Part B Microeconomic Foundations of Incomplete Nominal Adjustment
Some type of incomplete nominal adjustment appears essential to understanding why monetary changes have real effects. Although individuals really care about real prices and quantities, and it doesn't cost much to update prices, these little nominal frictions have a large effect in the aggregate. This makes it worth analyzing, seeing when the microeconomic conditions causing changing price costs lead to significant nominal stickiness in response to onetime monetary shocks. In particular, we will focus on a static model where firms face a menu cost of price adjustment -- a small fixed cost for changing a nominal price.
6.5 A Model of Imperfect Competition and Price-Setting
Before turning to stickiness, we'll examine an economy of imperfectly competitive price-setters with complete price flexibility. We are interested in causes and effects of barriers to price adjustment, but it is essential to see the foundation first.
Assumptions
There is a continuum of differentiated goods indexed by \(i \epsilon [0,1]\). Each good is produced by a single firm with monopoly rights. Firm i's production function is just
\(Y_i = L_i\)
Firms hire labor in a perfectly competitive labor market and sell output in an imperfectly competitive goods market. Thus firms can set their prices freely and any profits they earn are accrued to the households.
The utility of the representative household depends positively on its consumption and negatively on the amount of labor it supplies. It takes the form
\(U = C - \frac{1}{\gamma} L^{Y}, \gamma > 1\)
Crucially, C is not the household's total consumption of all goods (since all goods would be perfect substitutes for one another and firms would have no market power). It is rather an index of the household's consumption of the individual goods, taking the constant-elasticity-of-substitution form:
\(C = [\int _{i = 0} ^1 C_i ^{(\eta - 1)/ \eta}]^{\eta / (\eta - 1)}\)
This formulation parallels the production function in the Romer model of endogenous technological change. Government purchases and international trade are absent, so output matches consumption.
Households choose their labor supply and purchases of consumption goods to maximize utility. Firms choose prices to maximize profits, taking demand curves and wages as given. The aggregate demand side of the model will be:
\(Y = \frac{M}{P}\)
which implies an inverse relationship between price level and output.
Household Behavior
It is easiest to start by considering how households allocate their consumption spending among the different goods. Consider a household that spends S. The Lagrangian for its utility-max problem is
\(\mathcal{L} [\int_{i = 0} ^1 C_i ^{(\eta - 1)/ \eta} di] ^{\frac{\eta}{\eta - 1}} + \lambda [ S - \int _{i = 0} ^{1} P_i C_i di]\)
The first-order condition for Ci is
\(\frac{\eta}{\eta - 1} [\int_{j = 0} ^1 C_j ^{(\eta - 1)/ \eta} dj] ^{\frac{1}{\eta - 1}} \frac{\eta -1 }{\eta} C_i ^{-1/ \eta} = \lambda P_i\)
The only terms that depend on i are consumption and price. Thus it must take the form:
\(C_i = AP_i ^{- \eta}\)
Finding A in terms of the given variables, we can substitute it into the budget constraint and solve, yielding:
\(A = \frac{S}{\int_{j = 0} ^1 P_j ^{1 - \eta} dj}\)
Finally, substituting this to solve for the definition of C we find:
\(C = \frac{S}{(\int_{i = 0} ^1 P_i ^{1 - \eta} di) ^{1/(1 - \eta)}}\)
This means that when households allocate their spending across goods optimally, the cost of obtaining one unit of C is the denominator (which is the price index corresponding to the utility function). These expressions can simplify to
\(C_i = (\frac{P_i}{P}) ^{- \eta} \frac{S}{P} \Longrightarrow (\frac{P_i}{P}) ^{- \eta} C\)
Thus the elasticity of demand for each individual good is \(\eta\).
The household's other choice variable is its labor supply. Its spending equals \(WL + R\) where W is wage, R is profit income and so the problem for choosing L is therefore
max \(\frac{WL + R}{P} - \frac{1}{\gamma} L^\gamma\)
The first-order condition for L is
\(\frac{W}{P} - L^{\gamma - 1} = 0\)
Implying that
\(L = (\frac{W}{P}) ^{\frac{1}{\gamma - 1}}\)
Thus labor supply is san increasing function of the real wage, with an elasticity equal to the exponent. This represents the aggregate value of L.
Firm Behavior
The real profits of the monopolistic producer of good i are its real revenues minus its real costs:
\(\frac{R_i}{P} = \frac{P_i}{P} Y_i - \frac{W}{P} L_i\)
Recall that the amount of the good produced must equal the amount consumed. Substituting in the values of Yi and Li from the previous equations yields:
\(\frac{R_i}{P} = (\frac{P_i}{P}) ^{1 - \eta} Y - \frac{W}{P} (\frac{P_i}{P}) ^{- \eta} Y\)
Solving the first-order condition for Pi/P gives us:
\(\frac{P_i}{P} = \frac{\eta}{\eta - 1} \frac{W}{P}\)
That is, a producer with market power sets price as a markup over marginal cost, with the size of the markup determined by the elasticity of demand. This is our intuitive assumption.
Equilibrium
Because the model is symmetric, its equilibrium is also symmetric. This means that all households supply the same amount of labor and have the same demand curves. Similarly, the fact that all firms face the same demand curve and the same real wage implies that they all charge the same amount and produce the same amount. That is, in equilibrium, \(Y = C = L\). Starting with the expression of real wage as a function of output:
\(\frac{W}{P} = Y^{\gamma - 1}\)
Substituting this expression into the price equation yields an expression for each producer's desired relative price as a function of aggregate output:
\(\frac{P_i ^*}{P} = \frac{\eta }{\eta - 1} Y^{\gamma - 1}\)
We know each producer charges the same price, and that the price index equals this common price. Thus we can write:
\(Y = (\frac{\eta - 1}{\eta}) ^{\frac{1}{\gamma - 1}}\)
This is the equilibrium level of output. Using the aggregate demand equation, \(Y = \frac{M}{P}\), we can find the equilibrium price level:
\(P = \frac{M}{Y}\)
\(P = \frac{M}{ (\frac{\eta - 1}{\eta}) ^{\frac{1}{\gamma - 1}}}\)
\(P = \frac{M}{Y}\)
Implications
When producers have market power, they produce less than the socially optimal amount. The fact that producers face downward-sloping demand curves means that the marginal revenue product of labor is less than its marginal product. As a result, real wage is less than the marginal product of labor. Thus the gap between equilibrium and optimal levels of output is greater when producers have more market power.
The fact that equilibrium output is inefficiently low under imperfect competition has important implications for fluctuations. The booms and busts will still be away from the optimum. The gap between the actual and ideal also implies that pricing decisions have externalities. Suppose there is a marginal reduction in all prices. Marginal productivity rises and so aggregate output does too. The real wage increases and the demand for each god shifts out. Thus under imperfect competition, there is an aggregate demand externality where pricing externalities operate through the overall demand for goods.
Finally, an example of this can be seen in the graph below.
Remember, small frictions in economy are enough to cause dramatic swings.