The representative martian household utility founction is equal to:
\(\mathcal{U}_j = C_j - \frac{1}{\gamma} L_j^\gamma\) ; \(\gamma > 1\)
Where C is an index of the martian household's consumption. In order to make our derivation more intuitive, our martian economy is limited to three goods: oxygen, food and water. C is given by the following equation:
\(C = [C_1^{\frac{(\eta -1)}{\eta}} + C_2^{\frac{(\eta -1)}{\eta}} + C_3^{\frac{(\eta -1)}{\eta}}]^{\frac{\eta}{\eta -1}}\)
We therefore assume constant elasticity of substitution [SHOW THAT]
Setting the lagrangian:
\(\mathcal{L} = [C_1^{\frac{\eta -1}{\eta}} + C_2^{\frac{\eta -1}{\eta}} + C_3^{\frac{\eta -1}{\eta}}]^{\frac{\eta}{\eta -1}} + \lambda[S - P_1C_1 - P_2C_2 - P_3C_3]\)
FOC:
\(\frac{\partial \mathcal{L}}{\partial C_1} = 0 = \frac{\eta}{\eta -1}[C_1^{\frac{\eta -1}{\eta}} + C_2^{\frac{\eta -1}{\eta}} + C_3^{\frac{\eta -1}{\eta}}]^{\frac{1}{\eta -1}}*\frac{\eta -1}{\eta}C_1^{-\frac{1}{\eta}} + \lambda P_1\)
\(-\lambda P_1 = [C_1^{\frac{\eta -1}{\eta}} + C_2^{\frac{\eta -1}{\eta}} + C_3^{\frac{\eta -1}{\eta}}]^{\frac{1}{\eta -1}}*C_1^{-\frac{1}{\eta}}\)
\(-\lambda P_2 = [C_1^{\frac{\eta -1}{\eta}} + C_2^{\frac{\eta -1}{\eta}} + C_3^{\frac{\eta -1}{\eta}}]^{\frac{1}{\eta -1}}*C_2^{-\frac{1}{\eta}}\)
\(-\lambda P_3 = [C_1^{\frac{\eta -1}{\eta}} + C_2^{\frac{\eta -1}{\eta}} + C_3^{\frac{\eta -1}{\eta}}]^{\frac{1}{\eta -1}}*C_3^{-\frac{1}{\eta}}\)
Therefore we have:
\(\frac{-\lambda P_1}{-\lambda P_2} = \frac{[C_1^{\frac{\eta -1}{\eta}} + C_2^{\frac{\eta -1}{\eta}} + C_3^{\frac{\eta -1}{\eta}}]^{\frac{1}{\eta -1}}*C_1^{-\frac{1}{\eta}}}{[C_1^{\frac{\eta -1}{\eta}} + C_2^{\frac{\eta -1}{\eta}} + C_3^{\frac{\eta -1}{\eta}}]^{\frac{1}{\eta -1}}*C_2^{-\frac{1}{\eta}}} = \frac{C_1}{C_2}^{-\frac{1}{\eta}}\)
\(C_2 = (\frac{P_1}{P_2})^{\eta}C_1\)
Doing the same thing for \(C_3\) yields:
\(C_3 = (\frac{P_1}{P_3})^{\eta}C_1\)
Since the constraint is equal to: \(S=P_1C_1 + P_2C_2 + P_3C_3\) we have:
\(S=P_1C_1 + P_2(\frac{P_1}{P_2})^{\eta}C_1 + P_3(\frac{P_1}{P_3})^{\eta}C_1\)
\(\frac{S}{P_1^\eta} = P_1^{1-\eta}C_1 + P_2^{1-\eta}C_1 + P_3^{1-\eta}C_1\)
\(C_1 = P_1^{-\eta}[\frac{S}{P_1^{1-\eta} + P_2^{1-\eta} + P_3^{1 - \eta}}]\)
Which we simply as follows \(C_1 = AP_1^{-\eta}\) where \(A = \frac{S}{P_1^{1-\eta} + P_2^{1-\eta} + P_3^{1 - \eta}}\)
Hence the elasticity of demand with respect to price for each good is equal to:
\(\frac{\partial C_1}{\partial P_1} = -\eta P_1^{-\eta - 1}[\frac{S}{P_1^{1-\eta} + P_2^{1-\eta} + P_3^{1 - \eta}}] - [\frac{(1-\eta)P_1^{-2\eta}}{(P_1^{1-\eta} + P_2^{1-\eta} + P_3^{1 - \eta})^2}]\)
\(\frac{P_1}{C_1} = P_1^{1+\eta}[\frac{P_1^{1-\eta} + P_2^{1-\eta} + P_3^{1 - \eta}}{S}]\)
\(\frac{\partial C_1}{\partial P_1} \frac{P_1}{C_1} = -\eta - [\frac{(1-\eta)P_1^{1-\eta}}{S(P_1^{1-\eta} + P_2^{1-\eta} + P_3^{1 - \eta})}]\)
Where the first part of the equation is the substitution effect and the second part takes into account the income effect.
