Understanding DSGE

0. Preface

Chapter 1 Introduction

Definition 1.0.1 (Adjustment principle). According to Kocherlakota (2007), by this principle, the better a model’s projected data adheres to observed data, the more preferable it is for policy analysis.

1.1 The idea of Representative Agents and Lifespan

• The solution found was to group economic agents into large categories, for example, in the case of a study regarding consumers, forming groups with similar consumption characteristics (high, average and low-income consumers) would be recommended. Representative Agent
  • Households
  • Firms
  • Government
  • Financial Sector
  • Foreign Sector
• Consider each agent’s lifespan
  • Obviously, firms and governments do not exist forever. However, when a government decides upon its budget, it does not expect that it will cease to exist. Firms act likewise; when deciding their budgets, they do not consider that they will go bankrupt in the future. This assumption in relation to consumers is simpler. Although it is assumed that each consumer has a finite lifespan, when considering the family structure in which members periodically are born and die, the "family" representative agent becomes infinite.

1. Basic Principles

Chapter 2 Real Business Cycle (RBC) Model

• Basic Assumptions
  • perfect competition
  • fully flexible prices
  • agent:
    • households
    • firms
    • (fiscal authorities)
    • (monetary authorities)
    • (foreign sector and financal institutions)

Real business cycle theory states that supply shocks (technological shocks) are what generate economic fluctuations, and uses a neoclassical growth model as a reference for the economy’s longterm behavior.

2.1 Brief theoretical review: Real Business Cycles

Three main problems:

• a model with two goods (consumption and leisure)
  • $u(C,H)$
    • $\text { Strictly increasing, } \frac{\partial u}{\partial C}>0 \text { and } \frac{\partial u}{\partial H}>0 ; \text { and }$
    • $\text { Diminishing marginal returns, } \frac{\partial^{2} u}{\partial C^{2}}<0 \text { and } \frac{\partial^{2} u}{\partial H^{2}}<0$
• intertemporal consumption-savings
• a firm’s profit maximization problem

2.1.1 Model with two "goods": consumption and leisure

Budget constraints

Assuming that an individual can work the amount of hours he/she likes (L), receiving an hourly wage W , total weekly income is:

$$Y=L W$$

As previously mentioned, the number of work hours (L) plus the number of leisure hours (H) per week, must be equal to 60 hours, $L = 60− H$. Thus, income can be written as a function of leisure:

$$Y=(60-H) W$$

Another simplifying assumption is that individuals spend all their income on consumption, not saving anything. Each consumer good, c, can be bought on the market for the price P. Thus, an individual’s consumption in each period is:

$$P c=Y$$

Combining the two previous expressions, we arrive at the following budget constraint:

$$P c=(60-H) W$$

Budget constraint:

$$c=\left(\frac{60 W}{P}\right)-\left(\frac{W}{P}\right) H$$

Individuals’ decisions regarding consumption and work

$$\max _{c, L} u(c, L)$$

subject to:

$$\max _{c, L} u(c, L)$$

2.1.2 Dynamic structure of consumption-savings

$$u\left(c_{1}, c_{2}, c_{3}, \ldots\right)=u\left(c_{1}\right)+\beta u\left(c_{2}\right)+\beta^{2} u\left(c_{3}\right)+\ldots$$

Two-Dates Model

$$u(c_1, c_2)$$

In this model, individuals receive income twice during their lives - once in period 1 and once in period 2. They start off in period 1 with a certain amount of wealth, $A_0$ . They choose consumption in period 1 ($c_1$) paying a price of $P_1$, and also decide how much wealth they will carry forward to period 2, $A_1$. Thus, an individual’s budget constraint in period 1 can be written as:

$$P_{1} c_{1}+A_{1}=R A_{0}+Y_{1}$$

where R is the gross nominal interest rate that represents the returns on each monetary unit held as a financial asset from one period to another.

Savings:

$$S_{1}=(R-1) A_{0}+Y_{1}-P_{1} c_{1}$$

Rearranging the period 1 budget constraint:

$$A_{1}-A_{0}=(R-1) A_{0}+Y_{1}-P_{1} c_{1}$$

$$S_1 = A_1 - A_0$$

And

$$P_{2} c_{2}+A_{2}=R A_{1}+Y_{2}$$

$$P_{2} c_{2}=R\left[R A_{0}+Y_{1}-P_{1} c_{1}\right]+Y_{2}$$

$$P_{1} c_{1}+\frac{P_{2} c_{2}}{R}=Y_{1}+\frac{Y_{2}}{R}+R A_{0}$$

Assume that $A_0=0$,

$$c_{2}=\left[\left(\frac{R}{P_{2}}\right) Y_{1}+\frac{Y_{2}}{P_{2}}\right]-\left[\frac{P_{1} R}{P_{2}}\right] c_{1}$$

Optimal intertemporal choice

$$\max _ { c _ { 1 } , c _ { 2 } , A _ { 1 } } u ( c _ { 1 } ) + \beta u ( c _ { 2 } )$$

subject to:

$$\left. \begin{array} { c } { P _ { 1 } c _ { 1 } + A _ { 1 } = R A _ { 0 } + Y _ { 1 } } \\ { P _ { 2 } c _ { 2 } = R A _ { 1 } + Y _ { 2 } } \end{array} \right.$$

Use Lagrangian---and get Euler Equation:

$$\underbrace { - \frac { \partial u / \partial c _ { 1 } } { \beta \partial u / \partial c _ { 2 } } } _ { \text { MRS } c _ { 1 } - c _ { 2 } } = \underbrace { - \frac { R } { \pi _ { 2 } } } _ { \text { Relative price } c _ { 1 } - c _ { 2 } }$$

Define $\pi_2=\frac{P_2}{P_1}$.

Economists call this mechanism the intertemporal substitution effect (Barro, 1997).

2.1.3 Input markets

Definition of input markets

• Generally speaking, it is assumed that inputs are physically equal.
• It is assumed that firms and households take input price levels as a given(Complete market).

$$Y = f ( K ^ { d } , L ^ { d } )$$

$$\text {Profit}= P Y - W L ^ { d } - R K ^ { d }$$

Demand for inputs

$$\max _ { K ^ { d } , L ^ { d } } P Y - W L ^ { d } - R K ^ { d }$$

Subject to:

$$Y = f ( K ^ { d } , L ^ { d } )$$

The 1st order condition:

$$\left. \begin{array} { l } { P \frac { \partial Y } { \partial L ^ { d } } - W = 0 } \\ { P \frac { \partial Y } { \partial K ^ { d } } - R = 0 } \end{array} \right.$$

Definition 2.1.5 (Problem of the firm). A profit-maximizing firm chooses input and production levels with the sole objective of maximizing economic profit. That is, firms wish to obtain the largest possible difference between total revenue and total costs.

Definition 2.1.6 (Marginal Rate of Technical Substitution). The negative slope of an isoquant curve consisting of two inputs, capital (K) and labor (L) is called the Marginal Rate of Technical Substitution (MRTS) at that point. That is,

$$M R T S = - \frac { \partial K } { \partial L } | _ { f ( K , L ) = f ( K , L ) _ { 1 } } = - \frac { P M g L } { P M g K } | _ { f ( K , L ) = f ( K , L ) _ { i } }$$

where $| f (K ,L)=f (K ,L)_i$ indicates the slope is calculated along the isoquant $f (K ,L)_i$.

2.2 The Model

Assumption 2.2.1. The economy is closed, with no government or financial sector.

Assumption 2.2.2. This economy does not have a currency. That is, it is a barter economy.

Assumption 2.2.3. Adjustment costs do not exist.

2.2.1 Households

Assumption 2.2.4. The economy in this model is formed by a unitary set of households indexed by $j ∈ [0,1]$ whose problem is to maximize a particular intertemporal welfare function. To this end, a utility function is used, additively separable into consumption (C) and labor (L).

Assumption 2.2.5. Consumption is intertemporally additively separable (no habit formation).

Assumption 2.2.6. Population growth is ignored.

Assumption 2.2.7. The labor market structure is one of perfect competition (no wage rigidity).

The representative household optimizes the following welfare function (Constant Relative Risk Aversion, CRRA):

$$\max _ { C _ { j , t } , L _ { j , t } , K _ { j , t + 1 } } E _ { t } \sum _ { t = 0 } ^ { \infty } \beta ^ { t } ( \frac { C _ { j , t } ^ { 1 - \sigma } } { 1 - \sigma } - \frac { L _ { j , t } ^ { 1 + \varphi } } { 1 + \varphi } )$$

where $E_t$ is the expectations operator, $\beta$ is the intertemporal discount factor, $C$ is the consumption of goods, $L$ is the number of hours worked, $\sigma$ is the relative risk aversion coefficient, and $\varphi$ is the marginal disutility in respect of labor supply.

$$U_C>0 \quad \text {and} \quad U_L<0 \\ U_{CC}<0 \quad \text {and} \quad U_{LL}<0$$

Budget Constraints (Two sectors: Households own firms):

$$P _ { t } ( C _ { j , t } + I _ { j , t } ) = W _ { t } L _ { j , t } + R _ { t } K _ { j , t } + \Pi _ { t }$$

where $P$ is the general price level, $I$ is level of investment, $W$ is the level of wages, $K$ is the capital stock, $R$ is the return on capital, and $\Pi$ is the firms’ profit (dividends).

Capital accumulation over time

$$K _ { j , t + 1 } = ( 1 - \delta ) K _ { j , t } + I _ { j , t }$$

where $\delta$ is the depreciation rate of physical capital.

Solve:

$$\left. \begin{array} { c } { L = E _ { t } \sum _ { t = 0 } ^ { \infty } \beta ^ { t } \{ [ \frac { C _ { j , t } ^ { 1 - \sigma } } { 1 - \sigma } - \frac { L _ { j , t } ^ { 1 + \varphi } } { 1 + \varphi } ] } \\ { - \lambda _ { j , t } [ P _ { t } C _ { j , t } + P _ { t } K _ { j , t + 1 } - P _ { t } ( 1 - \delta ) K _ { j , t } - W _ { t } L _ { j , t } - R _ { t } K _ { j , t } - \Pi _ { t } ] \} } \end{array} \right.$$

1st order conditions:

$$\left. \begin{array} { l } { \frac { \partial L } { \partial C _ { j , t } } = C _ { j , t } ^ { - \sigma } - \lambda _ { j , t } P _ { t } = 0 } \\ { \frac { \partial L } { \partial L _ { j , t } } = - L _ { j , t } ^ { \varphi } + \lambda _ { j , t } W _ { t } = 0 } \end{array} \right.$$

$$\frac { \partial L } { \partial K _ { j , t + 1 } } = - \lambda _ { j , t } P _ { t } + \beta E _ { t } \lambda _ { j , t + 1 } [ ( 1 - \delta ) E _ { t } P _ { t + 1 } + E _ { t } R _ { t + 1 } ] = 0$$

(Why)

