures how much disutility labor gives and determines the elasticity of labor to wages and taxes. Solution: Solution: 1. [4 points] An equilibrium for this economy is a wage (w), consumption and labor supply (C, Ns), dividends and labor demand ( π, Nd ) , and taxes (τ) such that: (a) Given w, τ and π the consumer chooses (C, Ns) to maximize her utility. max c,NS u (c, Ns) s.t.c = (1− τ)wNs + π (b) Given w and z the firm chooses labor demand so as to maximize profits: π = max Nd zNd − wNd 2 (c) Taxes are such that the government balances its budget taking G, w and N as given: G = τwN (d) Wage is such that labor market clears: Ns = Nd = N 2. [4 points] The firm’s optimal demand for labor is 0 if w > z, ∞ if w < z="" and="" if="" w="z" then="" the="" firm="" is="" indifferent="" between="" any="" level="" of="" demand.="" the="" demand="" is="" then="" perfectly="" elastic="" at="" w="z." 3.="" [2="" points]="" equilibrium="" profits="" are="" zero.="" 4.="" [4="" points]="" from="" the="" consumer’s="" foc="" we="" get:="" mrs="(1−" τ)w="" ns="((1−" τ)w)="" 1="" φ="" the="" only="" wage="" that="" guarantees="" the="" market="" to="" clear="" (="" ns="Nd" )="" is="" w="z." then="" equilibrium="" wage="" is:="" n="((1−" τ)="" z)="" 1="" φ="" 5.="" [4="" points]="" the="" equilibrium="" government="" revenue="" is:="" r="τwN" =="" τz="" ((1−="" τ)="" z)="" 1="" φ="τ" (1−="" τ)="" 1="" φ="" z="" 1+φ="" φ="" 3="" to="" find="" the="" value="" of="" τ="" that="" maximizes="" revenue="" take="" the="" foc:="" (1−="" τ)="" 1="" φ="" z="" 1+φ="" φ="" −="" 1="" φ="" τ="" (1−="" τ)="" 1="" φ−1="" z="" 1+φ="" φ="0" (1−="" τ)="" 1="" φ="" z="" 1+φ="" φ="1" φ="" τ="" (1−="" τ)="" 1="" φ−1="" z="" 1+φ="" φ="" (1−="" τ)="1" φ="" τ="" φ="" 1="" +="" φ="τ" the="" maximum="" revenue="" is:="" r="τ" (1−="" τ)="" 1="" φ="" z="" 1+φ="" φ="φ" 1="" +="" φ="" (="" 1="" 1="" +="" φ="" )="" 1="" φ="" z="" 1+φ="" φ="φ" (="" z="" 1="" +="" φ="" )="" 1+φ="" φ="" 6.="" [2="" points]="" the="" taxes="" that="" maximize="" the="" revenue="" are="" increasing="" in="" φ="" (="" ∂="" φ="" 1+φ="" ∂φ="1" (1+φ)2=""> 0 ) . The higher φ the lower is the elasticity of labor to changes in the effective wage, and hence the taxes. Then when the government increases taxes the tax base is reduced less. Thus increasing revenue. If the consumer is more sensitive to changes in taxes (more elastic) then the increase in taxes is overturned by a decrease in the tax base, due to lower labor supply. Alternatively one can find the revenue-maximizing tax as a function of the elas- ticity of labor supply with respect to the effective wage: ξ ≡ ∂ ln N s ∂ ln (1− τ)w = ∂ 1φ ln ((1− τ)w) ∂ ln (1− τ)w = 1 φ Then we have: τ = φ 1 + φ = 1 1 φ + 1 = 1 ξ + 1 showing that the tax is decreasing in the elasticity of labor supply. 4 2 [20 points] Ricardian Equivalence and Distortionary Taxes Consider an economy with consumers that live for two periods and choose con- sumption, savings, and labor supply. The economy also has a firm that operates for two periods using only labor. There is no capital in this economy, so there is also no investment. The firm has a productivity of z in the first period and of z ′ in the second period. Finally, there is a government that finances exogenously given government expenditure with distortionary labor taxes. The utility of the consumer is given by: U ( c, c ′ , n, n ′ ) = ln ( c− (n) 1+φ 1 + φ ) + β ln c′ − ( n ′ )1+φ 1 + φ This preferences ensure that there is no wealth effect in the labor supply choice. The marginal utilities are: ∂U ( c, c ′ , n, n ′ ) ∂c = 1 c− (n) 1+φ 1+φ ∂U ( c, c ′ , n, n ′ ) ∂n = nφ c− (n) 1+φ 1+φ ∂U ( c, c ′ , n, n ′ ) ∂c′ = β c′ − (n ′) 1+φ 1+φ ∂U ( c, c ′ , n, n ′ ) ∂n′ = β ( n ′ )φ c′ − (n ′) 1+φ 1+φ The production function of the firm is: F (N) = N 1. Find and interpret the optimality condition for current labor supply. Show a 5 diagram with the demand and supply of labor. 2. What is the effect of an increase in current government spending financed with an increase in current labor taxes on the equilibrium wage, labor, consumption and output? 