General Equilibrium
Christopher Makler
Stanford University Department of Economics
Econ 50: Lecture 24
Where We Are
Today:
General Equilibrium
Unit I:
Autarky
Given resource constraints, production functions, and utility functions, solve for the bundle the market would "choose" to produce in competitive equilibrium.
(endogenize all prices, income, wages)
Given resource constraints, production functions, and utility functions, solve for the bundle a single agent would choose to produce and consume.
The Circular Flow
In our consumer theory, we've treated income as exogenous.
In our producer theory, we've treated wages as exogenous.
We've also assumed firms are maximizing profits, but haven't said where those profits go.
Crazy thought: what if the money firms pay for labor becomes the income of workers?
...and their profits become the income of the owners/shareholders of the firm?
Consumers
Good 1 Firms
Market for Good 1
Market for Good 2
Market for Labor
Good 2 Firms
Money flows clockwise
Goods, labor flow counter-clockwise
General Equilibrium: Everyone optimizes, all markets clear simultaneously.
Competitive Equilibrium
Review: Autarky (Chuck on a desert island)
We sometimes call the autarky model the "centralized" model: if there were a single agent making a decision, what would they do?
Similarly, we call competitive equilibrium a "decentralized" model, because lots and lots of individuals are making small decisions that add up to what "society chooses"
1. Given prices \(p_1,p_2\), firms will choose the point \((Y_1^*,Y_2^*)\) along the PPF where \(MRT = \frac{p_1}{p_2}\)
2. All money received by firms \((p_1Y_1^* + p_2Y_2^*)\) will become income \(M\) for consumers.
3. Given prices \(p_1,p_2\) and income \(M\), the consumer will choose the point \((X_1^*,X_2^*)\) along the budget line where \(MRS = \frac{p_1}{p_2}\)
4. At equilibrium prices, markets clear (\(X_1^* = Y_1^*\) and \(X_2^* = Y_2^*\)) so \(MRS = MRT\).
5. In disequilibrium, there is a shortage in one market and a surplus in the other, pulling the system toward equilibrium.
Overview of General Equilibrium
1. Given prices \(p_1,p_2\), firms will choose the point \((Y_1^*,Y_2^*)\) along the PPF where \(MRT = \frac{p_1}{p_2}\)
2. All money received by firms \((p_1Y_1^* + p_2Y_2^*)\) will become income \(M\) for consumers.
3. Given prices \(p_1,p_2\) and income \(M\), consumers will choose the point \((X_1^*,X_2^*)\) along the budget line where \(MRS = \frac{p_1}{p_2}\)
Equilibrium in and Disequilibrium in the Short Run
MRS =
p_1 = MC_1
p_2 = MC_2
= MRT
If consumers and firms all face the same price, and if they choose the same quantity in response to that price, then MRS = MRT.
u(X_1,X_2) = \alpha \ln X_1 + (1-\alpha) \ln X_2
What happens in general equilibrium if consumers have perfect complements or perfect substitutes utility functions?
\text{Consumer utility function: }u(x_1,x_2) = \min\{x_1,x_2\}
\text{Good 1 production function: }Y_1(L_1) = 6L_1^{1 \over 3}
\text{Good 2 production function: }Y_2(L_2) = 12L_2^{1 \over 3}
Which good will have the higher price in equilibrium?
Key insight: remember that firms will produce at the point along the PPF where the MRT is equal to the price ratio. What point along the PPF must be chosen with this utility function?
pollev.com/chrismakler
\text{Consumer utility function: }u(x_1,x_2) = \min\{x_1,x_2\}
\text{Good 1 production function: }Y_1(L_1) = 6L_1^{1 \over 3}
\text{Good 2 production function: }Y_2(L_2) = 12L_2^{1 \over 3}
What is the equilibrium price ratio?
Key insight: remember that firms will produce at the point along the PPF where the MRT is equal to the price ratio. What point along the PPF must be chosen with this utility function?
pollev.com/chrismakler
\text{Consumer utility function: }u(x_1,x_2) = \min\{x_1,x_2\}
\text{Good 1 production function: }Y_1(L_1) = 6L_1^{1 \over 3}
\text{Good 2 production function: }Y_2(L_2) = 12L_2^{1 \over 3}
What is the equilibrium price ratio?
Key insight: remember that firms will produce at the point along the PPF where the MRT is equal to the price ratio. What point along the PPF must be chosen with this utility function?
Equation of the PPF:
Expression for the MRT:
{x_1^3 \over 6^3} + {x_2^3 \over 12^3} = \overline L
{{3x_1^2 \over 6^3} \over {3x_2^2 \over 12^3}}
= \left({12 \over 6}\right)^3 \times \left(x_1 \over x_2\right)^2
= 8 \left(x_1 \over x_2\right)^2
\text{Consumer utility function: }u(x_1,x_2) = \min\{x_1,x_2\}
\text{Good 1 production function: }Y_1(L_1) = 6L_1^{1 \over 3}
\text{Good 2 production function: }Y_2(L_2) = 12L_2^{1 \over 3}
Bonus question to do at home: suppose that in equilibrium, \(p_1 = 80\) and \(Y_1 = 24\). What must the wage rate be?
Key Takeaways
In general equilibrium, everything having to do with money has been endogenized.
We are left with the same things Chuck had on his desert island:
resources, production technologies, and preferences.
As an individual in autarky, Chuck solved his maximization problem by setting
the marginal benefit of any activity he undertook equal to its opportunity cost.
Markets solve the problem of how to resolve scarcity in the same way:
by having everyone equate their own MB or MC to a common price,
which represents the opportunity cost of using resources in some other way.
Econ 50 | Lecture 24
By Chris Makler
Econ 50 | Lecture 24
General equilibrium
