<p><strong>Competitive equilibrium</strong> is the price vector $(p_x, p_y, w =1, r)$ such that it solves the following system of equations:</p>

<ol>
<li>Demand for $X$  = Supply of $X$</li>
<li>Demand for $Y$ = Supply of $Y$</li>
<li>Demand for $L$ = Supply of $L$</li>
<li>Demand for $K$ = Supply of $K$</li>
</ol>

<p>where these demands and supplies are either exogenously given or are derived by solving utility maximization problem of the consumers, and profit maximization problem of the firm in a standard way.</p>

<p><strong>Method 1</strong>: One way to solve the posted problem is to find these demands and supplies and solve the resulting system of equations. </p>

<p><strong>Method 2</strong>: Another way is to use first welfare theorem. First welfare theorem says that if the utilities are increasing then competitive equilibrium is Pareto efficient. In the posted problem, both the consumers have increasing utility function. So, we can use it to determine the competitive equilibrium prices. Equilibrium prices and allocation satisfy the following:</p>

<ol>
<li>Efficiency in production: MRTS$^X_{L,K}$ = MRTS$^Y_{L,K}$ = $\frac{w}{r}$. Use this to derive Production Possibility Frontier (PPF).</li>
<li>Efficiency in consumption: MRS$^A_{X,Y}$ = MRS$^B_{X,Y}$ = MRT$_{X,Y}$ = $\frac{p_X}{p_Y}$.</li>
</ol>

<p><strong>Solution</strong> (using method 2): Given the data in question, we see that both the production functions exhibit constant returns to scale and the PPF will be of the form $4x + y =$  constant. Therefore, the price ratio $\frac{p_x}{p_y}$ in equilibrium will be 4 regardless of the initial endowments. This will be true as long as both the inputs $L$ and $K$ are available in positive quantities. Equilibrium can be fully determined once the details about endowments are specified. To find $\frac{w}{r}$ ratio we need to know the total endowment of labor and capital in the economy, and to find the absolute prices of $X$ and $Y$ given $w = 1$ we need the data on how much of the total endowment of labor and capital belongs to each consumer.</p>

CC BY-SA 3.0
2017-01-25T01:47:59.147Z
2017-01-25T10:11:55.740Z
2017-01-25T10:11:55.740Z
Amit
<p><strong>Competitive General Equilibrium</strong></p>

<p>The model you sketched is a standard general equilibrium model and hence any competitive equilibrium is Pareto efficient.</p>

<p>The other answer talks about consumers/firms being atomistic. I do not understand this. Standard assumption of the general equilibrium theory is that consumers and firms are price takers.</p>

<p>The other answer also talks about partial equilibrium since wages cannot adjust. Again, I do not understand this. Given a competitive equilibrium price vector and allocation, multiplying all prices by a constant produces another equilibrium with the same allocation. In other words, competitive equilibrium pins down only relative prices. Standard simplification, which is without loss of generality, is then to normalize price of one commodity to unity. Here, the normalization applies to price of labour, wage.</p>

<p>How to find the competitive equilibrium? I am afraid there is no other way than to derive Walrasian demands of the consumers and supply of the firms and find prices that clear all the markets, that is, the markets for the two consumption goods and the markets for production inputs.</p>

