Macroeconomic Theory, Fall 03 SEF, HKU Instructor: Dr. Yulei Luo October 03 Suggested Solutions to Problem Set. 0 points] Consider the following Ramsey-Cass-Koopmans model with fiscal policy. First, we assume that the private sector households-firms is modeled as a representative agent who maximizes its lifetime utility and has access to a technology to produce consumption goods using capital and labor as inputs. The single good in the economy can either be consumed or saved in the form of physical capital. In addition, the government in the economy collects taxes on output and consumption for financing its spending. The government does not print any money or does not issue bond. Therefore, the single-period government budget constraint is g t = τ y t y t + τ c tc t, t 0, where g t is government spending, τ y t is the output tax rate, τ c t is the consumption tax rate, y t is output, and c t is consumption. For simplicity, we assume that τ y t and τ c t are constant over time, i.e., τ y t = τ y and τ c t = τ c. With population growth equal to n, the resource constraint for the representative agent is + n k t+ = τ y t y t + τ c c t + δ k t, where y t = f k t = k α t, α 0,. The objective of the representative agent is to max c t,k t+ } where γ > 0 and subject to and given k 0. β t c γ t γ, 3 a Derive the Euler equation for this RCK model with fiscal policy. Solution: Given the balanced government budget, the resource constraint facing the representative agent can be written as: + n k t+ = τ y t y t + τ c c t + δ k t. 4 The Lagrangian for this problem is } L = β t c γ t + λ t τ y t γ y t + τ c c t + δ k t + n k t+ ].
The FOCs w.r.t. c t and k t+ yields: and the TVC is c γ t = + τ c λ t, 5 + n λ t = β τ y t αkα t+ + δ] λ t+ lim t βt λ t k t+ = 0. 6 Combining the two FOCs yields the following Euler equation: + n c γ t = β τ y t αkα t+ + δ] c γ t+. 7 b Find the steady state values of k and c. Solution: In the steady state, we have k t+ = k t = k and c t+ = c t = c. Using 7, we can pin down: τ y t k = α ] / α. + n /β δ Using the resource constraint, we can pin down: c = ] + τ c τ y t kα n + δ k. 8 c After linearizing the two-difference equation system around the steady state, show that the model economy is saddle-point stable in the neighborhood of the steady state. Solution: We can now linearize the two difference equation system around the steady state: ] ] ] ] 0 kt+ β +τ c +n kt =, 9 Q c t+ 0 c t }}}}}}}} J } } u u M v β +n τ y t α α kα c > 0. The where x t+ = x t+ x x = c, k and Q = γ above system can be rewritten as ] ] ] kt+ β +τ c +n kt = c t+ β Q + Q +τ. 0 c +n c t }}}} b β bi K = 0 = K=J M β Q +τ c +n v b Q +τ c +n = 0
b + Q + τ c + n + ] b + β β = 0 = trace K = b + b = + Q + τ c det K = b b = β > Hence, the discriminant should be positive because + n + β >, = trace K 4 det K = + Q + τ c + n + 4 β β > 0 which means that both roots are real. Also, because det = β > and trace >, the two roots must individually be positive. We can also judge the magnitudes of the two roots as follows: bi K = 0 p b = b b b b = 0 = p = b b = trace K + det K = Q + τ c + n < 0. This can only be true if one root say b is less than and the other root is greater than. We can then conclude and confirm the predictions of the PD that the equilibrium is saddle-point.. 6 points] Consider the following optimal growth model with quadratic utility and full depreciation: v k 0 = subject to the resource constraint max c t,k t+ } β t c t a c t k t+ = Ak t c t, and k 0 is given, Assume that β 0,, A > 0, a > 0, and k t 0, k ] and a < /k so that the utility function is always increasing in consumption. a Formulate this maximization problem as a dynamic programming problem. Solution: The dynamic programming problem can be written as v k = max c a } c} c + βv k, 3 3
where k denotes the next period s capital stock and k = Ak c. 3 can be rewritten as v k = max c a } c} c + βv Ak c. 4 b Solve explicitly for the value function v k and the consumption function c k determining the level of consumption as a function of the level of capital stock.solution: The FOC w.r.t. c for 4 can thus be written as ac βv Ak k c = 0, or 5 ac βv = 0 6 which gives optimal consumption as c = c k. Substituting it into 4: v k = c k a c k + βv Ak c k. 7 The envelop theorem means that k v k = βav, 8 v k = ac A 9 Combining 5 with 8 gives the consumption Euler equation: ac = βa a c, 0 where c denotes the next period s consumption. We now guess the value function takes the following form v k = α 0 + α k + α k, where α 0, α, and α are undetermined coeffi cients. Applying to 9 gives which means that Substituting it into 7 gives α 0 + α k + α k = +β α 0 + α A + α ac A α + α k = 0, c = α k + A α α k + A α k A α 4 a α ] + α k + A α A + α k A α ] } 3,