3. What is the effect of an increase in current government spending financed with debt and an increase in future labor taxes on the equilibrium wage, labor, con- sumption and output? 4. Does the Ricardian Equivalence result apply to this economy? Solution 1. [5 points] From the consumer’s FOC we get: MRS = (1− τ)w Ns = ((1− τ)w) 1 φ This defines the labor supply. The optimality condition equates the marginal cost of supplying additional labor with the marginal benefit of doing so. The marginal cost measured by the MRS, which gives how many consumption units must be given to compensate the consumer for the disutility of work- ing an additional hour. The marginal benefit is given by the after tax wage that the consumer receives for an additional hour worked. An alternative specification for the first order condition comes directly from the Lagrangian: −UN = (1− τ)wλ 6 The interpretation is very similar, but now the costs and benefits are in terms of utility. The left hand side (−UN) measures the cost of working in terms of the disutility it generates (an additional unit of labor decreases utility by UN utils). The right hand side ((1− τ)wλ) measures the value for the consumer of the additional income (1− τ)w that is obtained with one more unit of labor. λ measures the value to the consumer (in terms of utility) of an additional unit of income. Labor demand is perfectly elastic at w = z. The firm’s optimal demand for labor is 0 if w > z, ∞ if w < z and if w = z then the firm is indifferent between any level of demand. (a) the only wage that guarantees the market to clear ( ns = nd ) is w = z. then: n = ((1− τ) z) 1 φ 2. [5 points] first note that equilibrium wages do not change in either period. from the equilibrium condition in the current labor market it is clear current labor decreases because the effective wage decreases under higher taxes (this is the substitution effect in action, leisure is now cheaper). future labor does not change because the effective wage (( 1− τ′ ) w ′ ) does not change. cur- rent output decreases because labor goes down (y = f (n) = n). consump- tion has to decrease as well because of both income and substitution effects (it is now more expensive to consume, it requires more hours of work, and the present value of wealth of the consumer has decreased). alternatively one can deduce the reduction of consumption from the goods market clearing condition. note that in this economy there is not investment, 7 nor there are exports or imports, the goods market clearing condition is: y = c + g. we know g is increasing and y is decreasing, so it must be the case that c decreases today. the consumer is compelled to reduce consumption because of the income and substitution effects discussed above. 3. [5 points] as before, wages do not react, but now current labor and output remain unchanged (because the current effective wage does not change), and future labor and output decreasing (because of the substitution effect in the future in favor of leisure). consumption drops as before because the agent is poorer and needs to save more to smooth consumption (alleviate drop in future consumption). from the goods market clearing condition it is also clear that consumption must fall because y is constant and g is increasing. what happens is that the drop in consumption is matched with savings in the form of government debt. the government taxes the consumer in the following period to repay that debt. 4. [5 points] the ricardian equivalence does not apply. the effects of the increase in government expenditure are different depending on the timing of taxes be- cause taxes not only change the agent’s lifetime wealth, but they also distort (change) the relative price of labor. 8 3 [20 points] housing prices and aggregate economic activity consider the macroeconomic model of chapter 11. in addition to everything else consumers are subject to borrowing constraints that come from a problem of limited commitment. consumers own housing (h) that is sold in the second period for a price p. they use housing as collateral of their loans so that the price of housing will affect aggregate demand. 