CC BY-SA 3.0
2017-01-25T09:45:45.493Z
*Standard assumption of the general equilibrium theory is that consumers and firms are price takers.* - this is False. If you disagree, provide references for this claim.
FooBar
CC BY-SA 3.0
2017-01-25T09:48:53.220Z
For a simple counter-example, have a look at the Dixit-Stiglitz framework with monopolistic competition (firms that are not price takers), which has been implemented in many general equilibrium models.
FooBar
CC BY-SA 3.0
2017-01-25T11:11:30.507Z
@FooBar Would you please also provide a link to support your claim as I have not encountered GE models which assumed market power before.
Giskard
CC BY-SA 3.0
2017-01-25T11:22:29.667Z
@FooBar The price-taking assumption is definitely standard in general equilibrium theory, unless otherwise stated. For reference, see the first time GE is introduced in any standard graduate microeconomics text. (Ex. MWG Ch 10, New Kreps Ch 14) In fact, as denesp points out, the OP asks about the *competitive* general equilibrium, which is almost always taken to mean that consumers and firms are price takers, so the fact that there are general equilibrium models where agents have market power is irrelevant.
Theoretical Economist
CC BY-SA 3.0
2017-01-25T11:27:33.300Z
The notion of general equilibrium is independent of competitiveness. Sure, you might more often see competitive GE, since they're easier to solve and give you all the intuition you needed, but *clearing all markets* has nothing to do with the particular structures behind the markets.
FooBar
CC BY-SA 3.0
2017-01-25T11:30:53.597Z
@denesp, the most basic New Keynesian models use the Dixit-Stiglitz framework which implies market power. I would not considered them thoroughly micro funded, but I would consider them GE.
FooBar
CC BY-SA 3.0
2017-01-25T11:43:31.033Z
@FooBar Competitive GE models (1) are seen more often; (2) are easier to solve; and (3) give you all the intuition you need. Don't these three claims lend support to the price taking assumption being standard?
Theoretical Economist
CC BY-SA 3.0
2017-01-25T11:48:31.453Z
@TheoreticalEconomist There is a difference between "Almost everyone assumes X" and "If you don't state it, X is assumed". I fear that our disagreement mostly comes from different readings of a vaguely posted question.
FooBar
CC BY-SA 3.0
2017-01-25T11:55:18.693Z
I don't want to judge what is and what is not standard. I just want to point out that the original sin here is OP not specifying the model fully. Incompletely specified model is, well, an incompletely specified model.
Jan
CC BY-SA 3.0
2017-01-25T11:59:13.237Z
@FooBar There may be a difference, but that difference is fine enough for me to be willing to treat them as the same in cases like these. I also disagree that the question is vague; the OP specifies what kind of GE they are looking for. Also, this type of question is standard in early grad/advanced ug courses, which will always be asking about price-takers unless otherwise stated. In any case, even supposing we agreed that the question is vague: Jan's answer comes from a reasonable interpretation of the question and probably doesn't deserve a downvote (assuming that downvote was yours).
Theoretical Economist
CC BY-SA 3.0
2017-01-24T21:56:12.353Z
2017-01-25T12:16:00.657Z
2017-01-25T12:16:00.657Z
Jan
<p>There are two consumers $A$ and $B$, and two producers $X$ and $Y$. he consumers are endowed with labour(L) and capital(K) $L_A,K_A$, and $L_B,K_B$ respectively. The preferences of the two consumers have the utility functions $U_A=X_A^4Y_A$ and $U_B=X_BY_B$. The technologies of the two producers are given by $X=\sqrt{K_XL_X}/2$ and $Y=2\sqrt{K_YL_Y}$. Setting the wage rate $w=1$, what are the competitive general equilibrium relative prices?</p>

<p>This is inspired from a homework problem, but I wish to understand how to solve this general class of questions. Does this require pareto optimality? How does calculus feature in this question? Any help would be great! Thanks.</p>

<microeconomics><general-equilibrium><pareto-efficiency>
Does a General Equilibrium here require Pareto Optimality?
freebird
# Does a General Equilibrium here require Pareto Optimality?

(Asked by: username_0 on 2015-04-29)

There are two consumers $A$ and $B$, and two producers $X$ and $Y$. he consumers are endowed with labour(L) and capital(K) $L\_A,K\_A$, and $L\_B,K\_B$ respectively. The preferences of the two consumers have the utility functions $U\_A=X\_A^4Y\_A$ and $U\_B=X\_BY\_B$. The technologies of the two producers are given by $X=\sqrt{K\_XL\_X}/2$ and $Y=2\sqrt{K\_YL\_Y}$. Setting the wage rate $w=1$, what are the competitive general equilibrium relative prices?

This is inspired from a homework problem, but I wish to understand how to solve this general class of questions. Does this require pareto optimality? How does calculus feature in this question? Any help would be great! Thanks.

**Tags:** microeconomics, general-equilibrium, pareto-efficiency


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## 2 answers

### Answer by username_1 on 2017-01-25. Score: 5

**Competitive equilibrium** is the price vector $(p\_x, p\_y, w =1, r)$ such that it solves the following system of equations:

1. Demand for $X$ = Supply of $X$
2. Demand for $Y$ = Supply of $Y$
3. Demand for $L$ = Supply of $L$
4. Demand for $K$ = Supply of $K$

where these demands and supplies are either exogenously given or are derived by solving utility maximization problem of the consumers, and profit maximization problem of the firm in a standard way.

**Method 1**: One way to solve the posted problem is to find these demands and supplies and solve the resulting system of equations.

