Economics 306 Doyle Fall 2022 Practice Problems for Chapter 3: Solutions Coverage: Chapter 3, Static General Equilibrium with Production: Diagrams. - University of Guelph Economics 306
Doyle
Fall 2022
Practice Problem
...
Economics 306 Doyle Fall 2022 Practice Problems for Chapter 3: Solutions Coverage: Chapter 3, Static General Equilibrium with Production: Diagrams. - University of Guelph Economics 306
Doyle
Fall 2022
Practice Problems for Chapter 3: Solutions
Coverage: Chapter 3, Static General Equilibrium with Production: Diagrams
Problem 1
Answer this question using diagrams.
Consider an economy in which the representative household has preferences given by U(c; l) = maxf2 · c; lg.
The representative rm's production function, F (K; N ¯ d) is a standard, concave production function. Suppose that the capital input is xed.
There is no government in this question.
(a) Formally de ne a competitive equilibrium in this economy.
(b) i) the household chooses C, Ns, and l to maximize its utility given its budget constraint, while taking prices as given,
(c) ii) the rm chooses Y and Nd to maximize pro ts given its production technology,
(d) while taking prices as given, and
(e) iii) both the goods and labour markets clear. That is:
(f) Nd = Ns;
(g) and
(h) Y = C:
(i) 1
(j) (b) Write down the household's optimization problem. Depict the solution to the consumer's
(k) problem in a diagram.
Note that since the preferences are not convex, the household will choose a corner solution. There are two possible solutions, depending on which corner is optimal. Figures 1
and 2 depict the two possibilities.
u(C)¢` S-,t Q-I_wlvi-t` +|T
c2o oS+,,(±\
C
wV\+IT
1
Figure 1: Corner solution: zero leisure
(c) Depict a case where no competitive equilibrium exists. Explain carefully why no equilibrium exists in your example.
To get the rm to produce at point a, the wage must be w0. In this case, however, the
household is better o at the other end of the budget line. To get the rm to produce
at point b, the wage must be w1. In this case, however, the household is better o at
the other end of the budget line.
2
1
Figure 2: Corner solution: zero work
Therefore, no price will get the household and rm to choose the same point. Therefore, no equilibrium exists.
(d) Depict a case where a competitive equilibrium does exist.
(e) Show that your diagram satis es the de nition of a competitive equilibrium.
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