Since \(C = [C_1^{\frac{(\eta -1)}{\eta}} + C_2^{\frac{(\eta -1)}{\eta}} + C_3^{\frac{(\eta -1)}{\eta}}]^{\frac{(\eta -1)}{\eta}}\) and \(C_1 = AP_1^{-\eta}\) we have:
\(C = [(AP_1^{-\eta})^{\frac{(\eta -1)}{\eta}} + (AP_2^{-\eta})^{\frac{(\eta -1)}{\eta}} + (AP_3^{-\eta})^{\frac{(\eta -1)}{\eta}}]^{\frac{(\eta -1)}{\eta}}\)
\(C = A[P_1^{1-\eta} + P_2^{1-\eta} + P_3^{1-\eta}]^{\frac{(\eta -1)}{\eta}}\)
\(C = \frac{S[P_1^{1-\eta} + P_2^{1-\eta} + P_3^{1-\eta}]^{\frac{\eta -1}{\eta}}}{P_1^{1-\eta} + P_2^{1-\eta} + P_3^{1 - \eta}}\)
\(C = \frac{S}{[P_1^{1-\eta} + P_2^{1-\eta} + P_3^{1 - \eta}]^{\frac{1}{1-\eta}}}\)
This last equation just implies that the cost of obtaining one unit of C is equal to \(P=[P_1^{1-\eta} + P_2^{1-\eta} + P_3^{1 - \eta}]^{\frac{1}{1-\eta}}\). Where P is the price index. Hence we have:
\(A = \frac{S}{P^{1-\eta}}\)
\(C_i = AP_i^{-\eta}\)
\(C_i = S\frac{P_i^{-\eta}}{P^{1-\eta}}\)
\(C_i = \frac{S}{P}(\frac{P_i}{P})^{-\eta}\) and since \(C = \frac{S}{P}\)
\(C_i = C(\frac{P_i}{P})^{-\eta}\)
We have \(\mathcal{U} = C - \frac{1}{\gamma}L^\gamma\) and \(C=\frac{S}{P}\) with \(S = WL+R\) where "R" are profits.
\(Max \frac{WL+R}{P} - \frac{1}{\gamma}L^\gamma\)
\(0 = \frac{W}{P} - L^{\gamma - 1}\)
\(L = (\frac{W}{P})^{\frac{1}{\gamma -1}}\)
Firm Behavior
Real profits equal to real revenues minus real costs:
\(\frac{R_i}{P} = \frac{P_i}{P}Y_i - \frac{W}{P}L_i\)
Since by assumption we have \(L_i = Y_i\), \(Y = C\) and \(C_i=(\frac{P_i}{P})^{-\eta}C\) we have \(Y_i=(\frac{P_i}{P})^{-\eta}Y\) and therefore:
\(\frac{R_i}{P} = (\frac{P_i}{P})^{1-\eta}Y - \frac{W}{P}(\frac{P_i}{P})^{-\eta}Y\)
Taking the first order conditions for \(\frac{P_i}{P}\):
\(0= (1-\eta)(\frac{P_i}{P})^{-\eta}Y + \eta \frac{W}{P}(\frac{P_i}{P})^{-(\eta+1)}Y\)
\((1-\eta)(\frac{P_i}{P})^{-\eta}Y = -\eta \frac{W}{P}(\frac{P_i}{P})^{-(\eta+1)}Y\)
\((1-\eta)=-\eta\frac{W}{P}\frac{P_i}{P}^{-1}\)
\(\frac{P_i}{P} = \frac{\eta}{1-\eta}\frac{W}{P}\)
Equilibrium
Since \(\frac{W}{P} = L^{\gamma -1}\), we have by assumption \(\frac{W}{P} = Y^{\gamma -1}\) and:
\(\frac{P_i}{P} = \frac{\eta}{1-\eta}Y^{\gamma - 1}\)
From the quantity equation (normalizing velocity to one), we have \(Y=\frac{M}{P}\). Since each firm is producing in a symetrical way, in equilibrium we have \(P_i = P\) and therefore:
\(1 = \frac{\eta}{1-\eta}Y^{\gamma -1}\)
\(Y = [\frac{\eta}{1-\eta}]^{\frac{1}{1-\gamma}}\)
\(P=\frac{M}{[\frac{\eta}{1-\eta}]^{\frac{1}{1-\gamma}}}\)
The last two equations makes it obvious that money is neutral even there is monopolistic competition.
In equilibrium, the real wage is determined as follows:
\(\frac{\eta}{1-\eta}\frac{W}{P} = \frac{P_i}{P} = \frac{\eta}{1-\eta}Y^{\gamma - 1}\)
\(\frac{W}{P} = Y^{\gamma -1}\)
\(\frac{W}{P} = \frac{\eta}{1-\eta}\)
Since \(Y = L\), it is obvious that the marginal productivity of labor is equal to one and that the equilibrium wage, as we should expect in an environment with monopolistic competition is inferior to the marginal productivity of labor.
Incentive to change the price
\(\frac{R_i}{P} = (\frac{P_i}{P})^{1-\eta}Y - \frac{W}{P}(\frac{P_i}{P})^{-\eta}Y\)
\(\frac{R_i}{P} = (\frac{P_i}{P})^{-\eta}Y (\frac{P_i}{P} - Y^{\gamma -1})\)
\(\frac{R_i}{P} = \frac{M}{P}(\frac{P_i}{P})^{1-\eta} - (\frac{M}{P})^\gamma (\frac{P_i}{P})^{-\eta}\)
Let us now imagine a situation where other firms do not change their price and where initially, all firms charge the same price. If firm "i" decides not to change its price, then:
\(\pi_{FIXED} = \frac{M}{P} - (\frac{M}{P})^\gamma\)
As we know that the profit maximizing price is equal to \(\frac{P_i}{P} = \frac{\eta}{1-\eta}Y^{\gamma - 1}=\frac{\eta}{1-\eta}(\frac{M}{P})^{\gamma - 1}\), then if the firm does adjust its price, we have: