Two-period consumption-savings model and consumption-labor model

Real lifetime budget constraint and budget constraints

Consider a representative household in the two-period consumption-savings model who has well-behaved preferences over period-1 and -2 consumption given by u (c1, c2). The household is constrained by their nominal lifetime budget constraint (LBC):

 P1 (1 + τ) c1 +   = Y1 +      

Where the LBC above incorporates the assumptions of a zero initial wealth endowment and terminal condition so that A0 = A2 = 0, and τ > 0 is a sales tax on consumption.

(a) Use the Fisher Equation to transform the nominal LBC into real terms.

P₁c₁ (1 + I (τ_i)) + P₂c₂ = Y₁ (1 + I) + Y2

P₂c₂ = Y₁ (1 + i) + Y₂ − P₁c₁ (1 + I)

c₂ = Y₁ (1 + I) P₂ + Y₂ 1 P₂ − c₁  

⇒ = 

→ Intercepts is as follows;

Horizontal: Set c₂ = 0 in LBC, solve for c₁ → Y₁ () + Y₂ ( 

Vertical: Set c₁ = 0 in LBC, solve for c₂ → Y₁ () + Y₂ (  )

(b) Express the household’s real consumption-saving optimality condition in terms of the general utility function u (c1, c2).

=

(c) Use indifference curve analysis for this model to illustrate the following:

1. Locate the optimal choice (c^∗₁, c^∗₂). Label that point E.
2. Locate potential point on the LBC for the combination real income (y1, y2) that implies the household is a saver in period 1. Label that point A.

 (d) Suppose that the sales tax is eliminated for period 2, but kept in place for period 1. Write down the optimality condition in this new environment.

(e) Illustrate the policy change in a graph, assuming that the substitution effect dominates the income effect on c1. Labeling the new optimal choice (c ^∗∗₁, c^∗∗₂) point E’, conclude whether saving in period 1 increases or decreases.

4. Consider a household in three-period version of the dynamic consumption-savings model with the preferences: u (c1, c2, c3) = log c1 + β log c2 + β 2 log c3 where β is the household’s subjective discount factor. In each of the three periods t = 1, 2, 3, they enter with a predetermined amount of real wealth at−1, earn interest income rtat−1 and exogenous real labor income it. They also choose the flow of consumption CT and the stock of wealth at to carry to the next period. You may assume that the interest rate is constant so that r = r1 = r2 = r3.

Budget constraints for period-1: C1+s = Y1+A ……………… (1)
Budget constraints for period-2: C2 = Y2+ (1+r) s ……………… (2)

 Sum of eel (1) and eel (2) C1 + C2/1+r = Y1 + Y2/1+r + A = I ·

 Price of consumption good is normalized in 1st period.

Price of consumption good in period 2 is 1/1+r, it is relative price of consumption for this period 2 relative to consumption in period 1.

1+r is gross interest rate, it is relative price of consumption good for today to consumption good for tomorrow.

Sequential Lagrangian: c1, c 2, l ¿ λ) =u (c1) + Bu (c2) – λ [c1 + (1/1+r) c2-Y1- (1/1+r) Y^2

• All the changes of LBC are not affecting the utility function u (c1, c2)
• The indifference map is as it is without any change
• Budget constraints determine the individuals’ optimal consumption over time.
• C1 is for period-1 whereas c2 is for period-2
• Period-1 individually begins with Au=0

Since the numerical value of (c1, c2) and S 1 privy.

 S 1 ^privy = Rao + Y1 – C1

¿ Rao+6 – C1

 =0.1(0) + 6 – C1

=6- C1

C1=6

S2^priv = Rao + Y2 – C2

=0+11-C2

C2=11

Optimality conditions for consumption and labor

U (c1, c2) = (6, 11)

During the course of period-1, definition of real private savings can be written be written as

 S1^priv = rao + Y1 -C1

 Since it seems to be analogous to one statement of nominal private savings during the period-1 (which we recall as S 1 priv =Ao +y – P1c1)

1. a) S1^priv= rao + Y1 – C1

= 0 + 10- C1

 C1= 10

1. b) S 1 priv = rao +Y2- C2

 =0+ 6.6- C2

C2=6.6

1. c) S1^priv= rao + Y3 – C3

= 0 + 11 - C1

 C3= 11

U (c1, c2, c3) = (10, 6.6, 11)

During the course of period-1, definition of real private savings can be written be written as S 1 priv = rao + Y1 -C1 Since it seems to be analogous to one statement of nominal private savings during the period-1 (which we recall as S 1 priv =Ao +y – P1c1)

Max _{c1, c2, l1, l2, a1} u (γ₁c₁, l1) + β_u (γ₂c₂, l₂)

Subject to: at − (1 + r) at−1 − (1 − τt) wt (1 − lt) + ct = 0, for t = 1, 2

⇒ L = u (γ1c1, l1) + βu (γ2c2, l2) +λ1 (a1 − (1 + r1) a0 − (1 − τ1) w1 (1 − l1) + c1) + λ2 (a2 − (1 + r2) a1 − (1 − τ2) w2 (1 − l2) + c2)

