Agents of the Macro Economy

Agents §

We’re doing static analysis; i.e. it’s very LR or very SR.

Numerous agents of each sector are grouped into a representative agent.

1. Representative household

   Objective: maximize utility over consumption ($C$) and Leisure ($L$) within the utility function

   → Labor Supply $S^N$ and # of laborers $N$

   → Goods Demanded $D$

2. Representative Firm

   Objective: maximuze profit ($\pi$) over capital ($K$) and Labor ($N$) within the production function

   → Labor Demand $D^N$

   → Goods Supplied $S$

3. Government

   Role: collect taxes and pay transfers ($NT$); spend remaining on the goods market $G$

Household Optimization §

• For households the price of consumption goods and the price of leisure [=wage] is an exogenous variable; i.e. it cannot be controlled
• Households have a time endownment of $h$ which they can use for leisure ($L$) or consumption ($C$)

The Household Utility Function $u(C,L)$

Assumptions:

1. $\frac{\partial u}{\partial C},\frac{\partial u}{\partial L}>0$ …i.e. More consumption/leisure is always better
2. $\frac{{\partial }^{2}u}{\partial C^2},\frac{{\partial }^{2}u}{\partial K^2}<0$ …i.e. the Marginal Utility always diminishes
3. $\exists \frac{\partial u}{\partial C}$ [= utility function is differentiable] …i.e. $C,L$ are somewhat substitutable → $MRS=\frac{\partial u/\partial C}{\partial u/\partial L}{|}_{\bar{u}}$

Household Budget Constraint

$$wN+\pi -NT=C$$

• $w$ is the representative wage
• $N$ is the hours of labor a HH chooses
• $\pi$ is profit or dividends if the HH is also an owner of a firm
• $NT$ is the net transfers (+transfers-taxes)
• $C$ is consumption (in units: # of goods)

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Firm Optimization §

• The representative firm employs Factors of Production ($K,N$) to produce $Y^S=f(K,N)$ output
• Profit of the firm is $\pi =f(K,N)-wN-rK$
• The firm is the average firm of the whole economy, which is unlike a microecon firm in many ways

The Firm Production Function $f(z,K,N)$

Assumptions:

1. Constant Returns to Scale (M&A is useless) $x^{\lambda }f(z,K,N)=f(z,\lambda K,\lambda N)$ → This is probable, since output as a whole in general has CRS
2. More capital or more labor always means more output $\frac{\partial Y}{\partial K},\frac{\partial Y}{\partial N}>0$, equivalently $M{P}_K,M{P}_N>0$ → Probable; throw another pencil into the economy, it’ll produce more
3. Diminishing Marginal Product $\frac{{\partial }^{2}Y}{\partial K^2},\frac{{\partial }^{2}Y}{\partial K^2}<0$
4. Marginal product of capital increase with more labor, and vice versa $\frac{\partial Y}{\partial K\partial N}>0$ [= positive cross-derivative]

Profit Function

$$\frac{\partial \pi }{\partial K}=\frac{\partial f}{\partial K}-r=0$$ $$\frac{\partial \pi }{\partial N}=\frac{\partial f}{\partial N}-w=0$$

↑ Must satisfy both conditions

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⇒ The Cobb-Douglas Production Function satisfies all these assumptions:

$$f(z,K,N)=z{K}^{\alpha }{N}^{1-\alpha }\text{where }0<\alpha <1$$

• Profit $\pi =z{K}^{\alpha }{N}^{1-\alpha }-wN-rK$
• Capital Demand, $\frac{\partial \pi }{\partial K}=0$ at $z\cdot \alpha (\frac{K}{N}{)}^{\alpha -1}=r$
• Labor Demand, $\frac{\partial \pi }{\partial N}=0$ at $z\cdot (1-\alpha )(\frac{K}{N}{)}^{\alpha }=w$

Government Plans §

The government’s goals are more vague, threefold:

1. Allocative: resources are used in certain proportions
2. Distributive: income, wealth is distributed at tolerable equalities
3. Stabilization: business cycle is smoothed out
   1. Automatic stabilizers; e.g. unemployment benefits
   2. government purchases ($G$)