Nominal Rigidity

#economics #macro

Oh, Hyunzi. (email: wisdom302@naver.com)
Korea University, Graduate School of Economics.
2024 Spring, instructed by prof. Eo, Yunjong.

Imperfect Competition and Menu Cost Model

Small frictions are not enough to generate significant nominal rigidity
To generate nominal rigidity, frictions must be combined with real rigidity
Effect of shocks other than monetary policy changes
Aggregate-demand externality: demand curve moves due to the price changes, with leads to more profit and more utility.

Market Structure

	# of firms	Market Power	Ela of demand	Prod diff	Exc prof	Pricing
Perf comp	inf	none	perf ela	none	no	taker
Mono comp	many	low	high ela	high	yes(SR)/no(LR)	setter
Monopoly	one	high	relative inela	absolute	yes	setter

Assumptions

Model Setting:

Static model with no capital
Imperfectly competitive Goods market
Competitive labor market
No Government spending, but central bank implements monetary policy
Closed economy
Labor market with unemployment under real rigidity

Firms

Infinitely many firms on a differentiated good from .
- not identical, but substitutable goods.
Each firm has monopoly rights over production and sale.
- Firms can set price freely to maximize profits
- but not perfect monopoly, thus maximizing level of exists.
Each firm owned by the households.
Simple production function:
- : labor hired from competitive labor market at wage rate .

Households

Budget Constraint

: labor income
: dividends to be distributed back to households from the profit.
: total consumption over goods

Utility Function

Based on Dixit-Stiglitz preferences,

: utility of the consumption over .
: labor supply
: implies a positive elasticity of labor supply with respect to real wages.

Characteristics of

Additivity: optimization over and can be done separately.
- : meaning that there is no wealth effect.
- : decision on labor supply is independent of the consumption
Remark on Meaning of Lagrangian multiplier Meaning of Lagrangian multiplier, that means the marginal utility of wealth at optimum, i.e. and since there is no wealth effect, the one unit increase in wealth will be one .

Consumption

: consumption of goods
- consumer has taste for variate, meaning the consumer prefers to consume a diversified bundle of goods.
: elasticity of substitution among differentiated goods
- elasticity of substitution equals on every goods .
- as , : consumption goods are perfect substitutes.

Government

From the Quantity theory with velocity normalized to 1,

: real money balances
: central bank target money supply
Under nominal rigidity, monetary policy can affect :
- However, in the Long-Run, prices are flexible, i.e. no effect on the real GDP .

Equilibrium under Imperfect Competition

Household Behavior

Lagrangian function: F.O.C.

Consumption Behavior

Demand for each goods

From F.O.C.
From , we have Using the budget constraint, thus we have therefore, From the monetary equation where the third equation holds by the market clearing for all , and the last equation holds by the budget constraint.

Finally, we have

goes up when , , .
If , then the law of demand holds ()

Aggregate Demand and Price

Total Consumption is Since , the price of unit of is Therefore, implying that

increases when , , and , when .
Taking logs on both sides, showing that is relative sensitivity to react to relative price changes.
Since , we have meaning that the marginal utility of wealth at optimum is totally depending on the price of consumption, independent of the labor (or leisure).

Labor Supply Behavior

From F.O.C. By letting , and from the result of , we have

: real wage.
: Frisch elasticity.
- positive since (positive elasticity of labor supply with respect to the real wage)
- the labor supply has positive relation with the real wage .

Definition 1 (Frisch Elasticity).

The Frisch elasticity of labor supply measures the percentage change in hours worked due to the percentage change in wages, holding the marginal utility of wealth (i.e. the Lagrangian multiplier) as constant. where denotes real wage.

Proof.Since we are assuming the static model, we can simplify the formula as As we have already driven thus we have This completes the proof. □

Firm Behavior

Suppose that profit for firm is fully returned to the households.
From the previous result and , we have Thus the real dividends are where the firms's problem is to decide the amount of labor input () and its price . Notice that is given since the labor market is perfectly competitive, while is settable since the goods market is monopolistic competitive.