Solve the first 2 equations and we get:

$$C _ { j , t } ^ { \sigma } L _ { j , t } ^ { \varphi } = \frac { W _ { t } } { P _ { t } }$$

$$\underbrace { - C _ { j , t } ^ { \sigma } L _ { j , t } ^ { \varphi } } _ { \text { Consumption-leisure MRS } } = \underbrace { - \frac { W _ { t } } { P _ { t } } } _ { \text { Consumption-leisure relative price } }$$

and,

$$\lambda _ { j , t } = \frac { C _ { j , t } ^ { - \sigma } } { P _ { t } }$$

$$\lambda _ { j , t + 1 } = \frac { C _ { j , t + 1 } ^ { - \sigma } } { P _ { t + 1 } }$$

and we will get

$$\left. \begin{array} { c } { - C _ { j , t } ^ { - \sigma } + \beta E _ { t } \{ ( \frac { C _ { j , t + 1 } ^ { - \sigma } } { P _ { t + 1 } } ) [ ( 1 - \delta ) P _ { t + 1 } + R _ { t + 1 } ] \} = 0 } \\ { ( \frac { E _ { t } C _ { j , t + 1 } } { C _ { j , t } } ) ^ { \sigma } = \beta [ ( 1 - \delta ) + E _ { t } ( \frac { R _ { t + 1 } } { P _ { t + 1 } } ) ] } \end{array} \right.$$

The previous equation determines the household’s savings decision (in this model, savings is the acquisition of investment goods). Thus, when households decide their level of savings, they compare the utility rendered by consuming an additional amount today with the utility that would be rendered by consuming more in the future. Thus, if interest rate expectations rise, consuming "today" (at t) is more expensive and, ceteris paribus, future consumption (t+1) will rise.

To simplify, assume $\beta = 1$ and $\delta = 1$,

$$\underbrace { - E _ { t } [ \frac { 1 } { \pi _ { t + 1 } } ( \frac { C _ { j , t + 1 } } { C _ { j , t } } ) ^ { \sigma } ] } _ { \text { TMS } C _ { t } - C _ { t + 1 } } = \underbrace { - E _ { t } ( \frac { r _ { t + 1 } } { \pi _ { t + 1 } } ) } _ { \text { Preço relativo } C _ { t } - C _ { t + 1 } }$$

where $E_t r_{t+1} = E_t (\frac{R_{t+1}}{P_{t+1}})$ is the real rate of return on capital.

In short, the problem of the household boils down to two choices. The first is an intratemporal choice between acquiring consumption and leisure goods. The other is an intertemporal choice, in which the household must choose between present and future consumption.

2.2.2 Firms

Assumption 2.2.8. There is a continuum of firms indexed by $j$ that maximize profit observing a structure of perfect competition, this means that their profits will be zero ($\Pi_t = 0$, for every $t$).

To this end a Cobb-Douglas production function is used:

$$Y _ { j , t } = A _ { t } K _ { j , t } ^ { \alpha } L _ { j , t } ^ { 1 - \alpha }$$

Or, more popular one (CES):

$$F ( K _ { t } , L _ { t } ) = [ \alpha K _ { t } ^ { \rho } + ( 1 - \alpha ) L _ { t } ^ { \rho } ] ^ { \frac { 1 } { \rho } }$$

where $\rho\in (-\infty, 1)$

The problem of the firm is solved by maximizing the Profit function, choosing the amounts of each input $(L_t ,K_t)$:

$$\max _ { L _ { j , t } , K _ { j , t } } \Pi _ { j , t } = A _ { t } K _ { j , t } ^ { \alpha } L _ { j , t } ^ { 1 - \alpha } P _ { j , t } - W _ { t } L _ { j , t } - R _ { t } K _ { j , t }$$

Solve:

$$\underbrace { \frac { R _ { t } } { P _ { j , t } } } _ { \text { Real MCK } } = \underbrace { \alpha \frac { Y _ { j , t } } { K _ { j , t } } } _ { \text { MPK } }$$

$$\underbrace { \frac { R _ { t } } { P _ { j , t } } } _ { \text { Real MCK } } = \underbrace { \alpha \frac { Y _ { j , t } } { K _ { j , t } } } _ { \text { MPK } }$$

It is assumed that productivity shocks follow a AR(1) process, such that:

$$\log A _ { t } = ( 1 - \rho _ { A } ) \log A _ { s s } + \rho _ { A } \log A _ { t - 1 } + \epsilon _ { t }$$

where $A_{ss}$ is the value of productivity at the steady state, $ρ_A$ is the autoregressive parameter of productivity, whose absolute value must be less than one $(|ρ_A|< 1)$ to ensure the stationary nature of the process and $\varepsilon_t \sim N(0,σ_A )$.

Assumption 2.2.9. Productivity growth is ignored in this model.

Rearranging the previous expression,

$$L _ { j , t } = ( \frac { 1 - \alpha } { \alpha } ) \frac { R _ { t } } { W _ { t } } K _ { j , t }$$

Therefore,

$$L _ { j , t } = ( \frac { 1 - \alpha } { \alpha } ) \frac { R _ { t } } { W _ { t } } K _ { j , t }$$

$$K _ { j , t } = \frac { Y _ { j , t } } { A _ { t } } [ ( \frac { \alpha } { 1 - \alpha } ) \frac { W _ { t } } { R _ { t } } ] ^ { 1 - \alpha }$$

and,

$$L _ { j , t } = \frac { Y _ { j , t } } { A _ { t } } ( \frac { 1 - \alpha } { \alpha } ) \frac { R _ { t } } { W _ { t } } [ ( \frac { \alpha } { 1 - \alpha } ) \frac { W _ { t } } { R _ { t } } ] ^ { 1 - \alpha }$$

$$L _ { j , t } = \frac { Y_ { j,t } } { A _ { t } } [ ( \frac { \alpha } { 1 - \alpha } ) \frac { W _ { t } } { R _ { t } } ] ^ { - \alpha }$$

Total cost:

$$T C _ { j , t } = W _ { t } L _ { j , t } + R _ { t } K _ { j , t }$$

Therefore,

$$T C _ { j,t } = W _ { t } \frac { Y _ { j , t } } { A _ { t } } [ ( \frac { \alpha } { 1 - \alpha } ) \frac { W _ { t } } { R _ { t } } ] ^ { - \alpha } + R _ { t } \frac { Y _ { j , t } } { A _ { t } } [ ( \frac { \alpha } { 1 - \alpha } ) \frac { W _ { t } } { R _ { t } } ] ^ { 1 - \alpha }$$

Simplify and we get,

$$T C _ { t } = W _ { t } \frac { Y _ { j , t } } { A _ { t } } [ ( \frac { \alpha } { 1 - \alpha } ) \frac { W _ { t } } { R _ { t } } ] ^ { - \alpha } + R _ { t } \frac { Y _ { j , t } } { A _ { t } } [ ( \frac { \alpha } { 1 - \alpha } ) \frac { W _ { t } } { R _ { t } } ] ^ { 1 - \alpha }$$

And the marginal cost:

$$M C _ { j , t } = \frac { 1 } { A _ { t } } ( \frac { W _ { t } } { 1 - \alpha } ) ^ { 1 - \alpha } ( \frac { R _ { t } } { \alpha } ) ^ { \alpha }$$

As the marginal cost depends solely on productivity and the prices of the factors of production, it will be the same for all firms $(MC_{j,t}=MC_{t})$. Knowing that $P_t = MC_t$, we arrive at the general price level,

$$P _ { t } = \frac { 1 } { A _ { t } } ( \frac { W _ { t } } { 1 - \alpha } ) ^ { 1 - \alpha } ( \frac { R _ { t } } { \alpha } ) ^ { \alpha }$$

2.2.3 The model’s equilibrium conditions

Households decide:

• how much to consume (C)
• how much to invest (I)
• how much labor to supply (L), with the aim of maximizing utility, taking prices as given.

On the other hand, firms decide

• how much to produce (Y) using available technology
• choosing the factors of production (capital and labor), taking these prices as given.

2.2.4 Steady state

an endogenous variable $x_t$ is at the steady state in each $t$, if

$$E _ { t } x _ { t + 1 } = x _ { t } = x _ { t - 1 } = x _ { s s }$$

This is the case of productivity, which is the source of standard RBC models’ shocks - at the steady state $E(\varepsilon_t)=0$

Generally, assigning $A_{ss}=1$

Stuctrual Model

• Households

$$C _ { s s } ^ { \sigma } L _ { s s } ^ { \varphi } = \frac { W _ { s s } } { P _ { s s } }$$

$$1 = \beta ( 1 - \delta + \frac { R _ { s s } } { P _ { s s } } )$$

$$I _ { s s } = \delta K _ { s s }$$

• Firms

$$K _ { s s } = \alpha \frac { Y _ { S s } } { \frac { R _ { s s } } { P _ { s s } } }$$

$$L _ { s s } = ( 1 - \alpha ) \frac { Y _ { s s } } { \frac { W _ { s s } } { P _ { s s } } }$$

$$Y _ { s s } = K _ { s s } ^ { \alpha } L _ { s s } ^ { 1 - \alpha }$$

$$Y _ { s s } = K _ { s s } ^ { \alpha } L _ { s s } ^ { 1 - \alpha }$$

• Equilibrium Condition

$$Y _ { s s } = C _ { s s } + I _ { s s }$$

8 Variables:

$$( Y _ { s s } , C _ { s s } , I _ { s s } , K _ { s s } , L _ { s s } , W _ { s s } , R _ { s s } \text { and } P _ { s s } )$$

The first values that must be determined are the prices $(W_{ss}, R_{ss} \quad and \quad P_{ss})$.

---

Proposition 2.2.1 (Walras’ Law). For any price vector p, has $pz(p) \equiv 0$; i.e., the demand excess value is identically zero.

Proof. In simple terms, the definition of excess demand is written and multiplied by p:

$$p z ( p ) = p [ \sum _ { i = 1 } ^ { n } x _ { i } ( p , p w _ { i } ) - \sum _ { i = 1 } ^ { n } w _ { i } ] = \sum _ { i = 1 } ^ { n } [ p x _ { i } ( p , p w _ { i } ) - p w _ { i } ] = 0$$

since $x_i (p,p w_i)$satisfies the budget constraint $p x_i = p w_i$ for each individual $i=1,...,n$

In other words, Walras’ law states that if each individual satisfies his/her budget constraint, the value of his/her excess demand is zero, therefore the sum of excess demand must also be zero.

Walras’ Law implies the existence of k-1 independent equations in equilibrium with k goods. Thus, if demand is equal to supply in k-1 markets, they will also be equal in the $k_{th}$ market. Consequently, if there are k markets, only k-1 relative prices are required to determine equilibrium.

Price Normalization：

$$p _ { i } = \frac { \hat { p } _ { i } } { \sum _ { j = 1 } ^ { k } \hat { p } _ { j } }$$

The sum of the normalized prices $p_i$ must always be 1

• $k-1$ dimension degree of freedom

$$S ^ { k - 1 } = \{ p \in R _ { + } ^ { k } : \sum _ { i = 1 } ^ { k } p _ { i } = 1 \}$$

---

In short, the economy’s general price level can be normalized, $P_{ss}= 1$

$$R _ { s s } = P _ { s s } [ ( \frac { 1 } { \beta } ) - ( 1 - \delta ) ]$$

In the Dynare simulation, there is no need to substitute $R_{ss}$ in the other equations. It should just be shown before the other steady states.

$$W _ { s s } = ( 1 - \alpha ) P _ { s s } ^ { \frac { 1 } { 1 - \alpha } } ( \frac { \alpha } { R _ { s s } } ) ^ { \frac { \alpha } { 1 - \alpha } }$$

The next step is to satisfy the equilibrium condition.

• To this end, the variables that make up aggregate demand ($C_{ss}$ and $I_{ss}$) must be determined.
• The idea underlying the equilibrium condition is formed by the following proposition.

Proposition 2.2.2 (Market adjustment). Given $k$ markets, if demand is equal to supply in $k-1$ markets and $p_k > 0$, then demand must equal supply in the $k_{th}$ market.

AD & AS Diagram

$$C _ { S S } = \frac { 1 } { Y _ { S S } ^ { \frac { \varphi } { \sigma } } } [ \frac { W _ { s s } } { P _ { S s } } ( \frac { \frac { W _ { s s } } { P _ { s s } } } { 1 - \alpha } ) ^ { \varphi } ] ^ { \frac { 1 } { \sigma } }$$

$$I _ { s s } = ( \frac { \delta \alpha } { R _ { s s } } ) Y _ { s s }$$

$$Y _ { S S } = ( \frac { R _ { s s } } { R _ { s s } - \delta \alpha } ) ^ { \frac { \sigma } { \sigma + \varphi } } [ \frac { W _ { s s } } { P _ { s s } } ( \frac { \frac { W _ { s s } } { P _ { s s } } } { 1 - \alpha } ) ^ { \varphi } ] ^ { \frac { 1 } { \sigma + \varphi } }$$

To summarize,

2.3 Log-linearization (Uhlig's method)

• The problem is converting a non-linear model to a sufficiently adequate linear approximation such that its solution helps in the understanding of the underlying non-linear system’s behavior.

Uhlig (1999) recommends a simple method of log-linearization of functions that does not require differentiation: simply replacing a variable $X_t$ with $X_{ss}e^{\tilde{X}_t}$, where $\tilde{X}_t=logX-logX_{ss}$ represents the log of the variable’s deviation in relation to its steady state.

$$e ^ { ( \tilde { X } _ { t } + a \tilde { Y } _ { t } ) } \approx 1 + \tilde { X } _ { t } + a \tilde { Y } _ { t }$$

$$\tilde { X } _ { t } \tilde { Y } _ { t } \approx 0$$

$$E _ { t } [ a e ^ { \tilde { X } _ { t + 1 } } ] \approx a + a E _ { t } [ \tilde { X } _ { t + 1 } ]$$

2.3.1 Labor Supply

$$C _ { t } ^ { \sigma } L _ { t } ^ { \varphi } = \frac { W _ { t } } { P _ { t } }$$

$$C _ { s s } ^ { \sigma } L _ { s s } ^ { \varphi } e ^ { ( \sigma \tilde { C } _ { t } + \varphi \tilde { L } _ { t } ) } = \frac { W _ { s s } } { P _ { s s } } e ^ { ( \tilde { W } _ { t } - \tilde { P } _ { t } ) }$$

$$C _ { s s } ^ { \sigma } L _ { s s } ^ { \varphi } ( 1 + \sigma \tilde { C } _ { t } + \varphi \tilde { L } _ { t } ) = \frac { W _ { s s } } { P _ { s s } } ( 1 + \tilde { W } _ { t } - \tilde { P } _ { t } )$$

$$\begin{aligned} &\because C _ { s s } ^ { \sigma } L _ { s s } ^ { \varphi } = \frac { W _ { s s } } { P _ { s s } } \\ &\therefore \sigma \tilde { C } _ { t } + \varphi \tilde { L } _ { t } = \tilde { W } _ { t } - \tilde { P } _ { t } \end{aligned}$$

2.3.2 Euler equation for consumption

Rearrange Euler equation:

$$\frac { 1 } { \beta } E _ { t } ( \frac { C _ { t + 1 } } { C _ { t } } ) ^ { \sigma } = ( 1 - \delta ) + E _ { t } ( \frac { R _ { t + 1 } } { P _ { t + 1 } } )$$

$$\frac { 1 } { \beta } ( \frac { C _ { s s } ^ { \sigma } } { C _ { s s } ^ { \sigma } } ) e ^ { ( \sigma E _ { t } \tilde { C } _ { t + 1 } - \sigma \tilde { C } _ { t } ) } = ( 1 - \delta ) + \frac { R _ { s s } } { P _ { s s } } e ^ { [ E _ { t } ( \tilde { R } _ { t + 1 } - \tilde { P } _ { t + 1 } ) ] }$$

$$\frac { 1 } { \beta } [ 1 + \sigma ( E _ { t } \tilde { C } _ { t + 1 } - \tilde { C } _ { t } ) ] = ( 1 - \delta ) + \frac { R _ { s s } } { P _ { s s } } [ 1 + E _ { t } ( \tilde { R } _ { t + 1 } - \tilde { P } _ { t + 1 } ) ]$$

$$\frac { \sigma } { \beta } ( E _ { t } \tilde { C } _ { t + 1 } - \tilde { C } _ { t } ) = \frac { R _ { s s } } { P _ { s s } } E _ { t } ( \tilde { R } _ { t + 1 } - \tilde { P } _ { t + 1 } )$$

2.3.3 Return on capita

$$R_t=\alpha \frac{Y_t}{K_t}$$

$$\frac { R _ { s s } } { P _ { s s } } e ^ { ( \tilde { R } _ { t } - \tilde { P } _ { t } ) } = \alpha \frac { Y _ { s s } } { K _ { s s } } e ^ { ( \tilde { Y } _ { t } - \tilde { K } _ { t } ) }$$

$$\frac { R _ { s s } } { P _ { s s } } ( 1 + \tilde { R } _ { t } - \tilde { P } _ { t } ) = \alpha \frac { Y _ { s s } } { K _ { s s } } ( 1 + \tilde { Y } _ { t } - \tilde { K } _ { t } )$$

$$\tilde { R } _ { t } - \tilde { P } _ { t } = \tilde { Y } _ { t } - \tilde { \mathcal { K } } _ { t }$$

2.3.4 Wage levels

$$\frac{W_t}{P_t}=(1-\alpha)\frac{Y_t}{L_t}$$

$$\frac { W _ { s s } } { P _ { s s } } e ^ { ( \tilde { W } _ { t } - \tilde { P } _ { t } ) } = ( 1 - \alpha ) \frac { Y _ { s s } } { L _ { s s } } e ^ { ( \tilde { Y } _ { t } - \tilde { L } _ { t } ) }$$

$$\frac { W _ { s s } } { P _ { s s } } ( 1 + \tilde { W } _ { t } - \tilde { P } _ { t } ) = ( 1 - \alpha ) \frac { Y _ { s s } } { L _ { s s } } ( 1 + \tilde { Y } _ { t } - \tilde { L } _ { t } )$$

$$\tilde { W } _ { t } - \tilde { P } _ { t } = \tilde { Y } _ { t } - \tilde { L } _ { t }$$

2.3.5 Production function

$$Y _ { t } = A _ { t } K _ { t } ^ { \alpha } L _ { t } ^ { 1 - \alpha }$$

$$Y _ { s s } e ^ { \tilde { Y } _ { t } } = A _ { s s } K _ { s s } ^ { \alpha } L _ { s s } ^ { 1 - \alpha } e ^ { ( \tilde { A } _ { t } + \alpha \tilde { K } _ { t } + ( 1 - \alpha ) \tilde { L } _ { t } ) }$$

$$Y _ { s s } ( 1 + \tilde { Y } _ { t } ) = A _ { s s } K _ { s s } ^ { \alpha } L _ { s s } ^ { 1 - \alpha } ( 1 + \tilde { A } _ { t } + \alpha \tilde { K } _ { t } + ( 1 - \alpha ) \tilde { L } _ { t } )$$

$$\tilde { Y } _ { t } = \tilde { A } _ { t } + \alpha \tilde { K } _ { t } + ( 1 - \alpha ) \tilde { L } _ { t }$$

2.3.6 Law of motion of capital

$$K _ { t + 1 } = ( 1 - \delta ) K _ { t } + I _ { t }$$

$$K _ { s s } e ^ { \tilde { K } _ { t + 1 } } = ( 1 - \delta ) K _ { s s } e ^ { \tilde { K } _ { t } } + I _ { s s } e ^ { \tilde { I } _ { t } }$$

$$K _ { s s } ( 1 + \tilde { K } _ { t + 1 } ) = ( 1 - \delta ) K _ { s s } ( 1 + \tilde { K } _ { t } ) + I _ { s s } ( 1 + \tilde { I } _ { t } )$$

$$( 1 + \tilde { K } _ { t + 1 } ) = ( 1 - \delta ) + ( 1 - \delta ) \tilde { K } _ { t } + \frac { I _ { s s } } { K _ { s s } } + \frac { I _ { s s } } { K _ { s s } } \tilde { I } _ { t }$$

$$\tilde { K } _ { t + 1 } = ( 1 - \delta ) \tilde { K } _ { t } + \delta \tilde { I } _ { t }$$

2.3.7 Equilibrium condition

$$Y_t=C_t+I_t$$

$$Y _ { s s } e ^ { \tilde { Y } _ { t } } = C _ { s s } e ^ { \tilde { C } _ { t } } + I _ { s s } e ^ { \tilde { I } _ { t } }$$

$$Y _ { s s } ( 1 + \tilde { Y } _ { t } ) = C _ { s s } ( 1 + \tilde { C } _ { t } ) + I _ { s s } ( 1 + \tilde { I } _ { t } )$$

$$Y _ { s s } \tilde { Y } _ { t } = C _ { s s } \tilde { C } _ { t } + I _ { s s } \tilde { I } _ { t }$$

2.3.8 Technological shock

$$\log A _ { t } = ( 1 - \rho _ { A } ) \log A _ { s s } + \rho _ { A } \log A _ { t - 1 } + \epsilon _ { t }$$

$$\tilde { A } _ { t } = \rho _ { A } \tilde { A } _ { t - 1 } + \epsilon _ { t }$$

Here, we have $w_t = W_t − P_t$ and $r_t = R_t − P_t$, which represent wages and the real interest rate, respectively.

It can be seen that the inflection point for the wage level is 57% higher than its steady state level.