∂l/∂c1 = 0 −→ ∂u/∂c1 + λ1 = 0 → ∂u ∂c1 = −λ1

∂L/∂c2 = 0 −→ β ∂u/∂c2 + λ2 = 0 → β ∂u/∂c2 = −λ2

∂L/∂l1 = 0 −→ ∂u/∂l1 + λ1 (1 − τ1) w1 = 0 −→ ∂u/∂l1 1 / (1 − τ1) w1 = −λ1

∂L/∂l2 = 0 −→ β ∂u/∂l2 + λ2 (1 − τ2) w2 = 0 → β ∂u/∂l2 1/ (1 − τ2) w2 = −λ2

∂L/∂a1 = 0 → λ1 − λ2 (1 + r2) = 0 → −λ1 = −λ2 (1 + r2) (5)

Consumption-Labor Optimality Conditions:

→ for period 1, use Equations (1) and (3):

∂u/∂c1 = ∂u/∂l1 1 (1 − τ1) w1

⇒ ∂u/∂l1 / ∂u/∂c1 = (1 − τ1) w1

→ similarly for period 2, use Equations (2) and (4):

Β ∂u/∂c2 = β ∂u/∂l2 1 (1 − τ2) w2

⇒ ∂u/∂l2 / ∂u/∂c2 = (1 − τ2) w2 (7)

Consumption-Savings Optimality Conditions:

Use equations (1) and (2) into (5):

−λ1 = −λ2 (1 + r2)

∂u/∂c1 = β ∂u/∂c2 (1 + r2)

⇒ ∂u/∂c1 /β ∂u/∂c2 = (1 + r2)

Using the specific utility function in the optimality conditions (6), (7), and (8):

Consumption-savings optimality conditions

1/γ1 (l1/c1) ^−1/2 = (1 − τ1) w1

1/γ2 (l2/c2) ^−1/2 = (1 − τ2) w2

Γ1/βγ2 (c1/c2) ^−1/2 = (1 + r2) (11)

1/γt(lt/ct) ^−1/2 = (1 − τt)wt

Ct/lt = (γt (1 − τt) wt) 2

 ⇒ ∂ (CT/lt) ∂γt = 2 (γt (1 − τt) wt) (1 − τt) wt > 0

Profit/P1 = f (k1, n1) − inv^net₁ − W1/P1 n1 + P2f (K2, n2) / P1 (1 + I) − P2invnet 2/ P1 (1 + I) − W2n2 P1 (1 + I)

Profit / P1 = f (k1, n1) − inv^net₁ − w1n₁ + f (K2, n₂) 1 + π₂ 1 + I − inv^ent ₂ (1 + π2) / (1 + I) − w2n2 (1 + π2) / (1 + I)

Using the Fisher Equation

Profit / P1 = f (k1, n1) − inv^net₁ − w1n1 + f (K2, n₂) / (1 + r2) − inv^ent ₂/ (1 + r2) – (w2n2) / (1 + r2)/

 (b) Taking FOCs for n1 and n2:

 n1: ∂f ∂n1 − n1 = 0 ⇒ ∂f ∂n1 = n1

 n2: ∂f ∂n2 − n2 = 0 ⇒ ∂f ∂n2 = n2

(b) Let the demand for the firm’s output be represented by qt = k α t n 1−α t . Use the optimality conditions from part (a) to express the firm’s demand function for labor in periods 1 and 2, and the firm’s demand function for capital in period 2, as derived demand function.

(1 − α) k α t n −α t = wt

We want to express the above condition in terms of qt.

Recalling that q = k α t n (1−α) t:

 (1 − α)k α t n −α t nt = wt

(1 − α)k α t n 1−α t 1 nt = wt

 (1 − α) qt NT = wt

Re-arranging for nt yields the labor demand function for t = 1, 2: nt∗ = (1 − α) qt wt

Using the Cobb-Douglas production function into the optimality condition for capital demand in equation

 αkα−1 2 n 1−α 2 = r2 + δ

Again, using q = k α t n (1−α) t and

solving for the endogenous variable:

αk^α−1 ₂ n ^1−α ₂ k2/k2 = r2 + δ

αk^α₂ n^1−α ₂ 1/ k2 = r2 + δ

α (q2/k2) = r2 + δ

k²∗ = α q2 r2 + δ

k²∗ = α q2 r2 + δ

Y = K1/2L1/2 by L

: If we define y = Y/L,

we can rewrite the above expression as:

q = K1/2/L1/2. Defining k = K/L,

we can rewrite the above expression as: q = k ^1/2.

Taxation is meant to bring to balance the price in the market and the price of the imput and by so doing the government get their revenue. Additionally taxation can be used to bring a balance to the economy. However when taxation is increase the will be a general effect to the system in that the government intent to reduce the money in circulation and thus reduce economic activity. The demand of the commodity will reduce and thus the productivity of the company will reduce. This mean that there will be a shift for the demand for labor to the left. Unemployment rate will increase, the cost of labor will reduce and all of this will be caused by huge decline to the demand of the labor in the market.

With the shift of the demand curve to the left it mean that the labor demand in the system will decline. The decline will because by the fact that less people will be needed to produce goods as the demand of good would have lowered. Nevertheless the company will make less income which mean it will have even less income to pay the emeployee and thus it will shed off the employee who are not needed in the business.