Unformatted text preview: Problem Set 4 Solution Key Suppose in some market that a typical ﬁrm’s production function of two inputs, labor and capital, given by Q = Ll/ZK 1/2. The wage rate is represented by w and the rent rate is represented by r. a- Identify the type ofreturns to scale this production function represents? lfthe ﬁrm hires L units oflabor and hires K units ofcapital, then he can produce at most L l/2K ”2 scale up all inputs by X, A > 1. The ﬁrm now produces (101/2 (2101/2 = )t (Ll/ZKl/Z) units of output. Since, the output is units ofoutput. Let us also scaled up by it, we can conclude that this production function represents constant returns to scale. For parts b , c: assume that capital is ﬁxed in the short-run and it is equal to . b— Derive the short—run demand for labor.
When the capital level is ﬁxed, say 7?, the production function becomes Q = L “Fl/2. Ifwe solve for L, we can derive
the short—run demand for labor: L = ($7) = %2. So, the short—run demand for labor is LSR (Q) = 9?. c— Derive the short-run total production cost.
The short—run total production cost is 2 _
CSR(Q)=wX%+rXK. For parts d, e,f,g: assume that all inputs are variable.
d, e- Derive the long-run demand for labor and capital. Q : Ll/ZKl/Z MPI _ itﬂ/ZKVZ _ g
MPK _ %LI/2K—l/Z _ L We know that at an interior optimum: ﬁll; : 3 Hence, ,5 = 'f => K = rﬂL.
We also know that the production function holds. That is, Q = L l/2K1/2. Now, we must solve these two equations:
Q = LWKW : Q = L”2 (ml/2 = L( )”2 2 L = Q (LY/2 I' U.) %
Then, the capital level becomes K = % (Q (5)1/2) = Q (3)1/2. I‘ Therefore, the long-run demand for labor and the long—run demand for capital are
L” (Q) = Q (L)”2 and K” (Q) = Q g)”, respectively. U.)
f- Derive the long-run total production cost. 1 2 1/2
C”(Q>=wx Q(&)/ +rx Q(%) 2
c” (Q) = 2Q (wrW2
g- Does this cost structure exhibit scale economies? Explain.
First, we must calculate average cost. ACLR (Q) = 2 (wr)'/2. As quantity increases average cost does not change, so there are constant returns to scale. ...
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- Fall '07