F.O.C.

Price Decision

From F.O.C.

: mark-up. monopolistic power, since represents the elasticity of substitution.
if , relative price is less than the real marginal cost of production .
if , then . thus completely competitive market.

Equilibrium

Equilibrium condition: (by )
Real wage at equilibrium: , driven by labor supply equation,
Relative prices at equilibrium: , driven by the price decision and the real wage at equilibrium,
Symmetric equilibrium: , , , and .
Output at equilibrium: , driven by
Price level at equilibrium: , driven by the quantity theory (),
- No nominal friction: Price is flexibly determined by the and .

Welfare Implications

vs. Social Optimum

Under perfectly competitive market, . Thus the parameters under monopolistic competitions compared to the social optimal values,

: produce less then social optimum
: more expensive then social optimum

Effect of Shocks

Given the symmetric aggregate demand shocks,

Mono. comp: symmetric effects on output, asymmetric effects on welfare.
- since , booms can make the welfare increase if .
Perf. comp: both symmetric effects on output and welfare.
- since is already under optimum welfare, in either direction of the shock, the welfare always decreases.

Nominal Friction

We introduce a fixed cost (menu cost) to add nominal friction

So far, there was no nominal friction under price determination.
Now assume that the firms must pay a fixed cost to change their prices.
Thus the firm changes its price when , where is the menu cost.
Given friction, decrease in would decrease instead of until the friction is overcome.

Profit under Adjusted Price

Assume one firm decides its pay, while other firms keep prices fixed. Then the real profit for firm is, F.O.C.
then, Thus, the optimal price for the firm to maximize its profit when all others keep theirs fixed is and the maximized real profit is

Profit under Fixed Price

While if the firm does not adjust its price, then the optimal price is from the prior discussion is, under the equilibrium, and the maximized real profit under the fixed cost is

Change in Real Profit

Thus, the change in real profit for the firm changing its price when all others keep their prices fixed is,

Quantitative Implication

Parameter Value

: relatively inelastic labor supply.
: elasticity of demand
: markup over the marginal cost.

Equilibrium

: flexible-price equilibrium output is about below the social optimum output.

Monetary Policy under Nominal Friction

: central bank decrease in aggregate demand by decreasing , and remains unchanged given other firms not responding.
: menu cost more than of the revenue can induce the nominal rigidity. implying that the menu cost has to be huge.
Thus the nominal friction is not enough to explain the price rigidity.
- given only a nominal friction, monopolistically competitive firms will change their prices in response to all but the small demand shock.
- thus we need a real rigidity to make a large nominal rigidity.

Real Rigidity

Since the nominal rigidity is not sufficient enough to produce a real rigidity, we need a real rigidity to make that small frictions in price can lead to a large nominal rigidity.

There are some possible real rigidities we can introduce

capital market imperfections: external finance premium
labor market imperfections: efficiency wages

which are due to asymmetric information and principal-agent problems. However, we only look into a case of efficiency wages, based on Ball and Romer (1990).

Real Wage

Compared to the previous real-wage equilibria of we have a real wage above market-clearing level, due to efficiency wage:

where and are set to be holding
is the elasticity of the real wage with respect to .
unlike , now have less procyclical behavior, meaning less sensitive to the

Change in Profit

The real profit for firm when other prices are fixed is

F.O.C.
then, Thus the optimal price to adjust under the real rigidity is Therefore, the optimal profit under the adjustment is

While the profit under the fixed cost is

Finally, the change in profit for one firm changing its price is

Quantitative Implication

Without real rigidity,

Same as before: , then . and .
Output level:
Change in price:

With real rigidity,

, , and .
Steady State output level similar to before:
Real marginal cost of production(real wage):
: meaning that required menu cost can be small to induce nominal rigidity.