\(C_{ij} = C_i(\frac{P_i}{P})^{-\eta}\)
\(Y_i^D = \sum\limits_{j=1}^n C_{ij} = C(\frac{P_i}{P})^{-\eta}\)
Martian i's consumption equals his real revenues:
\(C_i=\frac{P_i}{P} Y_i\)
Since by assumption, \(Y_i=L_i\), we can pllug the three last equations in the utility function to find:
\(\mathcal{U}_i = C(\frac{P_i}{P})^{1-\eta} - \frac{\eta -1}{\gamma\eta}C^\gamma(\frac{P_i}{P})^{-\gamma\eta} - \mathcal{z}D_i\)
We assume that one unit of money can be used only once per period to make transactions (i.e. velocity is equal to one) and that money is necessary for transactions and therefore we have \(PC=M\). At the start, the central bank is distributing an equal amount of money M to each farmer. Substituting in the utility function gets us:
\(\mathcal{U}_i = (\frac{M}{P})(\frac{P_i}{P})^{1-\eta} - \frac{\eta -1}{\gamma\eta}(\frac{M}{P})^\gamma(\frac{P_i}{P})^{-\gamma\eta} - \mathcal{z}D_i\)
We derive with respect to \(P_i\) to to find the utility maximizing price:
\(0 = \frac{M}{P}(1-\eta)P_i^{-\eta}P^{\eta-1}-\frac{\eta-1}{\gamma\eta}(\frac{M}{P})^\gamma(-\gamma\eta)P_i^{-\gamma\eta-1}P^{\gamma\eta}\)
\(MP_i^{-\eta}P^{\eta -2}=M^\gamma P^{\gamma\eta - \gamma}P_i^{-\gamma\eta -1}\)
\(P_i^{\gamma\eta -\eta +1} = M^{\gamma -1}P^{\gamma\eta - \gamma - \eta +2}\)
\(P_i^\star = M^{\frac{\gamma -1}{\gamma\eta -\eta +1}}P^{\frac{\gamma\eta - \gamma - \eta +2}{\gamma\eta -\eta +1}}\)
\(P_i^\star = P^\phi M^{1-\phi}\)
with \(\phi = 1- \frac{\gamma -1}{\gamma\eta -\eta +1}\)
Where \(\phi\) is the elasticity of the representative martian's desired price with respect to the aggregate price level. Hence, in the absence of menu costs, symmetric equilibrium occurs when \(P_i=P=M\).
From equation (???) we have:
\(\frac{P_i^\star}{P} = (\frac{M}{P})^{1-\phi}\)
\(\frac{M}{P} = (\frac{P_i^\star}{P})^{\frac{1}{1-\phi}} \)
Substituting equation (???) in equation (???):
\(\mathcal{U}_i = (\frac{P_i^\star}{P})^{\frac{1}{1-\phi}}(\frac{P_i}{P})^{1-\eta} - \frac{\eta -1}{\gamma\eta}(\frac{P_i^\star}{P})^{\frac{\gamma}{1-\phi}}(\frac{P_i}{P})^{-\gamma\eta} - \mathcal{z}D_i\)
\(\mathcal{U}_i = P_i^{\star \frac{\gamma\eta-\eta+1}{\gamma-1}}P^{\eta-1-\frac{\gamma\eta-\eta+1}{\gamma-1}}P_i^{1-\eta}-\frac{\eta-1}{\gamma\eta}P_i^{\star \gamma\frac{\gamma\eta-\eta+1}{\gamma-1}}P^{\gamma\eta - \gamma\frac{\gamma\eta-\eta+1}{\gamma-1}}P_i^{-\gamma\eta} -\mathcal{z}D_i\)
\(\mathcal{U}_i = P_i^{\star \eta +\frac{1}{\gamma-1}}P^{-1-\frac{1}{\gamma-1}}P_i^{1-\eta}-\frac{\eta-1}{\gamma\eta}P_i^{\star \gamma\eta +\frac{\gamma}{\gamma-1}}P^{-\frac{\gamma}{\gamma-1}}P_i^{-\gamma\eta} -\mathcal{z}D_i\)
\(\mathcal{U}_i= (\frac{P_i^\star}{P})^{\frac{\gamma}{\gamma-1}}(\frac{P_i}{P_i^\star})^{1-\eta}-\frac{\eta-1}{\gamma\eta}(\frac{P_i^\star}{P_i})^{\gamma\eta}(\frac{P_i^\star}{P})^{\frac{\gamma}{\gamma-1}}\)
Since \((\frac{P_i^\star}{P})^{\frac{\gamma}{\gamma-1}} = ((\frac{M}{P})^{1-\phi})^{\frac{\gamma}{\gamma-1}}=(\frac{M}{P})^{\frac{\gamma}{\gamma\eta-\eta+1}}=(\frac{M}{P})^{\gamma(1-\eta+\eta\phi)}\)
\(\mathcal{U}_i= (\frac{M}{P})^{\gamma(1-\eta+\eta\phi)}[(\frac{P_i}{P_i^\star})^{1-\eta}-\frac{\eta-1}{\gamma\eta}(\frac{P_i}{P_i^\star})^{-\gamma\eta}]-\mathcal{z}D_i \equiv V(\frac{M}{P} , \frac{P_i}{P_i^\star}) - \mathcal{z}D_i\)
The first derivative of utility with respect to M/P is positive while the second derivative of utility with respect to M/P is negative.