**Method 2**: Another way is to use first welfare theorem. First welfare theorem says that if the utilities are increasing then competitive equilibrium is Pareto efficient. In the posted problem, both the consumers have increasing utility function. So, we can use it to determine the competitive equilibrium prices. Equilibrium prices and allocation satisfy the following:

1. Efficiency in production: MRTS$^X\_{L,K}$ = MRTS$^Y\_{L,K}$ = $\frac{w}{r}$. Use this to derive Production Possibility Frontier (PPF).
2. Efficiency in consumption: MRS$^A\_{X,Y}$ = MRS$^B\_{X,Y}$ = MRT$\_{X,Y}$ = $\frac{p\_X}{p\_Y}$.

**Solution** (using method 2): Given the data in question, we see that both the production functions exhibit constant returns to scale and the PPF will be of the form $4x + y =$ constant. Therefore, the price ratio $\frac{p\_x}{p\_y}$ in equilibrium will be 4 regardless of the initial endowments. This will be true as long as both the inputs $L$ and $K$ are available in positive quantities. Equilibrium can be fully determined once the details about endowments are specified. To find $\frac{w}{r}$ ratio we need to know the total endowment of labor and capital in the economy, and to find the absolute prices of $X$ and $Y$ given $w = 1$ we need the data on how much of the total endowment of labor and capital belongs to each consumer.


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### Answer by username_2 on 2017-01-24. Score: 1

**Competitive General Equilibrium**

The model you sketched is a standard general equilibrium model and hence any competitive equilibrium is Pareto efficient.

The other answer talks about consumers/firms being atomistic. I do not understand this. Standard assumption of the general equilibrium theory is that consumers and firms are price takers.

The other answer also talks about partial equilibrium since wages cannot adjust. Again, I do not understand this. Given a competitive equilibrium price vector and allocation, multiplying all prices by a constant produces another equilibrium with the same allocation. In other words, competitive equilibrium pins down only relative prices. Standard simplification, which is without loss of generality, is then to normalize price of one commodity to unity. Here, the normalization applies to price of labour, wage.

How to find the competitive equilibrium? I am afraid there is no other way than to derive Walrasian demands of the consumers and supply of the firms and find prices that clear all the markets, that is, the markets for the two consumption goods and the markets for production inputs.

> **Comments:**
> **username_3**: *Standard assumption of the general equilibrium theory is that consumers and firms are price takers.* - this is False. If you disagree, provide references for this claim.  
> **username_3**: For a simple counter-example, have a look at the Dixit-Stiglitz framework with monopolistic competition (firms that are not price takers), which has been implemented in many general equilibrium models.  
> **username_4**: @username_3 Would you please also provide a link to support your claim as I have not encountered GE models which assumed market power before.  
> **username_5**: @username_3 The price-taking assumption is definitely standard in general equilibrium theory, unless otherwise stated. For reference, see the first time GE is introduced in any standard graduate microeconomics text. (Ex. MWG Ch 10, New Kreps Ch 14) In fact, as denesp points out, the OP asks about the *competitive* general equilibrium, which is almost always taken to mean that consumers and firms are price takers, so the fact that there are general equilibrium models where agents have market power is irrelevant.  
> **username_3**: The notion of general equilibrium is independent of competitiveness. Sure, you might more often see competitive GE, since they're easier to solve and give you all the intuition you needed, but *clearing all markets* has nothing to do with the particular structures behind the markets.  
> **username_3**: @denesp, the most basic New Keynesian models use the Dixit-Stiglitz framework which implies market power. I would not considered them thoroughly micro funded, but I would consider them GE.  
> **username_5**: @username_3 Competitive GE models (1) are seen more often; (2) are easier to solve; and (3) give you all the intuition you need. Don't these three claims lend support to the price taking assumption being standard?  
> **username_3**: @TheoreticalEconomist There is a difference between "Almost everyone assumes X" and "If you don't state it, X is assumed". I fear that our disagreement mostly comes from different readings of a vaguely posted question.  
> **username_2**: I don't want to judge what is and what is not standard. I just want to point out that the original sin here is OP not specifying the model fully. Incompletely specified model is, well, an incompletely specified model.  
> **username_5**: @username_3 There may be a difference, but that difference is fine enough for me to be willing to treat them as the same in cases like these. I also disagree that the question is vague; the OP specifies what kind of GE they are looking for. Also, this type of question is standard in early grad/advanced ug courses, which will always be asking about price-takers unless otherwise stated. In any case, even supposing we agreed that the question is vague: username_2's answer comes from a reasonable interpretation of the question and probably doesn't deserve a downvote (assuming that downvote was yours).  

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