where we use the fact that Ak c k = A + α k A α. Matching the k, k, and constant terms on both sides of 3, respectively, we obtain: α = a α + βα α = α + α α 0 = A α a Using 4, we can determine α as follows: A + α, 4 A α + β α A + α α A + α ] A α,5 A α A α + β α 0 α + ] A α α.6 α = a α + βα A + α = = α + β A + α = 7 α = a βa. 8 β Note that here the other root for the quadratic equation, 7, is ruled out because it makes the quadratic term in the value function, A + α, be 0. Using 5, we can determine α as a function of α : α = α + α A α + β α A + α α A + α ] A α = α = α α + β A + α + α α α ] = a α = βa + α α + α a βa + α ] 9 ] βa = βa α a = βa β A. + α Similarly, using 4 and 5, we can determine α 0 using the expressions of α and α : α 0 = A α a A α + β α 0 α A α + α ] A α, 5
which can be rewritten as β α 0 = βα A α + A a + βα α = βα + ] A a + βα α A α = Aβ aβ A, which means that Substituting 8 and 9 into c = α k + A α that links k to c: α 0 = Aβ aβ A β. 30 c = βa k Aβ βa Aaβ A. gives the consumption function 3. 5 points] Consider the following AK model with distortionary taxes: max c t,k t+ } where γ > 0 and the resource constraint is β t c γ t γ, 3 k t+ = τ Ak t c t + T t, 3 where τ is the tax rate on capital income levied by the government and T t is a lumpsum transfer from the government. budget: Assume that the government runs balanced T t = τak t. 33 Derive the growth rate of consumption in terms of model parameters when the economy is on the balanced growth path and show that this growth rate is a decreasing function of the tax rate, τ. Solution: Note that the agent takes the transfer from the government T t as given. The Euler equation is c γ t = β τ Ac γ t+, 34 where τ A is the after-tax return return on capital. The growth rate of consumption can thus be written as c t+ c t = β τ A] /γ. 6
It clearly shows that ct+ c t τ < 0. 35 4. 9 points] In the optimal growth models discussed in class, we assume that the agent does not value leisure and labor supply is inelastic and fixed. Here we relax this assumption by allowing the agent to decide how to split his or her time between enjoying leisure and working. Specifically, we assume that the agent need to solve the following optimization problem: max c t,l t,k t+ } β t u c t, h t, subject to k t+ = δ k t + f k t, h t c t, given k 0, where k t is capital, c t is consumption, h t measures the units of labor used to produce goods, h t measures the units of leisure, and the production function and the utility function are defined as f k t, h t = k α t h α t and u c t, h t = ln c t + ln h t, 36 respectively. Here is some positive constant. a Find the first order conditions for c t, l t, and k t+. Solution:Set up the Lagrangian: L = E β t ln c t + ln h t + λ t δ kt + kt α h α ]} ] t c t k t+, where λ t 0 denote the multiplier on the resource constraint,??, at time t. The FOCs w.r.t. c t, l t, and k t+ are 37 = λ t, 38 c t α kt = λ t α, 39 h t h t ] α kt+ λ t = βe t δ + α λ t+. 40 h t+ b Using all the first-order conditions together with the resource constraint to derive the steady state values of c t, l t, and k t+. 7
Solution: Combining two intra-temporal first-order conditions, 38 and 39, gives = α kt α. 4 h t c t y the definition of steady sate, the steady state values of c t, h t, and k t+ must satisfy From 43, we can determine Substituting it into 44, we have h t h = α k α, 4 c h α k = β δ + α, 43 h 0 = δk + k α h α c. 44 ] k /α h = α β + δ. 45 c k = α Combining the expressions for k h and c k gives β + δ δ. ] ] c /α h = α β + δ δ α β + δ 46 Substituting the expressions for k h and c k into 4 gives Using 46 and 47, we can determine h: h h = = h = c h = α ] α/α α β + δ. 47 α β + δ δα + β + δ δα ] δ α ] α β + δ ] α = ] β + δ }, 48 α 8
which means that h < 0. Using 48, 45, and 46, we can easily determine k and c. c What is the effect of an increase in on the steady state value of capital? Explain briefly about the economic intuition behind your result about the effect of on the steady state capital stock. Solution: Since k = ] /α α β + δ h, and h < 0, an increase in will reduce the steady state level of capital stock: k < 0. The intuition behind this result is that an increase in means that the agent values leisure more and supplies less labor to the labor market; consequently the economy will produce less goods and services and then lead to less capital stock and consumption in the steady state. 9