1. pose the consumer’s problem with a borrowing constraint z="" and="" if="" w="z" then="" the="" firm="" is="" indifferent="" between="" any="" level="" of="" demand.="" (a)="" the="" only="" wage="" that="" guarantees="" the="" market="" to="" clear="" (="" ns="Nd" )="" is="" w="z." then:="" n="((1−" τ)="" z)="" 1="" φ="" 2.="" [5="" points]="" first="" note="" that="" equilibrium="" wages="" do="" not="" change="" in="" either="" period.="" from="" the="" equilibrium="" condition="" in="" the="" current="" labor="" market="" it="" is="" clear="" current="" labor="" decreases="" because="" the="" effective="" wage="" decreases="" under="" higher="" taxes="" (this="" is="" the="" substitution="" effect="" in="" action,="" leisure="" is="" now="" cheaper).="" future="" labor="" does="" not="" change="" because="" the="" effective="" wage="" ((="" 1−="" τ′="" )="" w="" ′="" )="" does="" not="" change.="" cur-="" rent="" output="" decreases="" because="" labor="" goes="" down="" (y="F" (n)="N)." consump-="" tion="" has="" to="" decrease="" as="" well="" because="" of="" both="" income="" and="" substitution="" effects="" (it="" is="" now="" more="" expensive="" to="" consume,="" it="" requires="" more="" hours="" of="" work,="" and="" the="" present="" value="" of="" wealth="" of="" the="" consumer="" has="" decreased).="" alternatively="" one="" can="" deduce="" the="" reduction="" of="" consumption="" from="" the="" goods="" market="" clearing="" condition.="" note="" that="" in="" this="" economy="" there="" is="" not="" investment,="" 7="" nor="" there="" are="" exports="" or="" imports,="" the="" goods="" market="" clearing="" condition="" is:="" y="C" +="" g.="" we="" know="" g="" is="" increasing="" and="" y="" is="" decreasing,="" so="" it="" must="" be="" the="" case="" that="" c="" decreases="" today.="" the="" consumer="" is="" compelled="" to="" reduce="" consumption="" because="" of="" the="" income="" and="" substitution="" effects="" discussed="" above.="" 3.="" [5="" points]="" as="" before,="" wages="" do="" not="" react,="" but="" now="" current="" labor="" and="" output="" remain="" unchanged="" (because="" the="" current="" effective="" wage="" does="" not="" change),="" and="" future="" labor="" and="" output="" decreasing="" (because="" of="" the="" substitution="" effect="" in="" the="" future="" in="" favor="" of="" leisure).="" consumption="" drops="" as="" before="" because="" the="" agent="" is="" poorer="" and="" needs="" to="" save="" more="" to="" smooth="" consumption="" (alleviate="" drop="" in="" future="" consumption).="" from="" the="" goods="" market="" clearing="" condition="" it="" is="" also="" clear="" that="" consumption="" must="" fall="" because="" y="" is="" constant="" and="" g="" is="" increasing.="" what="" happens="" is="" that="" the="" drop="" in="" consumption="" is="" matched="" with="" savings="" in="" the="" form="" of="" government="" debt.="" the="" government="" taxes="" the="" consumer="" in="" the="" following="" period="" to="" repay="" that="" debt.="" 4.="" [5="" points]="" the="" ricardian="" equivalence="" does="" not="" apply.="" the="" effects="" of="" the="" increase="" in="" government="" expenditure="" are="" different="" depending="" on="" the="" timing="" of="" taxes="" be-="" cause="" taxes="" not="" only="" change="" the="" agent’s="" lifetime="" wealth,="" but="" they="" also="" distort="" (change)="" the="" relative="" price="" of="" labor.="" 8="" 3="" [20="" points]="" housing="" prices="" and="" aggregate="" economic="" activity="" consider="" the="" macroeconomic="" model="" of="" chapter="" 11.="" in="" addition="" to="" everything="" else="" consumers="" are="" subject="" to="" borrowing="" constraints="" that="" come="" from="" a="" problem="" of="" limited="" commitment.="" consumers="" own="" housing="" (h)="" that="" is="" sold="" in="" the="" second="" period="" for="" a="" price="" p.="" they="" use="" housing="" as="" collateral="" of="" their="" loans="" so="" that="" the="" price="" of="" housing="" will="" affect="" aggregate="" demand.="" 1.="" pose="" the="" consumer’s="" problem="" with="" a="" borrowing="">