Deriving Cobb-Douglas Long-Run Costs

From Conditional Input Choices to Minimum Cost

A long-run cost function answers a practical question: what is the least amount a firm must spend on inputs to produce a specified quantity of output? For a Cobb-Douglas technology, the answer can be derived explicitly. The derivation connects the cost-minimizing input mix to the firm’s production requirement and yields a formula that shows how output, input prices, and productivity affect minimum cost.

Assume a firm can choose capital \(K\) and labor \(L\) freely in the long run. Let \(r\) be the rental price per unit of capital and \(w\) the wage per unit of labor. Both prices are positive. The production technology is \(q=A K^\alpha L^\beta\), where \(A>0\), \(\alpha>0\), and \(\beta>0\). The firm seeks to produce a positive target \(q\) at minimum cost. There are no fixed costs or other inputs in this model.

Definition: The long-run cost function \(C(q;r,w)\) is the minimum expenditure required to produce output \(q\), given input prices \(r\) and \(w\). For this technology, the firm minimizes \(rK+wL\) subject to \(A K^\alpha L^\beta\ge q\), with \(K,L\ge 0\). Conditional factor demands are the cost-minimizing quantities \(K^*(q;r,w)\) and \(L^*(q;r,w)\) needed to produce the specified output.

Because the target output is positive and the exponents are positive, producing it requires positive amounts of both inputs. With positive input prices, the cost-minimizing solution uses exactly enough inputs to meet the target: producing more than \(q\) with the same input bundle would not minimize expenditure. The earlier material on cost minimization by tangency gives the key interior condition: the marginal product per dollar spent must be equal across inputs.

Deriving the Cost Function

Write \(s=\alpha+\beta\). The marginal products are \(MP_K=\alpha A K^{\alpha-1}L^\beta\) and \(MP_L=\beta A K^\alpha L^{\beta-1}\). At an interior cost minimum, \(MP_K/MP_L=r/w\). Since \(MP_K/MP_L=(\alpha/\beta)(L/K)\), the optimal input ratio is \(K/L=\alpha w/(\beta r)\). This ratio determines the mix; the production target determines its scale.

\frac{K^*}{L^*}=\frac{\alpha w}{\beta r}

The same tangency condition implies expenditure shares. The condition \(MP_K/MP_L=r/w\) can be rearranged to \(rK/(\alpha)=wL/(\beta)\). Thus capital and labor expenditures are proportional to \(\alpha\) and \(\beta\), respectively. Since their sum is total cost, the shares are \(\alpha/s\) for capital and \(\beta/s\) for labor.

rK^*=\frac{\alpha}{s}C,\qquad wL^*=\frac{\beta}{s}C

Substitute \(K^*=\alpha C/(sr)\) and \(L^*=\beta C/(sw)\) into the production requirement. Solving the resulting equation for \(C\) gives the minimum-cost function. This derivation also shows why the exponents appear both in the cost shares and in the cost formula.

Express inputs in terms of cost.
Use the expenditure shares to write \(K^*=\alpha C/(sr)\) and \(L^*=\beta C/(sw)\).

Impose the output target.
Substitute those quantities into \(q=A(K^*)^\alpha(L^*)^\beta\), so \(q=A(\alpha C/(sr))^\alpha(\beta C/(sw))^\beta\).

Solve for minimum cost.
Collecting terms gives \(C^s=q s^s r^\alpha w^\beta/(A\alpha^\alpha\beta^\beta)\). Taking the positive \(s\)th root yields the cost function.

C(q;r,w) =s\left(\frac{q}{A}\right)^{1/s} \left(\frac{r^\alpha w^\beta}{\alpha^\alpha\beta^\beta}\right)^{1/s}, \qquad s=\alpha+\beta

The conditional factor demands follow from the expenditure shares by substituting this expression for \(C\). They are useful for recovering the actual input quantities rather than just the minimum expenditure.

K^*(q;r,w)= \left(\frac{q}{A}\right)^{1/s} \left(\frac{\alpha w}{\beta r}\right)^{\beta/s}, \qquad L^*(q;r,w)= \left(\frac{q}{A}\right)^{1/s} \left(\frac{\beta r}{\alpha w}\right)^{\alpha/s}

Theorem: For \(q>0\), \(A>0\), \(\alpha,\beta>0\), and positive factor prices \(r,w\), the Cobb-Douglas cost-minimization problem has an interior solution. Its minimum cost is the displayed \(C(q;r,w)\), with conditional demands \(K^*\) and \(L^*\) above. The formulas do not apply unchanged if an input price is zero or negative, if an exponent is zero, or if the production technology has a different form.

Worked Applications

Worked Example: Correcting for the Input Price Ratio

A hypothetical firm has production function \(q=\sqrt{KL}\). It must produce \(q=25\). Capital costs \(r=4\) per unit and labor costs \(w=1\) per unit. Find the cost-minimizing input quantities and minimum cost, then verify both the output and expenditure.

Method. Here \(A=1\), \(\alpha=\beta=0.5\), and \(s=1\). The cost formula becomes \(C=2q\sqrt{rw}\). Therefore,

C=2(25)\sqrt{4(1)}=50(2)=100

Each input receives half of total expenditure because \(\alpha/s=\beta/s=0.5\). Capital expenditure is \(100/2=50\), so \(K^*=50/4=12.5\). Labor expenditure is also \(50\), so \(L^*=50/1=50\).

Check. The production function gives \(\sqrt{12.5(50)}=\sqrt{625}=25\). Total expenditure is \(4(12.5)+1(50)=50+50=100\), matching the cost formula.