2.4 BOX 2.1 - Basic log-linear RBC moel on Dynare

// RBC model -Chapter 2
// note: W and R are real the simulation

var Y I C R K W L A;
varexo e;
parameters sigma phi alpha beta delta rhoa;

sigma = 2;
phi = 1.5;
alpha = 0.35;
beta = 0.985;
delta = 0.025;
rhoa = 0.95;

model(linear);
#Pss = 1;
#Rss = Pss*((1/beta)-(1-delta));
#Wss = (1-alpha)*(Pss^(1/(1-alpha)))*((alpha/Rss)^(alpha/(1-alpha)));
#Yss = ((Rss/(Rss-delta*alpha))^(sigma/(sigma+phi)))*(((1-alpha)^(-phi))*((Wss/Pss)^(1+phi)))^(1/(sigma+phi));
#Kss = alpha*(Yss/Rss/Pss);
#Iss = delta*Kss;
#Css = Yss - Iss;
#Lss = (1-alpha)*(Yss/Wss/Pss);
//1-Labor supply
sigma*C + phi*L = W;
//2-Euler equation
(sigma/beta)*(C(+1)-C)=Rss*R(+1);
//3-Law of motion of capital
K =(1-delta)*K(-1)+delta*I;
//4-Production function
Y=A + alpha*K(-1) + (1-alpha)*L;
//5-Demand for capital
R = Y - K(-1);
//6-Demand for labor
W = Y - L;
//7-Equilibrium condition
Yss*Y =Css*C + Iss*I;
//8-Productivity shock
A = rhoa*A(-1) + e;
end;

steady;
check;
model_diagnostics;
model_info;

shocks;
var e;
stderr 0.01;
end;

stoch_simul;

Chapter 3 Basic New-Keynesian (NK) model

• The previous assumption is made more flexible with the introduction of imperfect competition, the "heart" of New-Keynesian models.
• This kind of model was initially developed by Rotemberg (1982), Blanchard and Kiyotaki (1987), Rotemberg and Woodford (1997), among others.
• There will be no change to the structure of household behavior, but there will be significant alterations to the structure of the production sector.
  • Firms that produce final goods (Retailers) - in an environment of perfect competition
  • Firms that produce intermediate goods (Wholesalers) - where imperfect competition occurs; differentiated goods are sold.

3.1 Brief theoretical review: New-Keynesians

3.1.1 Differentiated Products and the Consumption Aggregator

Q: How to understand "the only one final comsumption good"?

A: One way to reconcile the use of a single consumption good is to assume that everything is made up of these many differentiated products. Specifically, it is assumed that the habitual notion of consumption is a function,

$$c = c ( c _ { 1 } , c _ { 2 } , c _ { 3 } , \ldots , c _ { N } )$$

If there are N different products, the consumption of all things is a function of N different types of consumer good, formally known as a consumption aggregator function.

The features of consumption function:

$$\frac { \partial c ( . ) } { \partial c _ { j } } > 0$$

$$\frac { \partial ^ { 2 } c ( . ) } { \partial c _ { j } ^ { 2 } } < 0$$

The aggregate consumption function most commonly used in New-Keynesian models is

$$c ( c _ { 1 } , c _ { 2 } , c _ { 3 } , \ldots , c _ { N } ) = [ ( c _ { 1 } ) ^ { \frac { \psi - 1 } { \psi } } + ( c _ { 2 } ) ^ { \frac { \psi - 1 } { \psi } } + ( c _ { 3 } ) ^ { \frac { \psi - 1 } { \psi } } + \ldots + ( c _ { N } ) ^ { \frac { \psi - 1 } { \psi } } ] ^ { \frac { \psi } { \psi - 1 } }$$

where $\psi$ is the elasticity of substitution between these differentiated goods, possessing great economic significance in New-Keynesian models. It determines to what degree, from a consumer’s point of view, products differ from one another.

• $\psi \to\infty$, results in the simple linear sum $c_1+c_2+c_3+...+c_N$. Thus, each consumer good is as good as any other, from the viewpoint of the representative consumer, that is, the goods are perfect substitutes.
• With $\frac{\psi}{\psi -1} > 1$, however, the goods are only imperfect substiψ −1 tutes, which means that they are differentiated to some degree, depending on the exact value of $\psi$. Generally, New-Keynesian models assume $\frac{\psi}{\psi -1} > 1$.

---

Definition 3.1.1 (Elasticity of substitution between two goods). For an aggregate function $c ( c _ { 1 } , c _ { 2 } ) = [ ( c _ { 1 } ) ^ { \frac { \psi - 1 } { \psi } } + ( c _ { 2 } ) ^ { \frac { \psi - 1 } { \psi } } ] ^ { \frac { \psi } { \psi - 1 } }$, the elasticity of substitution $\psi$ measures the proportional change in $c_1/c_2$ in relation to the proportional change in the Marginal Rate of Substitution (MRS) along an indifference curve. That is, (WHY?)

$$\psi = \frac { \% \Delta ( c _ { 1 } / c _ { 2 } ) } { \% \Delta M R S } = \frac { \partial ( c _ { 1 } / c _ { 2 } ) } { \partial M R S } \cdot \frac { M R S } { ( c _ { 1 } / c _ { 2 } ) } = \frac { \partial \ln ( c _ { 1 } / c _ { 2 } ) } { \partial \ln M R S }$$

Derive CES and elasticity of substitution:

$$MRS= -(\frac{c_1}{c_2})^{-\frac{1}{\psi}}$$

$$\frac { \partial \ln ( c_{ 1 }/c_{ 2 } ) } { \partial \ln M R S }=\psi$$

---

3.1.2 Firms in monopolistic competition

The fundamental idea of New-Keynesian models does not lie in the representative consumer, but in firms, each of the $N$ differentiated products presumed to be produced by a distinct monopolistically competitive firm.

---

Definition 3.1.2 (Monopolistic competition). A market is in monopolistic competition when it has many firms that produce very similar, but not identical products, and when new firms can freely enter the market. The causes of differentiation between products can be many and varied: products’ intrinsic qualities, location of firms, additional services provided by firms etc.

---

Defining $\psi \to\infty$ is a way of "turning off" the elements that New-Keynesian models use to include monopolistic competition.

3.1.3 Price stickness

• In New-Keynesian models, there is imperfect competition in the market for products. The previous Keynesian models assumed perfect competition.
• While previous Keynesian models consider nominal stickiness in monetary wages, New-Keynesian models also focus on the stickiness of the prices of products.
• Besides the factors that cause stickiness in nominal variables, New-Keynesian models introduce real stickiness.

---

Definition 3.1.3 (Price stickiness). This refers to a situation in which the price of a good does not change immediately to a new equilibrium price when demand and/or supply curves are altered. Therefore, it is a failure of buyers and sellers to adapt to new market conditions and arrive at an equilibrium price.

---

Generally, the stylized facts about alterations in prices and wages are:

• Prices and wages are temporarily rigid.
• Prices and wages are readjusted on average two or three times a year.
• Prices and wages being adjusted frequently are the main factor responsible for high inflation.
• Prices and wages are not readjusted simultaneously.
• Changes in prices of tradable goods are more frequent than with non-tradable goods.

Appendix 1 An Application of DSGE: NONRIVALRY AND THE ECONOMICS OF DATA 数据经济学

0 Abstract

• Data is nonrival.
• In equilibrium, firms may not adequately respect the privacy of consumers.
• Fearing creative destruction, firms may choose to hoard data they own, leading to the inefficient use of nonrival data.
• Instead, giving the data property rights to consumers can generate allocations that are close to optimal.

1 Introduction

• We are particularly interested in how different property rights for data determine its use in the economy, and thus affect output, privacy, and consumer welfare.
• The key finding in our paper is that policies related to data have important economic consequences. When firms own data, they may not adequately respect the privacy of consumers. But nonrivalry leads to other consequences that are less obvious. Because data is nonrival, there are potentially large gains to data being used broadly. Markets for data provide financial incentives that promote broader use, but if selling data increases the rate of creative destruction, firms may hoard data in ways that are socially inefficient.
• Another allocation we consider is one in which a government — perhaps out of concern for privacy — sharply limits the use of consumer data by firms. While this policy succeeds in generating privacy gains, it may potentially have an even larger cost because of the inefficiency that arises from a nonrival input not being used at the appropriate scale.
• Finally, we consider an institutional arrangement in which consumers own the data associated with their behavior. Consumers then balance their concerns for privacy against the economic gains that come from selling data to all interested parties. This equilibrium results in data being used broadly across firms, taking advantage of the nonrivalry of data. Across a wide range of parameter values explored in our numerical example, consumer ownership of data generates consumption and welfare that are substantially better than firm ownership.

2 A Simple Model

• Aggregate Utility:

$$\left. \begin{array}{l}{ Y = ( \int _ { 0 } ^ { N } Y _ { i } ^ { \frac { \sigma - 1 } { \sigma } } d i ) ^ { \frac { \sigma } { \sigma - 1 } } }\\{ = N ^ { \frac { \sigma } { \sigma - 1 } } Y _ { i } }\end{array} \right.$$

$$Y_i = A_i L_i$$

The nonrival nature of ideas means there are constant returns to labor and increasing returns to labor and ideas together (Romer, 1990).

• Data is used to improve the quality of ideas:

$$A_i = D_i^{\eta}$$

Putting these last two equations together,

$$Y _ { i } = D _ { i } ^ { \eta } L _ { i } = D _ { i } ^ { \eta } L / N = D _ { i } ^ { \eta } \nu$$

$\nu \equiv L/N$ is firm size measured by employment.

• 数据的正外部性（scale effect: 技术上$\alpha >0$, 法律上$\tilde{x}>0$）

$$\left. \begin{array}{l}{ D _ { i } = \alpha x Y _ { i } + ( 1 - \alpha ) B }\\{ = \alpha x Y _ { i } + ( 1 - \alpha ) \tilde { x } N Y _ { i } }\\{ = [ \alpha x + ( 1 - \alpha ) \tilde { x } N ] Y _ { i } }\end{array} \right.$$

the bundle B can be used by any number of firms simultaneously. $B \equiv \tilde { x } N Y _ { i }$ and $\tilde {x}$ is the fraction of other firms' data that Tesla gets to use. (In richer model, both $x$ and $\tilde{x}$ are endogenous, subject to privacy considerations)

$\therefore$ Multiplier: the more people consume your product, the more data you have.

$$Y _ { i } = ( [ \alpha x + ( 1 - \alpha ) \tilde { x } N ] ^ { \eta } \nu ) ^ { \frac { 1 } { 1 - \eta } }$$

Derive, and

$$Y = N ^ { \frac { \sigma } { \sigma - 1 } } ( [ \alpha x + ( 1 - \alpha ) \tilde { x } N ] ^ { \eta } \nu ) ^ { \frac { 1 } { 1 - \eta } }$$

$$y = N ^ { \frac { 1 } { \sigma - 1 } } ( [ \alpha x + ( 1 - \alpha ) \tilde { x } N ] \nu ) ^ { \frac { \eta } { 1 - \eta } }$$

3 Economic Environment