\(V_2 = \frac{\partial V(\frac{M}{P} , 1)}{\partial \frac{P_i}{P_i^\star}} = 0\)
\(V_{22} = \frac{\partial^2 V(\frac{M}{P} , 1)}{\partial \frac{P_i}{P_i^\star}} < 0\) since \(\gamma>0 ; \eta > 0\)
When martian i maintains a fixed price, then \(D_i = 0\). \(P_i=P\) which implies that \(\frac{M}{P} = M\). As \(\frac{P_i^\star}{P} = (\frac{M}{P})^{1-\phi}\) the numbers of utils the representative martian get is equal to \(V(M , \frac{1}{M^{1-\phi}})\). When the representative martian pays the menu cost to change his price despite that the other martians refuse to adjust their prices, then \(D_i = 1\). In this case, martians can adjust their price so as to choose their desired price and therefore \(P_i^\star = P_i\). Since \(P=1\) this means that the number of utils that the representative martian gets in a situation where he does pay the menu cost is equal to \(V(M , 1) -\mathcal{z}\). Hence the gain \(G_N\) that the martian man gets from changing his price must be superior to the menu cost \(\mathcal{z}\) if he is to change his price:
\(G_N = V(M, 1) - V(M , \frac{1}{M^{1-\phi}})\)
Rigidity is an equilibrium everytime that \(G_N < \mathcal{z}\).
We now take a second order Taylor series approximation of \(G_N\) when \(M=1\).
Remember that a second order Taylor series approximation takes the following form:
\(f(X)\approx f(X_0)+\frac{f'(X_0)}{1!}(X-X_0)+\frac{f''(X_0)}{2!}(X-X_0)^2\)
\(G_N \approx V(1 ; 1) - V(1 ; 1) + [V_1(1 ; 1) - V_1(1 ; 1) - V_2(1 ; 1)](M-1) + \frac{1}{2}[V_{11}(1 ; 1) -V_{11}(1 ; 1)- (1-\phi)^2 V_{22}(1 ; 1)](M-1)^2\)
\(G_N \approx \frac{(1-\phi)^2}{2} V_{22}(M-1)^2\)
As we specify that \(x = M - 1 \), rigidity is an equilibrium whenever:
\(\frac{(1-\phi)^2}{2} V_{22}\mathcal{x}^2 < \mathcal{z}\)
\(\mid\mathcal{x}\mid < \sqrt{\frac{\mathcal{z}}{\frac{(1-\phi)^2}{2} V_{22}}}\)
\(\mid\mathcal{x}\mid <\mathcal{x}_N\)
We now want to find for the gain \(G_A\) to a martian from adjusting given that the others adjust their prices. If martian i and all other martians are paying the menu cost, then \(D_i=1\), \(\frac{P_i}{P_i^\star} = 1\), \(P=M\) and therefore \(\frac{M}{P}=1\). In this case, the utility of the representative martian will be equal to \(V(1 , 1) - \mathcal{z}\). If martian i does not pay the menu cost while all other martians are paying it, then \(D_i = 0\), \(P_i = 1\) and \(\frac{P_i}{P_i^\star} = \frac{1}{M}\). In this case, the representative martian's utility is equal to \(V(1 , \frac{1}{M})\). Flexibility will be an equilibrium whenever \(G_A > \mathcal{z}\) where:
\(G_A = V(1, 1) - V(1 , \frac{1}{M})\)
Taking a second order Taylor series approximation as we did previously yields:
\(G_A = -\frac{1}{2} V_{22}x^2\)
Conclusion
As Blanchard and Kiyotaki (1987) themselves admit that altough they assumed all prices to be initially equal and set optimally:
In a dynamic economy and in the presence of menu costs, such a degenerate price distribution is unlikely. But, if prices are initially not all equal or optimal, it is no longer obvious that even a small change in nominal money will leave all prices unaffected. It is no longer obvious that money, or aggregate demand in general, will have large effects on output.
This model might be criticized on the ground that it does not take into account time. If the representative martian firm is maximizing the present value of its profits, then the incentive to choose price flexibility as an equilbrium might be much greater than in the model since choosing the sticky price equilibrium as an equilibrium incurs not only costs in the resent but also in the future. This criticism however is invalid as Martians do not have the standard humanoid conception of time. Although time is a category of human action, it is not a category of martian action.
References