Common mistake avoided. Equal expenditure shares do not mean equal input quantities when input prices differ. Choosing \(K=25\) and \(L=100\) would cost \(4(25)+100=200\), and produce \(\sqrt{25(100)}=50\), not the target of 25.

Worked Example: Productivity and Unequal Factor Prices

A hypothetical workshop uses \(q=2\sqrt{KL}\). It needs to produce \(18\) units. The rental price of capital is \(r=9\), and the wage is \(w=1\). Find minimum cost and both conditional factor demands, and check the result.

Method. Here \(A=2\), \(\alpha=\beta=0.5\), and \(s=1\). The cost formula gives

C=2\left(\frac{18}{2}\right)\sqrt{9(1)} =2(9)(3)=54

The expenditure shares are equal, so the firm spends \(54/2=27\) on each input. Capital costs 9 per unit, giving \(K^*=27/9=3\). Labor costs 1 per unit, giving \(L^*=27/1=27\).

Check. Output is \(2\sqrt{3(27)}=2\sqrt{81}=18\). Expenditure is \(9(3)+1(27)=27+27=54\). Both the production target and the derived minimum cost are satisfied.

Common mistake avoided. The firm does not buy equal quantities just because the exponents are equal. Equal exponents imply equal expenditure shares here; the higher capital price means the firm buys fewer units of capital.

Worked Example: Unequal Production Exponents

A hypothetical producer has technology \(q=K^{0.25}L^{0.5}\). It must produce \(q=4\), with \(r=1\) and \(w=8\). Find its conditional demands and minimum cost. Verify the cost shares and output.

Method. The parameters are \(A=1\), \(\alpha=0.25\), \(\beta=0.5\), and \(s=0.75\). Thus \(q/A=4\), \(1/s=4/3\), \(\beta/s=2/3\), and \(\alpha/s=1/3\). The capital demand is

K^*=4^{4/3} \left(\frac{0.25(8)}{0.5(1)}\right)^{2/3} =4^{4/3}(4)^{2/3}=4^2=16

The labor demand is

L^*=4^{4/3} \left(\frac{0.5(1)}{0.25(8)}\right)^{1/3} =4^{4/3}(0.25)^{1/3} =4^{4/3}4^{-1/3}=4

Therefore, cost is \(C=1(16)+8(4)=16+32=48\). The expenditure shares are \(\alpha/s=1/3\) for capital and \(\beta/s=2/3\) for labor, which match \(16/48=1/3\) and \(32/48=2/3\). Output is \(16^{0.25}4^{0.5}=2(2)=4\).

Common mistake avoided. Do not treat the exponents as expenditure shares without dividing by their sum. The shares are \(\alpha/(\alpha+\beta)\) and \(\beta/(\alpha+\beta)\), not \(\alpha\) and \(\beta\) unless those exponents sum to one.

Interpreting the Cost Formula

The formula separates several influences on minimum expenditure. Holding technology and prices fixed, output enters as \(q^{1/s}\). The scale of productivity \(A\) enters as \(A^{-1/s}\): a larger \(A\) means fewer inputs are needed for a given target. Factor prices enter with weights \(\alpha/s\) and \(\beta/s\). In particular, doubling \(r\) while holding other values fixed multiplies cost by \(2^{\alpha/s}\); doubling \(w\) multiplies it by \(2^{\beta/s}\).

The exponents also determine how the firm allocates its budget: capital’s fraction of cost is \(\alpha/s\), and labor’s fraction is \(\beta/s\). These are spending shares, not the fractions of physical inputs. The physical input ratio also depends on relative prices: \(K^*/L^*=\alpha w/(\beta r)\). For example, a higher wage relative to the rental price makes the cost-minimizing bundle more capital intensive, all else equal.

A useful verification routine is to check three things after calculating a cost-minimizing bundle: the production function reaches the target, input expenditures sum to the stated cost, and the expenditure shares agree with the exponents. These checks catch arithmetic errors and mistakes in interpreting a price ratio.

Key takeaway: Derive minimum cost by combining the optimal input ratio with the production requirement. For Cobb-Douglas technology, the resulting cost shares are \(\alpha/(\alpha+\beta)\) and \(\beta/(\alpha+\beta)\), while the input quantities depend on both those exponents and the relative input prices.

Check Your Understanding

For each problem, find the requested cost or input choice and check the result using the production requirement or cost formula.

A hypothetical firm produces \(q=\sqrt{KL}\), must make 12 units, and faces \(r=1\), \(w=4\). Find its minimum cost and conditional demands. (Answer: \(C=48\), \(K^*=24\), \(L^*=6\).)
A producer has \(q=3\sqrt{KL}\), a target of 30, and prices \(r=4\), \(w=9\). Find minimum cost and the conditional demands. (Answer: \(C=120\), \(K^*=15\), \(L^*=20/3\).)
A technology has \(\alpha=0.25\), \(\beta=0.5\), with \(A,r,w\) fixed and positive. By what factor does minimum cost change if target output doubles? (Answer: It is multiplied by \(2^{4/3}\), approximately \(2.5198\).)
A hypothetical firm produces \(q=\sqrt{KL}\) and needs 10 units. Initially \(r=1,w=4\); then the capital price rises to \(r=4\), while the wage stays at 4. Find the minimum cost and conditional demands before and after the price change. (Answer: Initially \(C=40\), \(K^*=20\), \(L^*=5\); afterward \(C=80\), \(K^*=10\), \(L^*=10\).)
A producer has \(q=2\sqrt{KL}\), target output 16, and prices \(r=1,w=4\). Find the minimum cost and input quantities, then verify output. (Answer: \(C=32\), \(K^*=16\), \(L^*=4\); \(2\sqrt{16(4)}=16\).)

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