<p>You are right. The problem here is a corner solution.</p>

<p>Let us define the optimal combination $X^* = \beta_1e_1^* + \beta_2e_2^*$.</p>

<p>The first derivative gives you </p>

<p>$$X^* = \dfrac{l_1}{2\beta_1} $$ </p>

<p>whereas the second gives you</p>

<p>$$X^* = \dfrac{l_2}{2\beta_2}$$ </p>

<p>They can only be true "by chance". This is, <strong>the existence of an interior solution cannot be assured for every possible parameterisation allowed by the parameter space.</strong> </p>

<p>Actually, if you take a look at the two functions, this is clear. The cost function generate <strong>linear</strong> isoquants. You can check this <a href="https://www.wolframalpha.com/input/?i=plot+(x%2By)%5E2%3D5">here</a>. This is because the cost function is equivalent to $c = X^2$ and $X$ is a linear combination of both $e_1$ and $e_2$.</p>

<p>Similarly, the value function also gives <strong>linear</strong> isoquants. This is because the function has the form $e_1 = a + be_2$. </p>

<p>Then, the most likely case is that the isoquants will join in a corner. The interior solution happens just by chance. Notice that when this is the case, <strong>any interior solution is optimal!</strong> The equilibrium set is then infinite.</p>

CC BY-SA 3.0
2017-02-03T18:08:34.723Z
2017-02-03T18:08:34.723Z
luchonacho
<p>Consider the following cost function:</p>

<p>$$c(e_1, e_2) = (\beta_1e_1 + \beta_2e_2)^2$$</p>

<p>The value function is:</p>

<p>$$v = v_0 - [l_1(1-e_1) + l_2(1-e_2)]$$</p>

<p>How do I compute the optimum efforts $e_1$ and $e_2$, with $e_{i} \in [0,1]$ ? I seem to be ending up in an inconsistency. </p>

<p>If I look at $v - c = v_0 - [l_1(1-e_1) + l_2(1-e_2)] -  (\beta_1e_1 + \beta_2e_2)^2$, and then I take the partial derivative with respect to $e_1$ and $e_2$ respectively, I obtain:</p>

<p>$$\beta_1e_1^* + \beta_2e_2^* = \dfrac{l_1}{2\beta_1} = \dfrac{l_2}{2\beta_2}$$ </p>

<p>What am I missing? Does the inconsistency simply mean that the optimal efforts are corner points?</p>

<optimization><nash-equilibrium>
Computing optimum efforts
esperanto
# Computing optimum efforts

(Asked by: username_0 on 2017-02-02)

Consider the following cost function:

$$c(e\_1, e\_2) = (\beta\_1e\_1 + \beta\_2e\_2)^2$$

The value function is:

$$v = v\_0 - [l\_1(1-e\_1) + l\_2(1-e\_2)]$$

How do I compute the optimum efforts $e\_1$ and $e\_2$, with $e\_{i} \in [0,1]$ ? I seem to be ending up in an inconsistency.

If I look at $v - c = v\_0 - [l\_1(1-e\_1) + l\_2(1-e\_2)] - (\beta\_1e\_1 + \beta\_2e\_2)^2$, and then I take the partial derivative with respect to $e\_1$ and $e\_2$ respectively, I obtain:

$$\beta\_1e\_1^\* + \beta\_2e\_2^\* = \dfrac{l\_1}{2\beta\_1} = \dfrac{l\_2}{2\beta\_2}$$

What am I missing? Does the inconsistency simply mean that the optimal efforts are corner points?

**Tags:** optimization, nash-equilibrium

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## 1 answers

### Answer by username_1 on 2017-02-03. Score: 4 | Accepted answer

You are right. The problem here is a corner solution.

Let us define the optimal combination $X^\* = \beta\_1e\_1^\* + \beta\_2e\_2^\*$.

The first derivative gives you

$$X^\* = \dfrac{l\_1}{2\beta\_1} $$

whereas the second gives you

$$X^\* = \dfrac{l\_2}{2\beta\_2}$$

They can only be true "by chance". This is, **the existence of an interior solution cannot be assured for every possible parameterisation allowed by the parameter space.**

Actually, if you take a look at the two functions, this is clear. The cost function generate **linear** isoquants. You can check this [here](https://www.wolframalpha.com/input/?i=plot+(x%2By)%5E2%3D5). This is because the cost function is equivalent to $c = X^2$ and $X$ is a linear combination of both $e\_1$ and $e\_2$.

Similarly, the value function also gives **linear** isoquants. This is because the function has the form $e\_1 = a + be\_2$.

Then, the most likely case is that the isoquants will join in a corner. The interior solution happens just by chance. Notice that when this is the case, **any interior solution is optimal!** The equilibrium set is then infinite.


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